text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
|---|---|---|
Active Advantage
Get VIP deals on events, gear and travel with ACTIVE's premium membership.
Algebra: Factoring and Working with Quadratics (F-8:00)
Sep 07 - Dec 13
Ages 11-15
The REALM
Starting at
$360.00
Meeting Dates
From Sep 07, 2016 to Dec 13, 2016
About This Activity
MUST MEET ELIGIBILITY REQUIREMENTS. Please contact The REALM (cyndy@realmacademy.com) prior to registration. If you register but are not on the eligibility list, you will be un-enrolled; this will result in a $5 charge. In 'Algebra: Factoring and Working with Quadratics', students who have experience in working with polynomials, as well as with solving and graphing linear equations, will advance to the next algebraic level and discover what happens when linear expressions are multiplied together. Students will learn basic factoring, completing the square and how to derive and use the quadratic equation. Along the way, students will gain experience working with exponents and radical expressions, and be introduced to the concept of imaginary numbers. Topics and concepts covered and additional resources are available | 677.169 | 1 |
Course Origin:This course is developed at the ADE Schools's department of Mathematics from the Ontario curriculum document: "Ontario curriculum, mathematics Grades 11 and 12, 2007 (revised)"
Course Rational and Description:Prerequisite: Functions, Grade 11, University Preparation, or Mathematics for College
Technology, Grade 12, College Preparation
Strands:
The course consists of four strands, exponential and logarithmic functions, trigonometric functions, polynomial and rational functions and characteristics of functions.
Strand 1: EXPONENTIAL AND LOGARITHMIC FUNCTIONS
OVERALL EXPECTATIONS
By the end of this course, students will:
1.Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
2.Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
3.Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
Strand 2:TRIGONOMETRIC FUNCTIONS
OVERALL EXPECTATIONS
By the end of this course, students will:
1.Demonstrate an understanding of the meaning and application of radian measure;
2.Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
4.Demonstrate an understanding of solving polynomial and simple rational inequalities
Strand 4:CHARACTERISTICS OF FUNCTIONS
OVERALL EXPECTATIONS
By the end of this course, students will:
1. demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point; 2. determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems; 3. compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
Course Content:
This course is clustered into four units of study relating directly to the four strands of the curriculum document. These units, sequenced in the order they are to be delivered, and not the order of their actual order, are as follows: (NB: the first strand or unit, Exponential and Logarithmic Functions is the last to be delivered)
Unit 3
Polynomial and Rational Functions (1st to be delivered)
28 hours
Unit 4
Characteristics of Functions( 2nd to be delivered)
28 hours
Unit 2
Trigonometric Functions ( 3rd to be delivered)
28 hours
Unit 1
Exponential and Logarithmic Functions (4th to be taught)
28 hours
Total Hours
112hours
Teaching and Learning Strategies and Tools
Because the course the nature of the teaching strategies of the course is electronic and internet-basedAcademy of Distance Education has in place a policy that requires students to write an obligatory supervised final evaluation worth 30 % of the overall grade of each course as well as an obligatory supervised midterm worth 20% of the overall grades which takes place midway through the course.
At ADE, Assessment and Evaluationis done in such a way that for each course, every other unit test is written at the ADE site with full supervision to allow for a maximum control on the fairness of reporting student achievement of curricular expectations and performance standards. The proportion of the final evaluation that the supervised, on-site tests cover are clearly mentioned within the breakdown (along the different units) of the 70% of the overall grade section of the course outline. The final Examination (or the exam and another assessment instrument) worth 30% will also be written at the ADE office in a supervised environment. This mechanism will leave at least 50% of the final overall grade to have resulted from assessments conducted directly at on-site supervised settings.
ADE also assures students teacher-led session, fully executed through virtual, synchronous settings that allow for maximum student-teacher interaction and information-sharing through live sessions. Students will have the opportunity to ask questions before the teacher goes to the next section or topic and not wait until time interval has lapsed. This will assure students that delivery of learning requirements will be similar to the physical class settings except that the challenges of space, time and schedule limitations have all been erased.
Assessment and Evaluation processes completed at ADE heavily rely on the newly legislated Ontario dual standard system (Growing Success) wherecontent standardsandperformance standards are assessed and linked properly; content standards being thecurricular expectationsfrom the discipline documents and performance standards coming from theachievement chart categories and the levels that reflect how effectively such categories are linked. Each assessment will clearly mention how the interconnection and inter=relations of the assessment components and types take placeis a continuous process of gathering evidence to facilitate and enhance student learning, provide feedback, and improve instructional strategies.Evaluationiswill be used to determine prior learning, students' strengths and for planning purposes and therefore will not be used to determine term or final grades for report card.
Assessment Methods:TheAnTwo parts make up the evaluation of student achievement through the different assessment strategies mentioned above.
a) The term work 2008.
Evaluation
Breakdown of the 70% course evaluation among the units
This unit's work will account for 18% of the 70 marks for coursework.
Unit 4: CHARACTERISTICS OF FUNCTIONS
Tests 10% (2 tests)
Quizzes 5%
Assignments 2.5%
______________________________________
Total Unit 1 Evaluation 17.5%
This unit's work will account for 17.5% of the 70 marks for coursework.
Unit 2:EXPONENTIAL ANDLOGARITHMIC FUNCTIONS
Tests 10%
Quizzes 5%
Assignments 2.5%
______________________________________
Total Unit 2 Evaluation 17.5%
This unit's work will account for 18% of the 70 marks for coursework.
Unit 3: TRIGONOMETRIC FUNCTIONS
Tests 10% (2 tests)
Quizzes 5%
Assignments 2.5%
____________________________________
Total Unit 3 Evaluation 17.5%
This unit's work will account for 17.5% of the 70 marks for coursework.
Unit 4: POLYNOMIAL AND RATIONAL FUNCTIONS
Tests 10% (2 tests)
Quizzes 5%
Assignments 2.5%
______________________________________
Total Unit 4 Evaluation 17.5%
This unit's work will account for 17.5% of the 70 marks for coursework | 677.169 | 1 |
Karnataka Class 12 Commerce Maths Unit III – Calculus :Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.
2. Applications of Derivatives
Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
3. Integrals
Integration as inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic propertiesof definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).
5. Differential Equations
Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
Karnataka Class 12 Commerce Maths Unit III – Calculus : Definition of continuity for English Language Learners. : the quality of something that does not stop or change as time passes : a continuous quality. : something that is the same or similar in two or more things and provides a connection between them. Continuity is 'the state of being continuous' and continuous means 'without any interruption or disturbance'. For example, the price of a commodity and its demand are inversely proportional. The graph of demand curve of a commodity is a continuous curve without any breaks or gaps.
Continuity, in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable—say x—is associated with a value of a dependent variable—say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question "How close?" is asked, difficulties arise.
For close x-values, the distance between the y-values can be large even if the function has no sudden jumps. For example, if y = 1,000x, then two values of x that differ by 0.01 will have corresponding y-values differing by 10. On the other hand, for any point x, points can be selected close enough to it so that the y-values of this function will be as close as desired, simply by choosing the x-values to be closer than 0.001 times the desired closeness of the y-values. Thus, continuity is defined precisely by saying that a function f(x) is continuous at a point x0 of its domain if and only if, for any degree of closeness ε desired for the y-values, there is a distance δ for the x-values (in the above example equal to 0.001ε) such that for any x of the domain within the distance δ from x0, f(x) will be within the distance ε from f(x0). In contrast, the function that equals 0 for x less than or equal to 1 and that equals 2 for x larger than 1 is not continuous at the point x = 1, because the difference between the value of the function at 1 and at any point ever so slightly greater than 1 is never less than 2.
Karnataka Class 12 Commerce Maths Unit III – Calculus : A function is said to be continuous if and only if it is continuous at every point of its domain. A function is said to be continuous on an interval, or subset of its domain, if and only if it is continuous at each point of the interval. The sum, difference, and product of continuous functions with the same domain are also continuous, as is the quotient, except at points at which the denominator is zero. Continuity can also be defined in terms of limits by saying that f(x) is continuous at x0 of its domain if and only if, for values of x in its domain,
A more abstract definition of continuity can be given in terms of sets, as is done in topology, by saying that for any open set of y-values, the corresponding set of x-values is also open. (A set is "open" if each of its elements has a "neighborhood," or region enclosing it, that lies entirely within the set.) Continuous functions are the most basic and widely studied class of functions in mathematical analysis, as well as the most commonly occurring ones in physical situations.
CONTINUITY OF FUNCTIONS OF ONE VARIABLE
The following problems involve the CONTINUITY OF A FUNCTION OF ONE VARIABLE. Function y = f(x) is continuous at point x=a if the following three conditions are satisfied :
i.) f(a) is defined ,
ii.) exists (i.e., is finite) ,
and
iii.) .
Function f is said to be continuous on an interval I if f is continuous at each point x in I. Here is a list of some well-known facts related to continuity :
1. The SUM of continuous functions is continuous.
2. The DIFFERENCE of continuous functions is continuous.
3. The PRODUCT of continuous functions is continuous.
4. The QUOTIENT of continuous functions is continuous at all points x where the DENOMINATOR IS NOT ZERO.
5. The FUNCTIONAL COMPOSITION of continuous functions is continuous at all points x where the composition is properly defined.
6. Any polynomial is continuous for all values of x.
7. Function ex and trigonometry functions and are continuous for all values of x.
Karnataka Class 12 Commerce Maths Unit III – Calculus :: Most problems that follow are average. A few are somewhat challenging. All limits are determined WITHOUT the use of L'Hopital's Rule. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by using the above step-by-step definition of continuity at a point and the well-known facts, and by giving careful consideration to the indeterminate form during the computation of limits. Knowledge of one-sided limits will be required. For a review of limits and indeterminate forms click here.
Karnataka Class 12 Commerce Maths Unit III – Calculus : Applications Of Derivatives is the rate of change of one quantity in terms of another quantity. So, for example, if you want to know how something changes in time, you usually need to invoke a derivative. This could be something like the position of an object, or the strength of an electric field, or quantity of water in a container, or the amount of money in your bank account…
Suffice it to say that it would be essentially impossible to do any physics without derivatives, which would mean no modern engineering, which would mean essentially no technology developed after the 1700s.
In the previous chapter we focused almost exclusively on the computation of derivatives. In this chapter will focus on applications of derivatives. It is important to always remember that we didn't spend a whole chapter talking about computing derivatives just to be talking about them. There are many very important applications to derivatives.
Karnataka Class 12 Commerce Maths Unit III – Calculus : The two main applications that we'll be looking at in this chapter are using derivatives to determine information about graphs of functions and optimization problems. These will not be the only applications however. We will be revisiting limits and taking a look at an application of derivatives that will allow us to compute limits that we haven't been able to compute previously. We will also see how derivatives can be used to estimate solutions to equations.
Here is a listing of the topics in this section.
Rates of Change The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter. Namely, rates of change.
Critical Points In this section we will define critical points. Critical points will show up in many of the sections in this chapter so it will be important to understand them.
Minimum and Maximum Values In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.
Finding Absolute Extrema Here is the first application of derivatives that we'll look at in this chapter. We will be determining the largest and smallest value of a function on an interval.
The Shape of a Graph, Part I We will start looking at the information that the first derivatives can tell us about the graph of a function. We will be looking at increasing/decreasing functions as well as the First Derivative Test.
The Shape of a Graph, Part II In this section we will look at the information about the graph of a function that the second derivatives can tell us. We will look at inflection points, concavity, and the Second Derivative Test.
The Mean Value Theorem Here we will take a look at the Mean Value Theorem.
Optimization Problems This is the second major application of derivatives in this chapter. In this section we will look at optimizing a function, possibly subject to some constraint.
More Optimization Problems Here are even more optimization problems.
L'Hospital's Rule and Indeterminate Forms This isn't the first time that we've looked at indeterminate forms. In this section we will take a look at L'Hospital's Rule. This rule will allow us to compute some limits that we couldn't do until this section.
Linear Approximations Here we will use derivatives to compute a linear approximation to a function. As we will see however, we've actually already done this.
Differentials We will look at differentials in this section as well as an application for them.
Newton's Method With this application of derivatives we'll see how to approximate solutions to an equation.
Business Applications Here we will take a quick look at some applications of derivatives to the business field.
3) Karnataka Class 12 Commerce Maths Integrals
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral
{\displaystyle \int _{a}^{b}\!f(x)\,dx}
is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.
Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
{\displaystyle F(x)=\int f(x)\,dx.}
The integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by
{\displaystyle \int _{a}^{b}\!f(x)\,dx=F(b)-F(a).}
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves. For that we shall need some concepts of Integral Calculus. In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms only). We shall also deal with finding the area bounded by the above said curves.
A differential equation is a mathematical equationthat relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.
A differential equation is a mathematical equationCAKART provides Indias top faculty each subject video classes and lectures – online & in Pen Drive/ DVD – at very cost effective rates. Get video classes from CAKART.in. Quality is much better than local tuition | 677.169 | 1 |
Share this Page
Sketchpad Math Software Goes Universal
11.06.2006—Geometer has released an update to Sketchpad, a software tool for teaching and learning math. The new 4.07 update adds native compatibility for Intel-based Macs and continues to support PowerPC-based Macs and Windows systems as well.
Sketchpad is a suite designed for both students and educators, with separate editions for each. It allows students to explore mathematics by providing tools for them to create diagrams and figures. Educators can also use the software to generate teaching aids. It includes classroom activities, presentation and sketch samples, learning guides and reference materials. And it offers modules for various math courses.
Specific curriculum modules include:
Exploring Algebra 1;
Exploring Algebra 2;
Exploring Geometry;
Pythagoras Plugged In: Proofs and Problems;
Rethinking Proof;
Exploring Conic Sections;
Exploring Precalculus;
Exploring Calculus;
Geometry Activities for Middle School Students;
Shape Makers: Developing Geometric Reasoning in Middle School;
And Geometry in Action.
The latest release, version 4.07, is now a Universal Binary for Macintosh systems, supporting both Intel and PowerPC hardware. It also adds Web links to the Sketchpad Resource Center and adds sample documents to the Help menu. Several bug fixes are also included in the update.
The new 4.07 update is available free for current users. The student version of Sketchpad is available now for Mac OS X and Windows for $39.95. The full edition runs $129.95. Multi-license versions are also available, as is an evaluation version for instructors. See the company's Web site, below, for more details | 677.169 | 1 |
Elementary Algebra
by Joan DykesTo accompany the textbooks currently in use in Elementary Algebra courses, this outline provides information and sample problems in these areas: number systems and the basics, exponents and polynomials, factoring, rational expressions | 677.169 | 1 |
Am I A Function?
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|22 pages
Share
Product Description
Part of Common Core objective 8.F.1 Determine functions.
Contains mappings, discrete point graphs, ordered pairs (relations), tables, graphs (both linear and non-linear) for students to determine and explain which are functions. The problems are made large so they can be posted around the room if desired. | 677.169 | 1 |
Overview
Introductory Complex Analysis by Richard A. Silverman
Introductory Complex Analysis is a scaled-down version of A. I. Markushevich's masterly three-volume "Theory of Functions of a Complex Variable." Dr. Richard Silverman, the editor and translator of the original, has prepared this shorter version expressly to meet the needs of a one-year graduate or undergraduate course in complex analysis. In his selection and adaptation of the more elementary topics from the original larger work, he was guided by a brief course prepared by Markushevich himself. The book begins with fundamentals, with a definition of complex numbers, their geometric representation, their algebra, powers and roots of complex numbers, set theory as applied to complex analysis, and complex functions and sequences. The notions of proper and improper complex numbers and of infinity are fully and clearly explained, as is stereographic projection. Individual chapters then cover limits and continuity, differentiation of analytic functions, polynomials and rational functions, Mobius transformations with their circle-preserving property, exponentials and logarithms, complex integrals and the Cauchy theorem , complex series and uniform convergence, power series, Laurent series and singular points, the residue theorem and its implications, harmonic functions (a subject too often slighted in first courses in complex analysis), partial fraction expansions, conformal mapping, and analytic continuation. Elementary functions are given a more detailed treatment than is usual for a book at this level. Also, there is an extended discussion of the Schwarz-Christolfel transformation, which is particularly important for applications. There is a great abundance of worked-out examples, and over three hundred problems (some with hints and answers), making this an excellent textbook for classroom use as well as for independent study. A noteworthy feature is the fact that the parentage of this volume makes it possible for the student to pursue various advanced topics in more detail in the three-volume original, without the problem of having to adjust to a new terminology and notation . In this way, IntroductoryComplex Analysis serves as an introduction not only to the whole field of complex analysis, but also to the magnum opus of an important contemporary Russian mathematician.
About the AuthorRead an ExcerptFirst ChapterTable of ContentsReading Group GuideInterviewsRecipe hisThis concise introduction to the methods and techniques of vector analysis is suitable for college
undergraduates in mathematics as well as students of physics and engineering. Rich in exercises and examples, the straightforward presentation focuses on physical ideas rather than ...
Calculus is an extremely powerful tool for solving a host of practical problems in fields
as diverse as physics, biology, and economics, to mention just a few. In this rigorous but accessible text, a noted mathematician introduces undergraduate-level students to ...
This classroom-tested volume offers a definitive look at modern analysis, with views of applications to
statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. Upper-level undergraduate students with a background in calculus will benefit from its ...
This volume in Richard Silverman's exceptional series of translations of Russian works in the mathematical
science is a comprehensive, elementary introduction to real and functional analysis by two faculty members from Moscow University. It is self-contained, evenly paced, eminently readable, ...
This outstanding text and reference applies matrix ideas to vector methods, using physical ideas to
illustrate and motivate mathematical concepts but employing a mathematical continuity of development rather than a physical approach. The author, who taught at the U.S. Air ...
2013 Reprint of 1933 Edition. Full facsimile of the original edition, not reproduced with Optical
Recognition Software. Sigmund Freud's New Introductory Lectures on Psycho-Analysis recapitulates core tenets of his earlier work on psychoanalytic practice and theory, but in each case ... | 677.169 | 1 |
> Wonderful post.
> Can you answer this question, please:
> What is the URL for the online RK math store? Do they ship
> to international locations?
> Thanks,
Here's the URL where you can order books online.
They accept master/visa card for sure. They ship books to the US - am
not quite sure about other international locations. You can check with
them by email.
Hope this helps,
---ravi | 677.169 | 1 |
TRIANGLE: A Tri-modal Access Program for Reading, Writing, and Doing Math
1. Introduction
TRIANGLE is a DOS and Windows 95 computer program intended for print-impaired students and professionals in math, science, and engineering. It includes:
a math/science word processor
a graphing calculator
a viewer for y versus x plots
a table viewer
the Touch-and-Tell Program for audio and/or braille-assisted reading of tactile figures on an external digitizing pad.
The keyboard or any assistive device/software that emulates a keyboard may be used for input. TRIANGLE output may be viewed visually, audibly, and/or by braille.
DOS TRIANGLE[1] has an on-line help file describing all editing, calculating, graph-viewing, table-browsing, and figure-reading commands. Several tutorials are also included with the distribution files. DOS TRIANGLE is available to anyone interested in trying it.[2] The expanded symbol set used with the mathematical word processor can be accessed with DOS screen readers only if the appropriate character tables are installed. Support is included for Vocal-Eyes speech screen reader and TSI braille displays. Instructions are included for use with other screen readers, but some expertise and effort on the part of the user is required. TRIANGLE menus and help files are in English, but the program should work with most languages using the roman alphabet. The new Windows 95 TRIANGLE (beta release expected in summer 1998) has all the features of DOS TRIANGLE but is self-voicing through any MS SAPI-compliant speech engine and will also be accessible in braille through any on-line screen display that supports the new MS Braille API. This version of the TRIANGLE program will be demonstrated during the presentation.
2. Windows TRIANGLE mathematical word processor
The Windows TRIANGLE word processor is an RTF (rich text format) word processor whose character set includes typographic and foreign characters and the math symbol fonts of math editors bundled with MS Word or Word Perfect. TRIANGLE also utilizes special math and markup symbols to permit all scientific expressions (including fractions, superscripts, subscripts, and tabular arrays) to be written in a linear form. These characters may be entered through a Windows menu, with several of the most common characters having single-stroke short cuts. This is a particularly convenient format for blind users.
Expressions may be entered and manipulated with the usual kinds of editing capabilities found in any text processor, such as the Windows clipboard for cutting and pasting. In addition TRIANGLE has a number of special editing and browsing capabilities that make it particularly convenient for reading and writing math and scientific expressions. For example, there are ten specially-addressable clipboards that allow a user to cut and paste several text selections without losing the last one every time a new item is saved. This facility provides significantly increased flexibility that is handy when manipulating math, solving algebraic equations, etc.
There are several browsing features designed for ease of reading equations. These include "read enclosed expression" commands that jump to the beginning of the next or previous enclosure and read aloud either the entire expression or the portion of that expression to the next enclosure. Enclosures include standard items such as parentheses, brackets, and braces, as well as markup characters defining numerator and denominator of fractions, complex superscripts, subscripts, or radicals. The user has a number of audio templates for representing math symbols and can custom- design them to personal preferences.
Braille access poses a difficult problem, since there is no "accepted" braille representation for anything except letters. We have adopted the GS braille representation used in DOS TRIANGLE[2]. Although DOS TRIANGLE is restricted to 8-dot GS, Windows TRIANGLE can use either 8-dot or 6-dot GS codes.
GS is a dual 6/8-dot braille representation developed by John Gardner and Norberto Salinas (Prof. of Mathematics, Univ. of Kansas). GS was inspired by the current on-going unified braille code (UBC) development effort by the International Committee on English Braille[3]. GS adopts the UBC philosophy of retaining as much as possible of current literary braille. TRIANGLE includes a GS tutorial for braille users.
Users have a number of options for printing from the TRIANGLE editor. An ink copy for sighted people can be printed on any standard printer. The sighted reader must learn the GS markup symbols for such things as subscript, superscript, fractions, and arrays, but the representations are straightforward and the authors believe that such copy should be acceptable for almost any academic purpose.
There are a number of options for making tactile hard copies. A GS6 braille copy may be made on any braille embosser. A GS8 copy may be made on any braille embosser that has the ability to load the GS8 font set. Copies printed in DotsPlus[4], GS6, or GS8 may also be printed using the new TIGER printer[5]. Another output option is to make a font change to an on-screen braille dot pattern and print using swell paper[6] or with the Tektronix Phasor wax jet printer[7].
3. The Windows TRIANGLE graphing calculator
TRIANGLE includes the equivalent of a scientific graphing calculator. The calculator allows the user to input or define constants and expressions for convenience and accepts several types of notation for operations - e.g. one can use a GS multiply symbol or the * which is commonly used to indicate multiplication on computers. The result of the computation is displayed in a calculator window. Results may then be copied and pasted into an editor window.
TRIANGLE also has a graphing calculator that computes and displays a y vs. x function (or several functions simultaneously) on the screen. A number of screen display choices are available. The graph may be printed for sighted people on any standard printer. TIGER[5] can print a tactile copy directly, or indirectly with swell paper[6] or the Phasor[7]. A graphics braille embosser can be used to make a low resolution copy of the graph (without any text or labels of course). However the graph may also be viewed on-line with the x-y graph viewer.
4. The DOS and WINDOWS TRIANGLE x-y graph viewer
A graphed function is scaled so that it can be displayed as a convenient size picture on the screen. It is "viewed" audibly by a blind user through the use of a tone plot. The user may move a pointer along the independent variable axis, and the value of the function is indi-cated by the pitch of a tone. The function is scaled so that the full range from the minimum and maximum of the function corresponds to a pitch well within the normal range of human hearing. The function is also displayed with a moving icon on the bottom line of the screen. As the pointer is moved along the independ-ent-variable axis, the icon moves to the right as the function becomes larger and to the left as it becomes smaller. This icon is primarily intended for deaf blind users who would view it with an on-line braille display.
The graph's pointer may be moved one point at a time or allowed to scan automatically from left to right, or from right to left. In addition to the tone indicating the function, the values of the independent variable and the function are displayed on the screen, and may be read aloud if desired. The viewer functions include the ability to find both relative and absolute maxima and minima, zeros, etc. The simple tone reader gives a reasonable qualitative overview, and a user may look individually at various points for quantitative information.
5. The DOS and Windows TRIANGLE table viewer
Tables may be included as part of a text file or saved as table files with a defined file extension. In either case, they should be marked up with the GS markup indicators that tell where the table begins and ends and where each element and line ends.
The table view has two modes. A formatted mode is intended primarily for small tables being read using an on-line braille display. In this mode, tables are displayed on the screen much as they would be if formatted for sighted readers.
The formatted table mode is clumsy if the table is large or if the user (of DOS TRIANGLE) is reading with a speech synthesizer. The cell-by-cell mode is intended for such cases. The reader views one cell at a time and can navigate right and left, up and down from cell to cell in the table. The screen also shows the title and the cell row and column number. Optional information such as the row or column labels may also be displayed. This mode permits blind readers to read extremely large or complex tables.
The table viewer may be entered at will, and a table remains in the viewer until it is replaced by another one. Tables appearing in the text may be captured into the reader by placing the cursor anywhere within the table and pressing a "capture table" short cut key.
6. The DOS and Windows TRIANGLE Touch-and-Tell figure viewer
This feature requires use of an external digitizing tablet on which a tactile figure is mounted. A computer "map" file is required, so that whenever a user identifies an object on the figure, the computer can display information about that object. The information is read by speech and/or braille and may be arbitrarily large. The map files are produced by a sighted user to create the annotated pictures. Once the files are created, they can be read by the TRIANGLE program and printed to a graphics embosser.
The figure viewer can be entered at will, and once a map file is read in, the figure is scaled by indicating marks at the top right and bottom left of the figure. This information remains until a new figure is mounted and a new map file read into the table viewer.
7. Preparing materials for TRIANGLE
Both sighted and blind users can create scientific documents in the word processor, save expressions to be computed and graphed, and create tables in the table viewer. It is possible in principle to translate TRIANGLE files to and from other formats that typeset information in standard two-dimensional notation for sighted people. The authors expect to support the new XML world wide web language for this purpose.
A sighted person can create tactile figures and the accompanying computer map files with our Objectif[2] program. Objectif runs under Microsoft Windows and permits a sighted person easily to edit and simplify a bit mapped image file to be printed on a braille graphics embosser or one of the new direct technologies.[5-7]
The map file is created by selecting objects on the computer screen and entering labels or any desired text information to be displayed when the blind user selects that object. At present there is no way for a blind user to create TRIANGLE figures and map files.
8. Acknowledgments
This research was supported by National Science Foundation grant HRD-9452881.
[5] The TIGER (TactIle Graphics EmbosseR) is a Windows 95 tactile printer capable of printing from standard Windows 95 applications. Text can be printed in 6- or 8-dot DotsPlus or any braille font including American, DIN, or British computer braille or either GS6 or GS8. TIGER was invented and developed in this research group and has been licensed to ViewPlus Technologies, Inc. That company has a booth at this conference. TIGER will be available commercially in summer 1998 at $6,000. For more information, see
[6] Swell paper is the currently most common technique for making tactile printouts of arbitrary line and block graphics. See
[7] Tektronix Inc. is marketing a version of their Phasor wax jetcolor printer that allows extra wax to be deposited in order to make a tactile image. Cost for the lowest priced model is $14,000 | 677.169 | 1 |
Exam Board: EdexcelLevel: AS/A-levelSubject: MathematicsFirst Teaching: September 2017First Exam: June 2018Build your students' confidence in applying mathematical techniques to solving problems with resources developed with leading Assessment Consultant Keith Pledger and Mathematics in Education and Industry (MEI).- Build reasoning and problem-solving skills with practice questions and well-structured exercises that build skills and mathematical techniques.- Develop a fuller understanding of mathematical concepts with real world examples that help build connections between topics and develop mathematical modelling skills.- Address misconceptions and develop problem-solving with annotated worked examples.- Supports students at every stage of their learning with graduated exercises that build understanding and measure progress.- Provide clear paths of progression that combine pure and applied maths into a coherent whole | 677.169 | 1 |
Key Stage 5 – Mathematics
A Level Mathematics
A Level Mathematics is a challenging and immensely rewarding course, in which you will be able to take pride in your ability to problem solve, appreciate the applications of modelling techniques and wonder at the complexities of calculus.
In each of years 12 and 13 you will be taught by two teachers, each offering their own unique blend of experience, support, practical knowledge and skills. In the classroom, you will be introduced to the necessary skills and techniques needed to succeed in Mathematics. In your own study time you will be set regular tasks to complete, including written assignments, online tasks, and frequent exam paper practice. Each topic of work covered will be assessed with a summative test, and additional support will be readily available to ensure that high standards are maintained in all areas of the course.
A level Mathematics is one of St Crispin's most successful subjects, with over 70% of students typically achieving A* to B grade in their A level each year. We firmly believe that our exemplary teaching methods, our rigorous monitoring of student progress and our 'open door' support policy will continue to produce excellent mathematicians for the future.
Year 12 – AQA A Level Mathematics (7357) – Reformed
You will study a two year course which covers Core Mathematics and Applied Mathematics, studying them both as specific skills to be mastered and as topics that can be linked through functional questions and problem solving tasks.
Core Mathematics
The Core Mathematics section of the course continues from the algebra and trigonometry within the Higher GCSE course. Topics covered will include:
Proof
Algebra and Functions
Coordinate Geometry
Sequences and Series
Trigonometry
Exponential and Logarithms
Differentiation
Integration
Numerical Methods
Vectors
Mechanics
This applied Mathematics unit will develop many of the concepts already encountered in GCSE Physics. Topics covered will include:
Quantities and Units used in Mechanics
Kinematics
Forces and Newton's laws
Moments
Statistics
This applied Mathematics unit will develop many of the concepts already encountered in the data handling units of GCSE course. Topics covered will include:
Statistical Sampling
Data Presentation and Interpretation
Probability
Statistical Distributions
Statistical Hypothesis Testing
In accordance with AQA guidance, students will spend time working with a large data set, knowledge of which will be examined as part of the final examinations.
Calculators
It is a requirement of the reformed A level course that students are expected to use the new style calculators, such as the Casio Class-Wiz FX-991EX (pictured). Students will need to be in possession of this for the start of Year 12. They are available from the usual stockists or may be purchased from Mrs Prince.
Examination board
AQA Mathematics (7357)
For full details of the course specification please follow the link below.
Three written papers; each of two hours, all taken in the summer term of year 13
Year 13 – AQA A Level Mathematics (6360) – Unreformed
You will study these three, equally weighted, modules over the year, based on core and applied mathematical skills.
A2 Core Mathematics 3
This unit will cover the domain, range and inverse of a function, the modulus function, combining transformations of graphs, inverse and reciprocal trigonometric functions, exponential and logarithmic graphs, differentiation and integration of trigonometric functions, differentiation using the chain, product and quotient rules, integration by substitution, by parts and by the use of standard integrals, volumes of revolution, iterative methods, the mid-ordinate rule and Simpson's rule.
A2 Core Mathematics 4
This unit will cover algebraic division, partial fractions, parametric equations, binomial expansions, trigonometric addition and double angle formulae, exponential growth and decay, differential equations, vectors in two and three dimensions and the vector equation of a straight line.
A2 Statistics 1
This applied Mathematics unit will develop many of the concepts already encountered in GCSE Mathematics or statistics. The topics covered will include standard deviation, the laws of probability, the binomial distribution, the normal distribution, the use of unbiased estimators, confidence intervals, the product moment correlation coefficient, least squares regression and the interpretation of residuals.
Examination board
AQA Mathematics (6360)
For full details of the course specification please follow the link below.
A2: Three written papers; each of ninety minutes, all taken in the summer term
Requirements
To study A level Mathematics students are required to have achieved a minimum of grade 7 at GCSE
Future Opportunities
Mathematics A level is highly respected by universities and employers, and is required for a wide range of courses and careers such as accountancy, market research, games design, logistics, telecommunications and finance. University courses in Physics, Engineering and Computer Science often require an A level in Mathematics.
If you require any further information, please do not hesitate to email Mrs Bauwens, the head of key stage 5 Mathematics, on bauwensa@crispins.co.uk. | 677.169 | 1 |
Discrete Mathematics
Description: Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This book will help you think well about discrete problems: problems like chess, in which the moves you make are exact, problems where tools like calculus fail because there's no continuity, problems that appear all the time in games, puzzles, and computer science | 677.169 | 1 |
Calculus Foldable 7-1: Volumes of Solids
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|4 pages
Share
Product Description
Your students can practice the four methods for finding the volume of solids: cross section, washer, disk and shell. (Yes, I know the shell method is not a required topic on the AP test, but it is a great method in certain situations.) This foldable has been formatted so that it can be used alone or glued into an Interactive Notebook.
This is the first concept my students encounter in the Larsen Calculus textbook in Ch. 7 on Applications of Integration thus the numbering 7-1. Hope this is helpful. Enjoy! | 677.169 | 1 |
REDUCE is a system for general algebraic computations of interest to mathematicians, scientists and engineers, which is normally used interactively. It has been produced by a collaborative effort involving many contributors.
Basic features
The basic features of REDUCE, which might be useful to students and teachers from high school / secondary school up to university level, include:
exact arithmetic using integers and fractions;
arbitrary precision numerical approximation;
polynomial and rational function algebra;
exponential, logarithmic, trigonometric and hyperbolic functions;
differentiation and integration of functions of one or more variables;
factorization and expansion of polynomials and rational functions;
solution of single and simultaneous equations;
automatic and user controlled simplification of expressions;
plotting graphs of functions in 2 and 3 dimensions;
output of results in a variety of formats.
Advanced features
More advanced features, which allow calculations at university level and beyond, include:
Interactive and batch-mode use
REDUCE is often used as an interactive algebraic calculator for problems that are possible to do by hand. However, it is designed to support calculations that are not feasible by hand. Many such calculations take a significant time to set up and can run for minutes, hours or even days on the most powerful computers. REDUCE supports non-interactive batch-mode use for large computations.
General design characteristics
The REDUCE computer algebra system has been designed with the following general characteristics in mind:
Code stability
Various versions of REDUCE have been in use since the late 1960s. There has been a steady stream of improvements and refinements since then, with the source being subject to wide review by the user community. REDUCE has thus evolved into a powerful system whose critical components are highly reliable, stable and efficient.
Full source code availability
From the beginning, it has been possible to obtain the complete REDUCE source code, including the "kernel." Consequently, REDUCE is a valuable educational resource and a good foundation for experiments in the discipline of computer algebra. Many users do in fact effectively modify the source code for their own purposes.
Flexible updating
One advantage of making all code accessible to the user is that it is relatively easy to incorporate patches to correct small problems or extend the applicability of existing code to new problem areas. World Wide Web servers (currently SourceForge and its mirror sites) allow users to get such updates and completely new packages as they become available, without having to wait for a formal system release.
Portability
Careful design for portability means REDUCE is often available on new or uncommon machines soon after their release. This has led to significant user communities throughout the world. At the present time, REDUCE is readily available on essentially all computers.
Uniformity
Even though REDUCE is supported with different Lisps on many different platforms, much attention has been paid to making all versions perform in the same manner regardless of implementation. As a result, users can have confidence that their calculations will not behave differently if they move them to a different machine.
Algebraic focus
REDUCE aims at being part of a complete scientific environment rather than being the complete environment itself. As a result, users can take advantage of other state-of-the-art systems specializing in numerical and graphical calculations, rather than depend on just one system to provide everything. To this end, REDUCE provides facilities for writing results in a form compatible with common numerical programming languages (such as C or Fortran) or document processors such as LaTeX.
State-of-the-art algorithms
Another advantage of an "open" system is that there is a shared development effort involving both distributors and users. As a result, it is easier to keep the code up-to-date, with the best current algorithms being used soon after their development. At the present time, we believe REDUCE has superior code for solving non-linear polynomial equations using Gröbner bases, real and complex root finding to any precision, exterior calculus calculations and optimized numerical code generation among others. Its simplification strategy, using a combination of efficient polynomial manipulation and flexible pattern matching is focused on giving users as natural a result as possible without excessive programming.
Specialist packages
REDUCE has a wide user base, which has led to a wide range of packages for specific purposes. A particular algebra system is often chosen for a given calculation because of its widespread use in a particular application area, with existing packages and templates being used to speed up problem solving. As evidenced by approximately 1000 reports listed in the current bibliography, REDUCE has a large and dedicated user community working in just about every branch of computational science and engineering. A large number of special purpose packages are available in support of this, with many contributed by users. | 677.169 | 1 |
The Department of Mathematics has a general expectations statement, which we are assumed to follow in this class.
Reading
In this class we use the book
Steven J. Leon; Linear algebra with applications, eighth edition Pearson Prentice Hall.
The book is really unavoidable, and cannot be replaced with another textbook! There are, however, many linear algebra textbooks which may be useful. Linear Algebra by K. Hoffman and R. Kunze, Prentice Hall, is of particular interest. It covers much more material than this class.
Course Objective
To learn basic ideas and notions of linear algebra. It is particularly important to learn to operate with these ideas and notions. This includes both the ability to conduct a specific calculation, and to prove or disprove a statement.
Since this is a writing intensive class, to produce a mathematically rigorous argument (aka a proof) means to write it down in proper English.
The course contains a combination of concepts, ideas and techniques, which the students must be able to apply in solving specific problems. Some of these problems require to either prove or disprove a certain statement.
At the end of the day, your grade will reflect your ability to solve specific problems and to properly write your solutions down. This assumes your ability to read and understand the textbook. To understand, in this context means to be able to both create mathematical arguments (proofs) which are similar to those provided in the text, and to perform specific calculations similar to those in the exercises and examples. More specifically, the following rules are to be taken.
Final exam will count for 30% of the final grade. The exam is cumulative (it covers all the material).
The exam will take place on ???
Four Writing Homework Assignments count together for 40% of the final grade.
It will be possible to make up these assignments. After an assignment is graded, one may redo and resubmit it (one can do that only once with every assignment). These assignments are devoted to writing proofs. A Writing Homework Assignment is always due the class after the corresponding chapter from the book is finished.
Four Quizzes count together for 30% of the final grade.
Every quiz consists of problems similar (maybe even identical) to those from the regular homework. For this reason, it is highly recommended to solve homework problems. There will be no make-ups for quizzes. Note that regular homework is never collected and checked. The only way it contributes to the final grade is by means of the quizzes.
The following are not part of the grading scheme:
Attendance and regular homework
Contents and Homework Assignments
This table is only approximate. It may and will be updated regularly. In particular, some dates will be entered. | 677.169 | 1 |
Posts Tagged "High School Math"
The Charles A. Dana Center's Advanced Mathematical Decision Making (AMDM) is a comprehensive product that supports instruction in Advanced Quantitative Reasoning courses in Texas high schools. Content is available in print format. Instruction is discovery based and engages students in authentic problem solving that emphasizes real-world contexts, including statistics and finance. Algebra II is a prerequisite for the course. [Read more…]
Learning List has reviewed Cosenza & Associates' Algebraic Reasoning. This comprehensive product supports instruction in Texas algebraic reasoning courses. Content is available in print and digital formats and includes additional online tutorials. Instruction is inquiry based and covers the attributes of functions, systems of equations, modeling with functions and data, and approaches to solving equations. Teacher and student materials contain clear links to the Texas Essential Knowledge and Skills (TEKS) and English Language Proficiency Standards (ELPs).
Resources include blackline masters of four versions of mid-chapter and end-of-chapter assessments, including a modified version for students with special learning needs. Editable versions (i.e., Word versions) of a mid-term and final exam and a culminating course project are available. Each test includes multiple-choice and free response items. The culminating project requires students to independently identify and analyze a data set related to a topic of interest. Teachers also have access to the ExamView® assessment software that allows users to create customized assessments.
Learning List has reviewed Apex Learning's Texas Geometry. This is a comprehensive, online geometry product for Texas students. Core content is presented online and includes some printable materials. The course focuses on developing students' conceptual understanding, computational skills, and proficiency in solving problems. Resources support instruction in self-paced, remediation, and credit-recovery programs.
Texas Geometry is organized in two semesters. The first semester's content is presented in five units that cover the foundations of geometry, triangles, right triangles, quadrilaterals and other polygons, and circles without coordinates. The second semester presents content in four units that address coordinate geometry, constructions and transformations, three-dimensional solids, and applications of probability. Across semesters, students learn to reason mathematically and to use mathematical models and tools to solve real-world problems.
Each unit begins with a short video introduction that frames the real-world applications of what students will learn and connects new content to prior learning. Subsequent instruction is provided through a set of online activities and, where appropriate, accompanying worksheets. Each unit and semester ends with two versions of a unit/semester exam—one version presents open-ended questions and the second is made up of multiple choice items.
About Apex Learning*
Apex Learning's digital curriculum is designed to support all students in achieving their potential, from those struggling with grade-level content to those capable of accelerating their learning. The curriculum is designed to actively engage students in learning—combining embedded supports and scaffolds to meet diverse student needs, actionable data to inform instruction, and success management, to ensure students get the outcomes they need.
*Information in this section is provided by or adapted from Apex Learning.
Subscribe to Learning List for access to the spec sheet, full editorial review and detailed alignment report for this material, and thousands of other widely used Pk-12 resources.
McGraw Hill's ALEKS (Assessment and Learning in Knowledge Spaces) is a comprehensive, online program that provides adaptive math instruction for students in grades 3-12. Content is available online and includes some printable resources (e.g., worksheets). Instruction is presented using individualized, interactive tutorials that focus on a specific concept. Learning List recently reviewed ALEKS resources for middle and high school math courses.
Students take an "Initial Assessment" that evaluates they know and the concepts they are ready to learn. Based on the results of the Initial Assessment, each student receives a "Student Assessment Report" that clarifies the topics they will be assigned in "ALEKS Learning Mode."
In ALEKS Learning Mode, students receive direct instruction and have opportunities for guided and independent practice. Instruction focuses on a single topic or concept (e.g., the multiplicative property of equality with decimals) and begins with an example problem, its solutions, and an explanation of the strategy used to find the solution. After the example problem, students are presented with a set of similar practice problems.
ALEKS supports instruction in a variety of learning environments, including teacher-led, self-directed, and distance learning programs. Content is fully customizable and available in English and Spanish.
About McGraw Hill Education and ALEKS*
McGraw-Hill Education, a leading digital learning company, acquired ALEKS Corporation in 2013 At the time of the acquisition, McGraw-Hill Education had marketed and sold ALEKS for math in the higher education space for more than 10 years. | 677.169 | 1 |
Mathematics
207 Algebraic Foundations
Credit: 1
An introduction course in preparation for Algebra I. Outlines mastery of mathematical
concepts in preparation for the operations and equations of Algebra I. Also works with
the students to master formulations of word problems and translation of such materials
into mathematical formulas.
213 Honors Algebra I
Credit: 1 (weighted – Honors)
The students begin with a review of basic algebraic concepts of operations with
fractions, exponents, and simplification of algebra expressions. Students continue
through factoring, theory of line and slope, solution of quadratic equations by factoring,
formula and completing the square, simplifying radicals and graphing linear and
quadratic equations.
215 Algebra I
Credit: 1
Students study the Basic Algebraic concepts of set variables and properties,
progressing through operations with fractions, algebraic operations, exponents,
simplification of algebra expressions, including factoring, the theory of lines and slope,
solution of quadratic equations by formula and graphing linear equations.
The student will learn the basic postulates of Euclidean Geometry using the 2 column
proof structure of logic involving angles, congruent and similar triangles, parallel lines,
circles and arcs and Pythagorean Theorem integrated with constructions. Formulas for
areas of all basic geometric plane figures will be learned and formulas for volumes of
prisms, pyramids, cylinders, cones and spheres will be explored. Trigonometric rations will be
used plus sine and cosine laws, to solve new right triangles.
225 Geometry
Credit: 1
Prerequisite: Algebra I & Algebra II
Students will explore the concepts of plane geometry integrated with space and
coordinate geometry linked with algebra. Angles, congruent and similar triangles,
parallel lines, circles, properties of quadrilaterals and Pythagorean Theorem will be
studied. Formulas for the areas of all basic plane figures will be used plus an
exploration of logic structures, constructions, probability and ratios and proportions will
be covered. An introduction to trig ratios will be used to solve all right triangles.
225 Informal Geometry
Credit: 1
Prerequisite: Algebra I & Algebra II
Students will explore the concepts of plane geometry integrated with space and coordinate geometry linked with algebra. Angles, congruent and similar triangles, parallel lines, circles, properties of quadrilaterals and Pythagorean Theorem will be studied. Formulas for the areas of all basic plane figures will be used plus an exploration of logic structures, constructions, probability and ratios and proportions will be covered.
The student begins with a review of Algebra I material, moving into graphing of rational functions, methods of substitution, determinants, analysis and solution of word problems, roots of polynomials and complex numbers, arithmetic and geometric progressions plus exponential and logarithmic
functions. Trigonometric function, solutions of right triangles, radian and degree angle
measure, double angle, half angle and addition formulas, law of sines and cosines plus
inverse trigonometric functions and solutions of trigonometric functions are also
covered.
235 Algebra II
Credit: 1
Prerequisites: Algebra I
The student will review Algebra I material, then explore graphing and solving of higher polynomials, rational functions, methods of substitutions, determinants, analysis and solution of word problems, finding roots of polynomials and using complex numbers. Also covered will be arithmetic and geometric progressions and probability, exponential and logarithmic functions and an introduction to
Trigonometry.
Students will study real numbers, limits, continuity, compute the derivatives of algebraic
functions, explore tangent and normal lines, extremes of functions, mean value
theorem, related rates, definite integrals, areas and volumes and arc length. Some
applications to physics and mechanics will occur. This course will prepare students for
the Advanced Placement exam in the spring.
245 Trigonometry
Credit: 0.5
Prerequisite: Algebra II & Geometry
In this course the student is presented with solid geometry, trigonometry and if possible an
introduction to calculus. The emphasis is on trigonometric functions, solution of right triangles,
radian and degree angular measure, law of sines and cosines, inverse trip functions and the
solution of trigonometric equations. Analytic geometry, limits, continuity, derivatives of
algebraic function and tangents and normal lines will be explored as time permits. | 677.169 | 1 |
General Math
The goals of this class are for students to learn to value mathematics, become problem solvers confident in their ability, to reason mathematically and to communicate mathematically. Units covered may include: whole numbers, decimals, number theory, fractions, expressions and equations, integers, rational numbers, ratios and proportions, percents, and equalities and inequalities. | 677.169 | 1 |
guidelines mathematics | 677.169 | 1 |
Dimasuay, Lynie B.
Name: LYNIE B. DIMASUAY
Rank: ASSOCIATE PROFESSOR 5
Highest Degree: MS MATHEMATICS
Specialization:PURE MATHEMATICS
Research Interest:MATHEMATICS EDUCATION
Publications: | 677.169 | 1 |
Often calculus and mechanics are taught as separate subjects. It
shouldn't be like that. Learning calculus without mechanics is
incredibly boring. Learning mechanics without calculus is missing
the point. This textbook integrates both subjects and highlights
the profound connections between them.
This is the deal. Give me 350 pages of your attention, and I'll
teach you everything you need to know about functions, limits,
derivatives,... more...
Success in your calculus course starts here! James Stewart's
CALCULUS texts are world-wide best-sellers for a reason: they are
clear, accurate, and filled with relevant, real-world examples.
With CALCULUS, Eighth Edition, Stewart conveys not only the utility
of calculus to help you develop technical competence, but also
gives you an appreciation for the intrinsic beauty of the subject.
His patient examples and built-in learning aids will help you... more...
This book is written to be a convenient reference for the working
scientist, student, or engineer who needs to know and use basic
concepts in complex analysis. It is not a book of mathematical
theory. It is instead a book of mathematical practice. All the
basic ideas of complex analysis, as well as many typical applica
tions, are treated. Since we are not developing theory and proofs,
we have not been obliged to conform to a strict logical ordering... more...
In order not to intimidate students by a too abstract approach,
this textbook on linear algebra is written to be easy to digest
by non-mathematicians. It introduces the concepts of vector
spaces and mappings between them without dwelling on statements
such as theorems and proofs too much. It is also designed to be
self-contained, so no other material is required for an
understanding of the topics covered.
As the basis for courses on... more...
Get ready to ace your AP Calculus AB Exam with this
easy-to-follow, multi-platform study guide
5 Steps to a 5: AP Calculus AB introduces an easy to
follow, effective 5-step study plan to help you build the skills,
knowledge, and test-taking confidence you need to achieve a high
score on the exam. This wildly popular test prep guide matches
the latest course syllabus and the latest exam. You'll get online
help, four full-lengthThe book summarizes several mathematical aspects of the vanishing
viscosity method and considers its applications in studying
dynamical systems such as dissipative systems, hyperbolic
conversion systems and nonlinear dispersion systems. Including
original research results, the book demonstrates how to use such
methods to solve PDEs and is an essential reference for
mathematicians, physicists and engineers working in nonlinear
science.... more...
This book gives a comprehensive and thorough introduction to ideas
and major results of the theory of functions of several variables
and of modern vector calculus in two and three dimensions. Clear
and easy-to-follow writing style, carefully crafted examples, wide
spectrum of applications and numerous illustrations, diagrams, and
graphs invite students to use the textbook actively, helping them
to both enforce their understanding of the material and to... more...
PREMIUM PRACTICE FOR A PERFECT 5! Ace the AP Calculus AB Exam
with this Premium version of The Princeton Review's comprehensive
study guide.
In addition to all the great material in our
classic Cracking the AP Calculus AB
Exam guide—which includes thorough content reviews,
targeted test strategies, and access to online extras via our AP
Connect portal—this edition includes extra exams, for a total
of 6 full-length practice... more...
The main goal of this third edition is to realign with the
changes in the Advanced Placement (AP ) calculus syllabus and the
new type of AP exam questions. We have also more carefully
aligned examples and exercises and updated the data used in
examples and exercises. Cumulative Quick Quizzes are now provided
two or three times in each chapter. | 677.169 | 1 |
Digital electronics homework help
Provides on demand homework help and tutoring services that connect students to a professional tutor online in math, science, social studies or English.Get instant access to our step-by-step Digital Electronics solutions manual.
Whatever your feelings about what some have called the digital.Your problems are now mine,your assignment will be completed before the deadl. Digital Electronics (Grade 11-College Undergraduate).Personalize instruction with digits, a Pearson Middle Grades Math Curriculum.I mean I get the fact that there are some people out there who.Mywordsolution offers you exactly you need in your Digital Electronics courses if you think that you cannot cope with all the complexities of Digital Electronics.The Digital Electronics Basics series present the fundamental theories and concepts taught at entry level electronics courses at both 2 year and 4 year institutions.
3-Bit Binary Adder Truth Table
The readings are provided on a digital display and can be stored in the memory of the monitor or.Homework Help If you have a question about an electronics exam,.
Digital Electronics: A Practical Approach (8th - Course Hero
Here you can get homework help for Electronic BP Apparatus,.
Solved: Chapter 5 Problem 10ME Solution | 9th Edition | Chegg.com
Tablets contain many technological features that cannot be found in print textbooks.It explains how electricity can be used to carry or process information.Even though digital electronics homework help her chances of persuasion improve dramatically.
Tablets give users the ability to highlight and edit text and write notes without.You can hire us for Electrical Engineering Homework Help and.
Electrical Engineering and Technology Welcome to this open and free electrical engineering study site.Yet every week or biweekly, provide the title of the communication.
Solved: Chapter 5 Problem 23ME Solution | 9th Edition | Chegg.com
Apps to Organize ADHD Students at School Tools for children with ADHD that provide homework help and aid in organization and time management.
TO DIGITAL CONVERTERS Electronics Assignment Help and Homework Help ...
Types of Flip Flop Digital Electronics
There are two basic types of electrical signals: analog and digital.Electronics Forum (Circuits, Projects and Microcontrollers) Home Forums.Homework Solutions Fundamentals Power Electronics, Essay price.Use our interactive solutions player to walk you through ten steps to help you better understand your homework. | 677.169 | 1 |
This is the one we have been given, our teacher doesn't seem to like it terribly. We have missed a few lessons due to inset days etc. and some of the poofs in this book don't make sense (to me at least) unless I work at them for a long time. It wasn't so bad for the matrices unit but now we have moved on to matrix transformations and we've had only one lesson.
I mean to say I understand I will find it difficult I just expected the explanations to be explanations and not expect many things to be left unsaid- its not easy to go through the book studying by yourself.
Just so you know, this text goes into more depth than you actually need to, learn the basics, do some questions, do some practice papers, if you dont understand a question look for a solution in the textbook
(Original post by Kasc)
Just so you know, this text goes into more depth than you actually need to, learn the basics, do some questions, do some practice papers, if you dont understand a question look for a solution in the textbook
(Original post by Killjoy-)I don't think you're likely to find much better to be honest. Mathematical proofs tend to take ages to read compared to other things; you just have to get used to them!
(Original post by Killjoy-)
Any suggestions would be greatly appreciated.
You could send a private message to me containing your email address and requesting my set of OCR FP1 matrices powerpoints. | 677.169 | 1 |
This week students continued to solve equations using algebraic properties. Equations are becoming more complex with variables on both sides.
QUIZ FRIDAY!
Week 10
This week students will be writing and solving equations. We will discuss using inverse operations and reciprocals as solving methods.
Week 9
This week students will take a chapter 2 test demonstrating their learning throughout the first grading period. Friday is the last day to turn in any missing assignments from this quarter. Check Pinnacle to see what you are missing!
Week 8
This week students will work on combining like terms and simplifying more complex expressions.
Homework:
Pg. 110 #1-14
Practice Test
Weeks 6&7
Students have been working on operations with real numbers and using the distributive property to simplify expressions.
Homework:
Pg. 74/75 #1-16, 41-46
Pg. 96 #1-5, 16-48 even
(plus other assignments from worksheets given in class)
Week 5
This week students will be representing data using line and bar graphs. Students will also interpret graphs. A studyguide for the test will be provided.
TEST FRIDAY (9/29) OVER CHAPTER 1!
Week 4
This week students will be learning to translate expressions into mathematic equations and inequalities. We will practice using mental math to solve real world problems and check for correctness. Students will have a TEST FRIDAY (9/22)!
Assignments:
1.5 Practice Worksheet
Page 39/40 #4-15
Study Guide for Test
Week 3
Students will be using the order of operations to evaluate expressions.
QUIZ WEDNESDAY (9/13) covering sections 1.1 to 1.3.
Assignments:
Page 18 #1-16
1.3 Practice Worksheet
All students need their own calculator! (model TI30XIIS or TI-84 Plus)
Week 2
During the week of September 5 students will work on sections 1.1 (variables in Algebra) and 1.2 (exponents).
Assignments: page 6 #1-16, 1.1 worksheet, 1.2 worksheet
Students need a calculator for class and completing homework assignments!
Welcome!
Welcome back to a new school year!
Supplies needed for a successful year:
3-ring binder or spiral notebook with a folder
Loose leaf paper
Pencils
Calculator (TI84 Plus or TI30XIIS)
Book Cover
This week we will get to know each other, sign out textbooks, learn classroom procedures, and take the STAR initial assessment | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
7 MB|116 pages
Share
Product Description
This r squared creation zip bundle contains all the PowerPoints necessary to teach Chapter 8 "Factoring Polynomials" from Holt Algebra in the method we have found most effective for student learning. (All of the files are also available for individual purchase as well). The materials covered in the review are standards based and are aligned to Chapter 8 "Factoring Polynomials" of the Holt Algebra textbook. Concepts covered are how to find the prime factorization, how to find the GCF of polynomials, how to factor by taking out the GCF, how to factor a quadratic expression where the "a" term does not equal one, how to factor a quadratic expression where the "a" term equals one, how to factor a special quadratic expression, and how to factor out the GCF first, and then factor the remaining quadratic expression.
The PowerPoints are also editable if you would like to make any changes (you may need to download MathType for free).
They have been reviewed for accuracy and rigor and taught in the classroom. This chapter addresses CA standard 11.0. | 677.169 | 1 |
AP's high school Physics 1: Algebra-Based course is a rigorous, college-level class that provides an opportunity to gain skills colleges recognize . community college chancellor california proposing rid the requirement all students take intermediate effort boost the. Algebra great fun - you get solve puzzles! With computer games play by running, algebra for college student gustafson frisk jumping or finding algebra for college student gustafson frisk secret things . Well, with letters new policy from state university system will soon allow some math classes pre-requisites other to. Tutorial explains how work exponents, radical expressions, solving equations, inequalities, functions and graphing our goal bring our own open courseware as well high-quality ones freely available online. | 677.169 | 1 |
PreCalculus Transformations Compositions & Inverse Functions
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB|2 + keys
Share
Product Description
This activity is designed to help your Algebra 2 Honors and Pre-Calculus students review key concepts at the end of the unit on Functions and Graphs. There are two different practice sets. The first is transformations of functions from a graph, the second is reading composition and inverse function notation to find values given a graph.
Students often have difficulty learning to read proper notation. This exercise will help them understand and grow in their knowledge of reading mathematics correctly. All answer keys are included. | 677.169 | 1 |
Problem solution topics
Pre-algebra and algebra lessons, from negative numbers through pre-calculus Grouped by level of study Lessons are practical in nature informal in tone, and contain. Problem solving consists of using generic or ad hoc methods, in an orderly manner, for finding solutions to problems Some of the problem-solving techniques developed. 12/28/2016 Welcome to the new FTcom The same global insight Faster than ever before on all your devices. Latest trending topics being covered on ZDNet including Reviews, Tech Industry, Security, Hardware, Apple, and Windows.
Experts at Grademinerscom take every "write my essay" request seriously and do the best job on your essay, term paper, or research papers Get an excellent paper. WebMath is designed to help you solve your math problems Composed of forms to fill-in and then returns analysis of a problem and, when possible, provides a step-by.
Problem solution topics
The Art of Problem Solving mathematics curriculum is designed for outstanding math students in grades 6-12 Our texts offer broader, deeper, and more challenging. Search the mainstream medical journals, even search the Internet, and you won't find this undeniably simple answer Everywhere you look, conventional medicine is. GATE Coaching at EII is Top Ranked GATE Coaching Institute with Highest Results EII GATE Coaching Institute run by Top Faculty from IITs & IES for quality GATE.
Speech topics lists with free persuasive and informative ideas and class writing tips on outlining your public speaking oral all under one website hosting roof. A community site with forums, blogs, photos, videos and more for both new and experienced aquaponic gardeners Let's learn together. Business strategy simulations for educating management and marketing students in global business markets using an intuitive business simulation game. 12/29/2016 Find out more about the history of New Year's, including videos, interesting articles, pictures, historical features and more Get all the facts on HISTORYcom.
The Math Forum is the comprehensive resource for math education on the Internet Some features include a K-12 math expert help service, an extensive database of. MAA's American Mathematics Competitions is the oldest (began in 1950) and most prestigious mathematics competition for high schools and middle schools. Serious Education plus the most fun you can have in metal finishing Solution for precipitating gold from HAuCl 4 solution (2005) Q Respected Sir.
12/20/2016 CNET news editors and reporters provide top Computers news, with in-depth coverage of issues and events. 9/2/2010 Children, eating a peanut paste, at a seaside tent camp in Haiti Credit Maggie Steber for The New York Times Plumpy'nut proved so palatable and so. Online homework and grading tools for instructors and students that reinforce student learning through practice and instant feedback.
Planning an out of state move? U-Pack Moving is the affordable solution You pack, load and unload, and U-Pack drives Compare to traditional moving companies and.
Find a Science Fair Project Idea Looking for inspiration for a science fair project? Science Buddies has over 1,150 Project Ideas in all areas of science.
Thought Of The Day ADVERTISEMENT. Find out more about the history of The Holocaust, including videos, interesting articles, pictures, historical features and more Get all the facts on HISTORYcom. Welcome to the KPMG knowledge base of research that demonstrates our understanding of complex business challenges faced by companies around the world. | 677.169 | 1 |
(Original post by a5a09)
I said that referring to actuarial science as a branch of applied mathematics (Actuarial Science is a branch of applied mathematics). Not the syllabus of an actuarial science degree as such. But what you've said about the mathematical theories may be what makes an AS degree different from a math degree.
(Original post by Schleigg)As someone studying a straight maths degree, with 1 actuarial exemption from 1 module and currently on an actuarial internship. I would not say an actuarial science degree is a "maths degree". It would contain a high level of numerical content but in no way represents a maths degree. I haven't seen number in my degree in years whereas actuarial science would deal with a lot of stats (my one exemption is CT3, the probability one) and a lot of financial maths and econ whereas my degree - bar my optional module - is a lot of algebra and analysis which you would not see in an AS degree at all... | 677.169 | 1 |
Get help with algebra 2 homework
Tutorvista is a one stop solution for all Math problems. With this learning process, students can schedule their sessions at any preferred time. There is also an errata sheet available for some courses get help with algebra 2 homework if you select "Launch a Full Course. However, my favorite part of the site is the "activities" area that offers some really high quality and valuable online applications that graphically show algebraic principles, such as this "pan balance" app that graphically displays what happens when you change different elements of equations. Understands this problem and created a website called Illuminations that brings the subject (including algebra) to life for students. Apart from this, with the help of whiteboard, they can get both answers as well as the explanations of each Math problem. The virtual tutors associated with TutorVista are exceptionally trained and are endowed with sound knowledge in each Math topic. I think a good site that no one thinks of using is cramster. Choose online sessions with TutorVista and make your learning process extremely useful. NROC makes editorial and digital engineering investments in the content to prepare it for distribution by HippoCampus. As an individual user, however, you may create a custom HippoCampus page and then link to an individual topic. Unfortunately, there is no way to download the video from our website. Users do not need to register or log in to use the site. Math concepts that are centered on clearly defined numbers and with all "factors" within the problem typically defined, are usually fairly easy for most students to follow and understand. Algebra 2 is an advanced level Math students need to practice the get help with algebra 2 homework problems regularly with the help of our online sessions instantly as per their convenience. As an open resource for personalized learning, HippoCampus. Students can use online calculators and can practice several worksheets to brush up the required Algebraic topic before final exams. First, there is a "maximize" button beneath the bottom left corner of the Media Window which will widen the screen. =P (says me, who relied on Cramster to survive college math T__T ) Due to the complexity of modifying the multimedia content, we cannot always correct errors within the video presentations. There is an Errata icon that appears with any topics in which a known error has been identified. The site includes high quality get help with algebra 2 homework activities and lessons that cover all aspects of both math and algebra. Org is a free, core academic web site that delivers rich multimedia content--videos, animations, and simulations--on general education subjects to middle-school and high-school teachers and college professors, and their students, free of charge. Nothing is quite as frustrating for many students as trying to understand get help with algebra 2 homework abstract concepts, and learning to do so by reading drab and boring textbooks that make you want to go to sleep. TutorVista offers free demo sessions and online tests to assess your understanding level in different algebraic sub-topics. However, it does have step by get help with algebra 2 homework step solutions for the odds to many different textbooks, get help with algebra 2 homework as well as a decently sized db of user submitted material. Detailed explanations are added in each session helping students cope with the subject. Com just because it's a guide to text books. Mathematics, and algebra in particular are critical skills for students to learn early on that can tremendously simplify their lives later on. S. " How do I report do my logic homework a course errata item? Students can opt for both algebra 1 and algebra 2 homework thesis statement homework help help and can take sessions simultaneously for their exam get help with algebra 2 homework preparation. Trained subject experts are associated with us and they design sessions that make learning each algebraic topic valuable and easier for students of all Grades to understand. Students can use Algebra calculators and get their answers instantly. The same great content available for free individual use at HippoCampus. Moreover, with this online learning process, students can learn from basic to advanced get help with algebra 2 homework concepts of Math in a jiffy. These sessions get help with algebra 2 homework improve your performance in exams when you understand each Algebra II topic thoroughly by taking adequate help from proficient online tutors. Teachers project HippoCampus content during classroom learning and assign it for computer labs and homework. For some content, such as that from Khan Academy, a small button in the lower right corner of the media control bar allows the content to be shown full screen. As long as the user were willing to dig, there's lots of good quality content on there. Can I change the size custom essays uk review of the video window? HippoCampus is powered by The NROC Project, a non-profit, member-driven project focused on new models of digital content development, distribution, and use. TutorVista gives you opportunity to avail learning help from our experienced tutors anytime. Yes, in multiple ways. We encourage our users to report any errors they discover so that we can notify everyone of the problem. For lessons, you select grade levels and topics (one area is devoted to just algebra), and you get access to dozens of valuable lessons that teach important algebra concepts. After you have created your custom page, there will be buttons in the upper right corner that allow you to view the text version (when available), bookmark, or link to the topic. Org was designed as part of a worldwide effort to improve access to quality education for everyone. These can be used simultaneously or independently. Many of the resources are buried within internal links, but the Algebra Section alone offers classroom materials for teachers, Internet projects, public forums and even links to algebra software throughout the net. Take unlimited where to buy homework sims 4 online should i do my homework right now tutoring for Algebra II with TutorVista. Membership fees sustain the operation of this non-profit endeavor to make quality educational content freely available to individual learners worldwide. Students use the site in the evenings for study and exam prep. Why won't the Environmental Science animations play? For other content, such as Algebra I--An Open Course, right-clicking the mouse over the video content will open a menu that offers Full Screen as english gcse creative writing essay an option. There is also a "hide column" button beneath the first column of content in the Browse Topics tab. This site offers exclusive online sessions for perplex algebraic topic. Org is also available for institutional use through membership in The NROC Project. Students can choose online Algebra 2 help and solve all Math problems in a step-by-step manner with appropriate guidance from our tutors. Furthermore, students can get free sessions for solving assessment and homework as well. Algebra is a method of calculating using simple equations to represent quantity and to show relation between them. HippoCampus. Luckily, the National Council of Teachers of Mathematics in the U. Its expertly sessions are useful for students of different Grades. The Math Forum is one of the largest math & algebra resources on the net, and if you can afford to donate it would be a valuable and worthy cause to contribute. Of all available free algebra homework help websites, this is definitely the one that's the most fun and interesting. The Math Forum is a valuable public service offered by Drexel University that offers both students and teachers dozens of resources. The grade level where children make the transition from math to algebra can be very easy for some students, but difficult and confusing for others. Hm... Any free algebra homework help websites that exist on the Internet are very positive resources that should be promoted and applauded by everyone. | 677.169 | 1 |
Monthly Archives: Ožujak 2013
Matamicus is a math step-by-step problem solver. It means that it can solve a math problem entered by the user and produces a step-by-step solution with explanations. The solution is consisted of a sequence of steps. The steps are described by mathematical expressions and explanations. On this web site you can find several examples and their solutions produced by Matamicus. You can also download a free trial version of the program and try it for yourself.
Matamicus is also a simple math text editor that can be used to edit mathematical text in a simple and intuitive way. | 677.169 | 1 |
An experienced maths teacher breaks down precalculus into a series of easy-to-follow lessons designed for self-teaching and rapid learning. The book features a generous number of step-by-step demonstration examples as well as numerous tables, graphs, and graphing-calculator-based approached. Major topics covered include: algebraic methods; functions and their graphs; complex numbers; polynomial and rational functions; exponential and logarithmic functions; trigonometry and polar coordinates; counting and probability; binomial theorem; calculus preview; and much more. Exercises at the end of each chapter reinforce key concepts while helping students monitor their progress.Charts, graphs, diagrams, instructive line illustrations, and where appropriate, amusing cartoons help to make learning E-Z. Barron's E-Z books are self-teaching manuals designed to improve students' grades in a wide variety of academic and practical subjects. For most subjects, the level of difficulty ranges between senior school and college-101 standards. E-Z books review their subjects in detail, and feature both short quizzes and longer tests with answers to help students gauge their learning progress. All exercises and tests come with answers. Subject heads and key phrases are set in a second colour as an easy reference aidDie Erw hlten has been writing in one form or another for most of life. You can find so many inspiration from Die Erw hlten also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Die Erw hlten | 677.169 | 1 |
2014 2015 COMMON COURSES OF Study IN Math These are the most common courses of Study but other options are available Students are encouraged to take the most challenging course of studypossible It is possible to accelerate your Math program and move to a more challenging course of Study Your Guidance Counselor and Math instructor willhelp determine what Math courses are best for you and your futur...
Math 40 Handout for Chapter 7 Study Guide Math 40Chapter 7The exam will have 20 or more of the following types of problemsSection s Number s7 1 101-177 every other odd201-217 odd221-241 every other odd7 2 101-190 every other odd201-220 every odd301-345 every odd7 3 101-187 every other odd201-334 every odd7 4 201-270 every odd301-333 every oddChapter Review 1-48 allChapter Test 1-25 allDON T FORGET...
orming and creative arts playing a part in group activities and Camp Hours Monday through Friday 9 00am 4 00pmattending special eventsExtended Care 6 30 am 9 00am 4 00 pm 6 00 pmOur campers have the opportunity to expand their horizons during the summerembarking on new adventures and having fun Our exceptional program and Week 1 June 4-8 Week 6 July 9 - 13dedicated staff create an environment that
Microsoft Word - $ASQteacher201002030007 The Great Indian ChiefSOH-CAH-TOAMany moons ago there lived a great Indian warriorHowever this warrior was angry - angry because hecould not complete much of the geometry the tribe swise woman prescribed In his anger the warriorkicked against a stone and crushed his big toe Fortunately he learnedfrom this experience and began to use Study and concentration ...
2012- 2013 2012- 2013 Middle School CatalogLanguage Arts 6This sixth-grade Language Arts course integrates the Study of literature vocabulary writing andgrammar Language Arts 6 is divided into five units Who Am I Who Are We Where Do I Fit InCan We All Get Along and How Can I Make a Difference In addition to their coursework studentsmaintain an online journal where they can blog and reflect on thei...
onal factors that are affecting my academic performancepoor health family pressuresfinancial problems easily distracted by friendstoo many commitments change in a relationship with someone special to melack of confidence in my abilities lonelinessOtherII EOU services that I have usedAcademic advisingLearning Center Group Study Sessions Math LabA quiet place to studyClassmates to work with when I n
business or the economyby a loving but competitive pair of siblings Evan and JessieTreski At the end of the summer Jessie the younger and more Economic concepts are defined throughout the book asacademically inclined of the two is delighted to learn that Jessie investigates researches and develops her own ideasbecause she is skipping third grade she will be in Evan s fourth- Each chapter contains
the features of thisdistance information book contentspage and index as well asglossaryShared story with Ms Ayi Making 100 posterLeanne Reading to learnIearing about Previewpurpose predict and priorknowledge Story MrGumpy s Motor CarShared Poem Up Look at Ms Yang skip countingnew words Focus on the u counting in 3 s First withsound and sight words number board Then onLook especially at the word co
Scope & Sequence of Courses and Materials Portable Assisted Study SequenceIntegrated Math ConceptsAvailable in English and SpanishSCOPE OF COURSEThis unique course provides a flexible concentrated step-by-step series of modules designed to enhance studentability to master the various components of secondary level Math skills This course offers a variety of ways inwhich it may be utilized either in...
uipment in use in industryParticipate in a hands-on activity oriented program that utilizes team effortsHave the opportunity to enroll in a sequence of courses covering essential topics in technologyTake courses that will apply and reinforce their Study of Math and scienceEnjoy a challenging program that incorporates and addresses the goal of raising standards of learningParticipate in a program t
Reluctant Moms and Dads Suppose I told you I could get your child to enthusiastically Study geography Math physics chemistry andpsychology After you had my head examined would you be interested Oh and as a bonus I can get him or herto hang out with highly motivated well educated older people who are good role models because they don t dodrugs graffiti or tattoos and they have a great work ethicYou... | 677.169 | 1 |
MatBasic 1.2
MatBasic is a calculating, programming and debugging environment...
Author's review
MatBasic is a calculating, programming and debugging environment using special high-level programming language designed for solving mathematical problems.
MatBasic programming language allows execution of sophisticated mathematical calculations, involving an exhaustive set of tools for the purpose of creation of algorithmic programs.
It also allows a user to abstract his mind from the type of working data which can be either real-valued, or complex numbers, or matrices, or strings, or structures, etc.
The MatBasic supports both the text and the graphical data visualization. MatBasic is fast language interpreter and its environment application field is wide: from solving the school problem to executing different engineering and mathematical calculations.
The MatBasic programming language combines; simplicity of BASIC language, flexibility of high-level languages such as C or Pascal and at the same time turns up to be a powerful calculation tool.
By means of a special operating mode, Matbasic it is possible to use as the powerful calculator. Also the MatBasic can be used for educational purpose as a matter of studying the bases of programming and raising algorithmization skills.
Matrix factorizations (LU, Cholesky, QR, SVD, Schur);Polynomial problems. Roots of the polynomial. Polynomials multiplication, division and other functions;Numerically evaluate integral of function;Solve non-linear system of equations;Find zero of a function of one variable (solve non-linear equation);Minimize a function of one and more variables;Solve initial value problems for ordinary differential equations (stiff and non-stiff);and others;.
Your review for MatBasic | 677.169 | 1 |
Currently Monday, last,day till senior checkout, I don't have all my credits, I failed math with like a 48%. Luckly I got the class online, gotta finish it Tuesday by 5pm, but my gosh algebra 2b is hard, my teachers never taught us any of this stuff
It's a really detailed calculator for checking answers. Just pop the equation in and it will give you solutions. I recommend buying the mobile version, it will give you step by step instructions on how to solve an expression. | 677.169 | 1 |
Classical Mechanics with MATLAB Applications
ISBN-10: 0763746363
ISBN-13: 9780763746Classical Mechanics with MATLAB Applications is an essential resource for the advanced undergraduate taking introduction to classical mechanics. Filled with comprehensive examples and thorough descriptions, this text guides students through the complex topics ofrigid body motion, moving coordinate systems, Lagrange's equations, small vibrations, the Euler algorithm, and much more. Step-by-step illustrations, examples and computational physics tools further enhance learning and understanding by demonstrating accessible ways of obtaining mathematical solutions. In addition to the numerous examples throughout, each chapter contains a section of MATLAB code to introduce the topic of programming scripts and theirmodification for the reproduction of graphs and simulations | 677.169 | 1 |
course details
Maths: Pre GCSE
Who is this course for?
This course is for those wishing to progress onto a maths GCSE course and is designed to prepare you for the GCSE. Learn Devon run the GCSE course over nine months so this is an opportunity to consolidate your understanding and ensure you have the key skills required before starting the GCSE in September.
Course content
You will be developing your knowledge and understanding of maths. You will cover the following topics:
1. Integers
2. Decimals
3. Approximation
4. Fractions
5. Percentages
6. Money
7. Time
8. Measures
9. Area and Perimeter
10. Volume
11. Tables and Charts
At the end of the course there will be an exam and if successful you will gain an Edexcel Level 1 award in Number and Measure. Aside from extending your knowledge of maths and gaining a qualification you will also benefit from growing confidence and be familiar with the structure of the GCSE.
Support and feedback
To find out how you are getting on, you will complete an individual learning plan, have one- to-one discussions with your tutor with written feedback and you will also create a portfolio to demonstrate your work including samples and finished pieces: something to be proud of.
Additional costs and requirements
You will need to be assessed by the tutor prior to joining the course to ensure that the course is suitable for your level. The tutor will then be able to explain in further detail the resources you may need for each part of the course. If you are interested in enrolling on this course please call 01822 613701 to make an appointment.
Will I have to do anything outside of the session?
As with the GCSE course you will be expected to complete homework and practice the techniques learned each week.
About the tutor
John Austen works as an Area Tutor (Maths) for Learn Devon and is based in West Devon. Prior to this his posts have included school teacher, literacy/numeracy tutor, management training consultant and Skills for Life numeracy tutor – amounting to over 40 years experience in education.
John has Qualified Teaching Status (QTS) in maths and arts. He also plays guitar in local bands – along with maths music is another passion.
"I like teaching, it is great when you see the learner understand and then look to progress further" | 677.169 | 1 |
An important feature that sets the TI-89 apart from many other graphing calculators is its computer algebra system. A computer algebra system includes not only the ability to calculate numerical expressions and produce graphs, but it can also manipulate symbols and perform exact calculations, which can facilitate discovery learning. With this type of learning, you can experience the thrill that mathematicians enjoy when discovering a theorem.
Lesson Index:
2.1 - Numerical Calculations
2.2 - Variables
2.3 - Pattern Recognition
After completing this module, you should be able to do the following:
Perform exact calculations on the TI-89
Understand the difference between AUTO, EXACT and APPROXIMATE modes
Use the TI-89 catalog
Define and delete variables
Create activities in which your students use computer algebra to discover theorems | 677.169 | 1 |
UNDERGRADUATE COMPUTATIONAL
ENGINEERING AND SCIENCES
An Educational Initiative in Computational Science
OBJECTIVE:
This project promtes the emerging field of computational
science as an interdisciplinary/multidisciplinary approach to scientific
analysis. Currently our emphasis is on collecting, developing and
distributing a set of educational materials in computational science.
EDUCATIONAL MATERIALS:
DESCRIPTION
In order to make these educational materials useful to as wide a range of
people as possible, we believe they should be problem driven, modular in
format, and as interactive as possible. These modular materials will be
logically grouped into courses, with the first planned course at the
introductory level targeted towards freshmen and sophomores. The
computational topics in each "course" will be introduced and developed in the context of specific problems and examples drawn from as wide a variety
of relevant fields as possible, thus stressing the interdisciplinary
nature of computational science. Each example will be in an independent
module that includes most or all of the necessary code, and which can be
executed in an interactive fashion. These interactive modules will be
designed to stress the journey from physical problem to computational
solution (with ongoing feedback for assessment) as illustrated in the
figure.
The course materials will also include descriptions of the various
"guiding themes" of computational science, such as high performance
computing architectures and scientific visualization. These modules will
be more "essay-like" in nature, and are meant to provide breadth and cohesion by intertwining separate examples and showing their
inter-relation.
Working in concert with the authors, all materials submitted to the
project will be assembled into modular form, and a "flow chart" of the modules will be created. However, this project is more than an electronic library where software is collected and distributed; the materials will be
organized around course outlines and provide completed course examples
with detailed outlines and associated modules. We recognize that "real
world" courses must differ, so the outlines will serve as a framework on
which similar, but not identical courses can be constructed. To help
tailor a course for specific needs we will provide a large number of
problem driven software modules drawn from many disciplines. Our paradigm
is that of an electronic textbook in which the instructor fills out
chapters by selecting appropriate modules from the database.
As the project proceeds we expect to offer additional modules at the
advanced undergraduate level, as well as special materials for both
teachers and advanced high school students. It is our hope that the
various modules will be contributed and utilized by scientists and
educators drawn nationwide from the broad range of disciplines. In
addition, the means we pursue in promoting this nascent field are not
rigidly confined to providing electronic educational materials, and we
welcome additional suggestions on advancing the general subject of
computational science.
AVAILABILITY:
The materials will be available to students and educators at all levels,
and will be distributed via the internet. The contributed materials will
be assembled at the Ames Laboratory-USDOE. At present the project has 25+
active members drawn from small colleges, research universities, and
government laboratories throughout the country.
As much as possible the archived materials will be platform independent
and where that is not possible, support will be provided for a variety of
platforms and software packages. To help meet this goal we have developed
an X-windows "teaching tool". The teaching tool provides an interactive tutorial environment well suited for the problem driven nature of our materials. Other interactive environments, such as Mathematica notebooks,
will also be supported.
SUBMISSION REQUEST:
We actively seek people interested in innovative ways of introducing
topics in computational science into the undergraduate curriculum. Those
interested in contributing additional materials or in using contributed
materials for teaching elements of computational science, are encouraged
to contact us and become involved in the project.
We encourage interested parties to submit materials that might further
these goals. We are particularly interested in assembling as wide a range
of "example" modules as possible, and actively seek problems drawn from engineering and the physical and life sciences.
For More Information. Specific information about this program can be
obtained from | 677.169 | 1 |
AS/SC/AK/ MATH 4250 6.0
Differential Geometry
Curves and surfaces
in 3-space, tangent vectors, normal vectors, curvature,
introduction to topology, manifolds, tangent spaces, multilinear
algebra and tensors. Normally offered in alternate years.
Differential geometry uses the methods of multivariable calculus and
linear algebra to study curves, surfaces, and higher-dimensional
"manifolds". The subject was initiated by Gauss, further developed by
Riemann, and has seen important advances in this century due to Cartan
and Lie. The concept of a manifold and the geometric structures
associated to it play central roles in such fields as dynamical
systems, topology, harmonic analysis and differential equations. Many
basic ideas are involved in differential geometry: how can space be
curved, what is the relationship between local and global information,
why don't you always end up where you started when going around a loop?
Differential geometry is now a central tool in theoretical physics;
this century has seen the "geometrization of physics" with significant
applications to the fields of general relativity , classical mechanics
and elementary particle theory.
The first part of the course will study surfaces, both in three- and
higher-dimensional spaces, using as its main tool vector fields on these
surfaces. We will treat geodesics, parallel transport and curvature. We
then discuss differential forms and a general form of Stokes's Theorem.
This will lead to a significant and beautiful result, the Gauss-Bonnet
Theorem, relating local geometric to global topological information. It
is a prototype of many important modern results. The later part of the
course will be an introduction to differentiable manifolds.
The course will be accessible to students with a background in
vector calculus and linear algebra. Some knowledge of differential
equations will also be useful. In fact, the course may be seen as
putting the finishing touches on these courses, unifying them and
showing their significant applications.
The grade will be based on term tests and assignments (70%) and a final
project (30%). The text has not been chosen. | 677.169 | 1 |
Book Description:
Instead of presenting the standard theoretical treatments that underlie the various numerical methods used by scientists and engineers, Using R for Numerical Analysis in Science and Engineering shows how to use R and its add-on packages to obtain numerical solutions to the complex mathematical problems commonly faced by scientists and engineers. This practical guide to the capabilities of R demonstrates Monte Carlo, stochastic, deterministic, and other numerical methods through an abundance of worked examples and code, covering the solution of systems of linear algebraic equations and nonlinear equations as well as ordinary differential equations and partial differential equations. It not only shows how to use Rs powerful graphic tools to construct the types of plots most useful in scientific and engineering work, but also: Explains how to statistically analyze and fit data to linear and nonlinear models Explores numerical differentiation, integration, and optimization Describes how to find eigenvalues and eigenfunctions Discusses interpolation and curve fitting Considers the analysis of time series Using R for Numerical Analysis in Science and Engineering provides a solid introduction to the most useful numerical methods for scientific and engineering data analysis using R.
Download Link:
Related Books:
Jun 30 2016 The book presents a representative selection of all publications published between 01/2009 and 06/2010 in various books, journals and conference proceedings by the researchers of the institute cluster: IMA - Institute of Information Management in Mechanical Engineering ZLW - Center for Learning and Knowledge Management IfU - Institute for Management Cybernetics, Faculty of Mechanical Engineering, RWTH Aachen University The contributions address the cluster's five core research fields: suitable processes for knowledge- and technology-intensive organizations, next-generation teaching and learnin...
Jun 30 2016 A muchneeded guide on how to use numerical methods to solve practical engineering problemsBridging the gap between mathematics and engineering, Numerical Analysis with Applications in Mechanics and Engineering arms readers with powerful tools for solving realworld problems in mechanics, physics, and civil and mechanical engineering. Unlike most books on numerical analysis, this outstanding work links theory and application, explains the mathematics in simple engineering terms, and clearly demonstrates how to use numerical methods to obtain solutions and interpret results.Each chapter is devote...
Jun 30 2016 State-of-the-art nanostructuring principles, methods, and aplicationsSynthesize, characterize, and deploy highly miniaturized components using the theories and techniques contained in this comprehensive resource. Written by a nanotechnology expert, this authoritative volume covers the latest advances along with detailed schematics and real-world applications in engineering and the life sciences. Inside, 37 different nanostructuring methods and 16 different kinds of nanostructures are discussed.Nanostructuring Operations in Nanoscale Science and Engineering explains how to manufacture high-puri...
Jun 30 2016 Advanced Java for Engineers and Scientists gives the reader all the information needed to use Java to create powerful, versatile, and flexible scientific and engineering applications. The book is full of practical example problems and valuable tips. Grant Palmer, a research scientist himself, goes in-depth into advanced technical programming concepts applicable to scientific-oriented applications, such as solving differential equations, data modeling, integration of functions, and creating generic class libraries. The last section of the book shows readers how Java can be used to develop GUI o... | 677.169 | 1 |
This course presents and develops all the necessary topics for an algebra course. It is written for average
and above average students who would like a solid preparation for high school mathematics. Algebra 1
is the basis for any other mathematics course you will take. Without the skills established in Algebra 1,
you will not be able to excel this year or beyond.
Objectives:
The objectives of this class are 1) to provide a strong foundation of Algebra 1, 2) to help you become a
more independent thinker, 3) to help present the topics in different ways to accommodate and show the
different types of learning styles, 4) to help you develop problem solving techniques and 5) to increase
the student's appreciation of mathematics through seeing a wide range of mathematical backgrounds in
different applicable fields.
Time Table:
1st Nine Weeks2nd Nine Weeks3rd Nine Weeks4th Nine Weeks-
Approach:
The major method of presentation is the lecture. Each topic is presented by showing the theoretical
relationship to the practically solved problem. Homework is utilized to reinforce newly presented ideas
with boardwork used to practice these methods. TI-Nspire is used to visualize the in-class work.
Quizzes and homework are used as the information upon which an effective evaluation is made each
quarter.
Grading:
Each students grade is calculated based on the number of points earned divided by the number of points
possible during that particular grading period.
Scoring:
Tests:
Quizzes:
Absences:
School Policy--It is the responsibility of the student to get and make up missed assignments or quizzes.
For each day absent, you will have one day to make up the assignment for full credit. (i.e. If you are
absent just once, on a b-day, then you have until the end of the next b-day to make up the missing
homework, quiz or test. If you are absent just once, on an a-day, then you have until the end of the next
a-day class to make up the missing homework, quiz or test.). Longer absences will allow for more time
to make up the school work.
Worth 100 points (we will have 3-4 each quarter)
Worth 20-75 points (we will have 3 or 4 each quarter)
-Pop quizzes may occur and will be worth significantly less
Homework: Worth 8 points (we will have around 20 each quarter)
Notebook:
Worth 2 points/section (will be checked occasionally)
Late Homework: If homework is not turned in on assigned date and an absence was not involved, the student has two weeks
from the due date to turn the assignment in for half credit. After two weeks, the student will receive a zero for
the assignment.
Computer:
The student is to have his computer with him for every class. If the student fails to have his computer with him
and does not have a note from the help desk, the student and teacher will work out an alternate plan to complete
the assignment. From the second instance on, the student will receive a zero for the assignment(s).
Class Policies
Late Work
Make-up
Work
Daily Work
Tests/Quizzes
(also see below)
Notebook/
Binder
Tutoring
On Task
Pencil ONLY
Dismissal
Late work may be turned in up to two weeks from the date it
was due. No late work will be accepted after the two week
window has passed. The only exception is in the case of an
extended excused absence.
Upon the student's return from an excused absence, it is his
responsibility to find out what he missed and to complete those
assignments in due time (one extra class per class missed).
Students need to also obtain missed warm-ups and notes from a
classmate or teacher before/after school.
Daily work such as warm-ups, classwork, and homework will
be graded for completeness. Homework will be given on a
daily basis.
Tests will be given at the end of each chapter. Tests dates will
be announced at least one week in advance to give time for
review and questions. Quizzes may be given without prior
notice. If a student misses a test, he is responsible for making it
up on his own time based on how long he was absent.
You will use OneNote to take your notes and to do your
homework, warm ups, etc. It is an excellent study guide for
review, quizzes, and tests. This organizational tool will be
graded
Students may obtain extra help outside of class time before
school any day or after school by making an appointment with
the teacher. NEVER hesitate to ask for help. I will do
everything to help you succeed. Do not wait until the day
before the test to ask for assistance! Mrs. Nay will also be
available throughout the school year.
It is expected that you will give 100% in all tasks. Any work
not related to the assigned task will be confiscated.
To receive credit on paper assignments/tests/quizzes, all work
MUST be in PENCIL unless stated otherwise.
I will dismiss the class at the end of the bell. Lining up at the
door is NEVER permitted. | 677.169 | 1 |
Home works
Schedule (Tentative)
26th August
Introduction: Algorithms, their design and
analysis, the measures of efficiency for algorithms; The algorithms to find
maximum, minimum, maximum and minimum together, the second smallest
element;Ideas of best case,
average case and worst case analysis, ideas on lower bounds;harmonic numbers.
28th August
Introduction: Best Case, worst case and
average case analysis;Linear
search and its average case analysis; Selection sort; Insertion sort and its
analysis;models of computation;growth of functions; efficient
algorithms | 677.169 | 1 |
Too often, computing students? first experiences of university mathematics will be of abstract theoretical concepts that appear irrelevant to their chosen course of study. This book is a concise introduction to the key mathematical ideas that underpin computer science, continually stressing the application of discrete mathematics to computing. It is suitable for students with little or no knowledge of mathematics, and covers the key concepts in a simple and straightforward way. The theoretical ideas are constantly reinforced by worked examples and each chapter concludes with a mini case study showing a particular application. This provides further motivation to the reader to engage with the mathematical ideas involved, as well as demonstrating how the mathematics can be applied within a computing context. The book also contains carefully selected exercises for which full worked solutions are provided Rod Haggarty is Deputy Head of the School of Computing and Mathematical Sciences at Oxford Brookes University. He has extensive experience of teaching undergraduate mathematics, and has taught discrete mathematics to computing students for many years | 677.169 | 1 |
Saxon Algebra 1 3rd Ed. Home Study Kit Homeschool
I'm pleased to offer you this outstanding homeschooling curriculum at this fantastic price! This kit is the 3rd edition brand new in the shrink wrap.
We sell a wide variety of homeschool and christian books, so if you don't see what you are looking for in our listings please contact us. Chances are we sell it.
Product Description: Saxon math programs produce confident students who are not only able to correctly compute, but also to apply concepts to new situations. These materials gently develop concepts, and the practice of those concepts is extended over a considerable period of time. This is called "incremental development and continual review." Material is introduced in easily understandable pieces (increments), allowing students to grasp one facet of a concept before the next one is introduced. Both facets are then practiced together until another one is introduced. This feature is combined with continual review in every lesson throughout the year. Topics are never dropped but are increased in complexity and practiced every day, providing the time required for concepts to become totally familiar. Algebra 1, third edition is made up of five instructional components: Introduction of the New Increment, Examples with Complete Solutions, Practice of the Increment, Daily Problem Set, and Cumulative Tests. Algebra 1 covers topics typically treated in a first-year algebra course. This set contains a student text, answer key, and test forms. A solutions manual is available separately. Grade 9. | 677.169 | 1 |
Algebra - Function Notation Worksheet
Be sure that you have an application to open
this file type before downloading and/or purchasing.
80 KB|2 pages
Share
Product Description
This practice worksheet covers basic function notation and linear functions and would make a great practice assignment in an Algebra I class.
Specifically, students will:
-Use function notation to find inputs or outputs.
-Describe a common error made using function notation.
-Find function values using a graph.
-Graph a linear function given a couple function values. | 677.169 | 1 |
Our secure web pages are hosted by Chrislands Inc, who use a Thawte SSL Certificate to ensure secure transmission of your information.
Fisher, George, Howard, Peter Listings
If you cannot find what you want on this page, then please use our search feature to search all our listings.
Click on Title to view full description
1
Excellence in Maths for Secondary Students 2 Fisher, George, Howard, Peter Australia Phoenix Education 1875695419 / 9781875695416 Years 9-10.First in a series of six books designed to help primary school children improve their mathematical awareness and ability. Provides practice in basic number work, exercises in problem solving, and enrichment activities with puzzles, riddles and problems. (ISBN: 9781875695416 - When referring to this item please quote our stock ID: 878) Price:
14.95 AUD
Maths Handbook A Reference for Secondary students Fisher, George Australia Phoenix Education Australia 31/Dec/1997 1875695877 / 9781875695874 Revised ed. Years 7-10 The Maths Handbook has all the essential mathematical facts required by junior and middle secondary students in one easy-to-use book. This second edition contains a number of improvements, including additional entries, corrections to the first edition, and expanded explanations.
How many times in an average school week does a maths student need to • look up a forgotten fact? • check a formula, theorem or definition? • revise a topic?
The Maths Handbook is for students in junior and middle secondary years, and anyone else using maths at this level. It allows the user to quickly find, and understand, important information such as: • essential facts • definitions and descriptions • theorems • formulas • procedures and methods • terminology
These features make the book easy-to-use: • all the information from several year levels is in one book: students don't have to search through a number of textbooks • organised into topic areas so that relevant information can be easily located • explanations are clear yet concise • concepts demonstrated by worked examples (ISBN: 9781875695874 - When referring to this item please quote our stock ID: 567) Price:
9.95 AUD | 677.169 | 1 |
Lesson Closing Narrative.docx - Section 3: Closing
Compound Interest Formula Exit Ticket.docx
Lesson Closing Narrative.docx
Compound Interest Formula
Compound Interest Formula
Unit 8: Exponential Functions
Lesson 23 of 26
Objective: Students will be able to develop a formula to calculate compound interest. Students will be able to explain how they know this formula is exponential and how to find each part of the formula using the given information. | 677.169 | 1 |
Upper School Scientific Calculus
Calculus methods are used as the introductory mathematical tools for discussing and designing scientific experiments. Students are tasked with designing, building, implementing, and presenting experiments in small groups. Varying statistical methods of error analysis are used at the conclusion of each experiment. Students are required to assess each other's' work as well as their own and are expected to give and receive critical feedback from peers. An example of a unit taught in the course includes Air Resistance, the unit culminated in modeling the two effects (air resistance and Magnus Effect) that allow MLB pitchers to create the motion on different pitches. This required learning an advanced method of approximation called Runge-Kutta 4. Students also shot baseballs out of a homemade air cannon to see some of these effects in action | 677.169 | 1 |
John Vince describes a number mathematical issues to supply a starting place for an undergraduate direction in machine technological know-how, beginning with a overview of quantity platforms and their relevance to electronic desktops, and completing with differential and indispensable calculus. Readers will locate that the author's visual approach will significantly increase their realizing as to why sure mathematical buildings exist, including how they're utilized in real-world applications.
Each bankruptcy comprises full-colour illustrations to elucidate the mathematical descriptions, and sometimes, equations also are colored to bare very important algebraic styles. the varied labored examples will consolidate comprehension of summary mathematical concepts.
Foundation arithmetic for desktop technology covers quantity structures, algebra, good judgment, trigonometry, coordinate platforms, determinants, vectors, matrices, geometric matrix transforms, differential and imperative calculus, and divulges the names of the mathematicians at the back of such innovations. in this trip, John Vince touches upon extra esoteric issues equivalent to quaternions, octonions, Grassmann algebra, Barycentric coordinates, transfinite units and major numbers. even if you need to pursue a profession in programming, medical visualisation, platforms layout, or real-time computing, you want to locate the author's literary type refreshingly lucid and interesting, and get ready you for extra complicated texts.
A completely modern method of educating crucial technical images talents has made Bertoline and Wiebe's basics of images communique the top textbook in introductory engineering photos courses. The 5th variation keeps to combine layout options and using CAD into its notable insurance of the fundamental visualization and sketching options that allow scholars to create and speak easy and simply understood.
This ebook is designed to explain basic algorithmic concepts for developing drawings of graphs. compatible as a publication or reference guide, its chapters provide an exact, obtainable mirrored image of the swiftly increasing box of graph drawing.
Interactive special effects with WebGL, 7th variation , is appropriate for undergraduate scholars in machine technology and engineering, for college kids in different disciplines who've stable programming abilities, and for pros drawn to machine animation and pix utilizing the newest model of WebGL. | 677.169 | 1 |
98102440Nevanlinna Theory and Its Relation to Diophantine Approximation
It was discovered recently that Nevanlinna theory and Diophantine approximation bear striking similarities and connections. This book provides an introduction to both Nevanlinna theory and Diophantine approximation, with emphasis on the analogy between these two subjects.Each chapter is divided into part A and part B. Part A deals with Nevanlinna theory and part B covers Diophantine approximation. At the end of each chapter, a table is provided to indicate the correspondence of theorems. | 677.169 | 1 |
Download and read online Intermediate Algebra in PDF and EPUB Larson IS student success. INTERMEDIATE ALGEBRA owes its success to the hallmark features for which the Larson team is known: learning by example, a straightforward and accessible writing style, emphasis on visualization through the use of graphs to reinforce algebraic and numeric solutions and to interpret data, and comprehensive exercise sets. These pedagogical features are carefully coordinated to ensure that students are better able to make connections between mathematical concepts and understand the content. With a bright, appealing design, the new Fifth Edition builds on the Larson tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. The text also continues coverage and integration of geometry in examples and exercises. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Download and read online Intermediate Algebra in PDF and EPUB Intended for developmental math courses in intermediate algebra Intermediate Algebra An Applied Approach in PDF and EPUB As in previous editions, the focus in INTERMEDIATE ALGEBRA remains on the Aufmann Interactive Method (AIM). Students are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. Student engagement is crucial to success. Presenting students with worked examples, and then providing them with the opportunity to immediately solve similar problems, helps them build their confidence and eventually master the concepts. Simplicity is key in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully constructed hierarchy of objectives. Each exercise mirrors a preceding objective, which helps to reinforce key concepts and promote skill building. This clear, objective-based approach allows students to organize their thoughts around the content, and supports instructors as they work to design syllabi, lesson plans, and other administrative documents. New features like Focus on Success, Apply the Concept, and Concept Check add an increased emphasis on study skills and conceptual understanding to strengthen the foundation of student success. The Ninth Edition also features a new design, enhancing the Aufmann Interactive Method and making the pages easier for both students and instructors to follow. Available with InfoTrac Student Collections Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Download and read online Intermediate Algebra Connecting Concepts through Applications in PDF and EPUB INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problem solving, and communication skills. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate solutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students. First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application. Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS takes an application-driven approach to algebra, using appropriate calculator Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Download and read online Intermediate Algebra Student Support Edition in PDF and EPUB Designed for first-year developmental math students who need support in intermediate algebra, the Fourth Edition of The purpose of this book is to provide students with an algebra text that they can read and understand. Explanations are written carefully in language that is familiar to the general population as well as to those students for whom English is a second language. Real-world applications of algebra has a practical importance to them.
Download and read online Intermediate Algebra in PDF and EPUB The main focus of INTERMEDIATE ALGEBRA, 5e, is to address the fundamental needs of today's developmental math students. Offering a uniquely modern, balanced program, essential in making these connections and it is emphasized in INTERMEDIATE ALGEBRA, 5e, with additional practice problems both in the text and Enhanced WebAssign. Give your students confidence by showing them how Algebra is not just about the x -- it's also about the WHY. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Download and read online Beginning and Intermediate Algebra An Integrated Approach in PDF and EPUB Get the grade you want in algebra with Gustafson and Frisk's BEGINNING AND INTERMEDIATE ALGEBRA! Written with you in mind, the authors provide clear, no-nonsense explanations that will help you learn difficult concepts with ease. Prepare for exams with numerous resources located online and throughout the text such as online tutoring, Chapter Summaries, Self-Checks, Getting Ready exercises, and Vocabulary and Concept problems. Use this text, and you'll learn solid mathematical skills that will help you both in future mathematical courses and in real life! Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Download and read online Elementary and Intermediate Algebra A Combined Course Student Support Edition in PDF and EPUB Developed to prepare students in the combined elementary and intermediate algebra course for a college-level curriculum, Elementary and Elementary and Intermediate Algebra in PDF and EPUB The main focus of ELEMENTARY AND INTERMEDIATE ALGEBRA, 5e, is to address the fundamental needs of today's developmental math students. Offering a uniquely modern, balanced program, ELEMENTARY AND essential in making these connections and it is emphasized in ELEMENTARY AND INTERMEDIATE ALGEBRA, 5e, with additional practice problems both in the text and Enhanced WebAssign. Give your students confidence by showing them how Algebra is not just about the x it's also about the WHY. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Download and read online Elementary and Intermediate Algebra A Combined Approach in PDF and EPUB Master the fundamentals of algebra with Kaufmann and Schwitters' ELEMENTARY AND INTERMEDIATE ALGEBRA: A COMBINED APPROACH, Sixth Edition. Learn from clear and concise explanations, many examples, and numerous problem sets in an easy-to-read format. The book's Learn, Use and Apply formula helps you learn a skill, use the skill to solve equations, and then apply it to solve application problems. This simple, straightforward approach helps you understand and apply the key problem-solving skills necessary for success in algebra and beyond. Access to Enhanced WebAssign and the Cengage YouBook is sold separately. To learn more and find value bundles, visit: and search for ISBN: 0840053142 | 677.169 | 1 |
You do not have to be a genius to develop into an algebra ace-you can do it in precisely quarter-hour an afternoon choked with brief and snappy classes, Junior ability developers: Algebra in quarter-hour an afternoon makes studying algebra effortless. it truly is precise: making feel of algebra does not need to take decades . . . and it does not need to be tricky! in precisely one month, scholars can achieve services and simplicity in all of the algebra thoughts that frequently stump scholars. How? each one lesson supplies one small a part of the larger algebra challenge, in order that each day scholars construct upon what was once discovered the day prior to. enjoyable factoids, catchy reminiscence hooks, and worthwhile shortcuts ensure that each one algebra inspiration turns into ingrained. With Junior ability developers: Algebra in quarter-hour an afternoon, sooner than you recognize it, a suffering pupil turns into an algebra pro-one step at a time. in exactly quarter-hour an afternoon, scholars grasp either pre-algebra and algebra, together with: Fractions, multiplication, department, and different simple math Translating phrases into variable expressions Linear equations actual numbers Numerical coefficients Inequalities and absolute values structures of linear equations Powers, exponents, and polynomials Quadratic equations and factoring Rational numbers and proportions and lots more and plenty extra! in precisely quarter-hour an afternoon, scholars grasp either pre-algebra and algebra, together with: Fractions, multiplication, department, and different simple math Translating phrases into variable expressions Linear equations genuine numbers Numerical coefficients Inequalities and absolute values platforms of linear equations Powers, exponents, and polynomials Quadratic equations and factoring Rational numbers and proportions and lots more and plenty extra! as well as the entire crucial perform that youngsters have to ace school room exams, pop quizzes, type participation, and standardized assessments, Junior ability developers: Algebra in quarter-hour an afternoon offers mom and dad with a simple and available strategy to support their young ones exce
Globalizing pursuits is an cutting edge examine of globalization "from inside," taking a look at the response of nationally constituted curiosity teams to demanding situations produced by means of the denationalization technique. The members concentrate on enterprise institutions, exchange unions, civil rights enterprises, and right-wing populists from Canada, Germany, nice Britain, and the U.S., and look at how they've got replied to 3 tremendous globalized factor parts: the web, migration, and weather switch.
Qxd:JSB 12/18/08 11:44 AM S E Page 21 C T I O N 1 algebra basics BEFORE WE CAN use algebra, we need to understand what it is. This section begins by explaining the vocabulary of algebra, so that when you see x in a problem, you will know what it is and why it's there. Once these definitions are out of the way, we will review how to perform basic operations (addition, subtraction, multiplication, and division) on real numbers, and then show how these same operations can be performed on algebraic quantities.
2. Parentheses are first in the order of operations. 10 + 3 = 13, and the expression becomes –2(13). Multiply: –2(13) = –26. 3. This expression contains an exponent, multiplication, and addition. Exponents come before multiplication and addition, so begin with 32: 32 = 9. The expression is now 9(7) + 1. Multiplication comes before addition, so multiply next: 9(7) = 63, and the expression becomes 63 + 1. Finally, add: 63 + 1 = 64. 4. There are two sets of parentheses in this expression, so work on each of them separately. | 677.169 | 1 |
For more information contact
Mrs S Williams
September 2017 is the first teaching of the new AS level syllabus – the content is the same as the old syllabus however the style of questions is going to change to include problem solving questions and some questions will both pure and applied topics. There is also a greater emphasis on use of technology in maths – students will be required to use spreadsheets and graphical calculators.
AS Course Content :
Pure
Problem Solving
Algebra Trigonometry
Co-ordinate Geometry
Calculus
Logs and exponentials
Vectors
Mechanics
Kinematics
Forces & Newton's Laws
Variable Acceleration
Statistics
Collect & Interpret Data
Probability
Binomial Distribution
Statistical Hypothesis Testing
Mechanics is useful for studying Physics & Chemistry in Post 16 and University courses such as Engineering & Architecture
Statistics is useful for students studying Biology & Geography in Post 16 & University courses such as Business ICT/ Computing & Marketing
Structure of A Level Mathematics
Papers 1 & 2
Section A Consists of shorter questions with minimal reading and interpretation – in these questions it should be fairly obvious to students what they are required to do; the aim of this is to ensure that all students feel as though they can do some of the questions on the paper.
Section B includes longer questions and more problem solving. Section B has a gradient of difficulty.
Mechanics/Statistics questions can be in either section A or B
Section A will have 20 to 25 marks with the remainder of the marks with the remainder of the marks (75 to 80) allocated to Section B
Overarching themes
A Level specifications in mathematics must require students to demonstrate the following overarching knowledge and skills. These must be applied, along with associated mathematical thinking and understanding, across the whole of the detailed content.
Mathematical argument, language and proof.
Mathematical problem solving, using a problem solving cycle.
Mathematical modelling.
Technology
The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A Level mathematics.
Calculators used must include the following features:
an iterative function
the ability to compute summary statistics and access probabilities from standard statistical distributions. | 677.169 | 1 |
Now in its 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining the key features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises.
Revised and updated version of the leading text in mathematical physicsa thorough handbook about mathematics that is useful in physics."--MAA.org, Mathematical Methods for Physicists, 7th Edition
"This volume is a great collection of essential mathematical tools and techniques used to solve problems in physics, very useful to any student of physics or research professional in the field. It is concentrated to problem-solving art and offers a large amount of problems and exercises."--Zentralblatt MATH 1239
From the Back CoverNew chapter on Green's functions
Extended discussion of partial differential equations
Expanded coverage of mathematical statistics
Thorough treatment of contour integration
More systematic treatment of orthogonal polynomials
|There was a problem filtering reviews right now. Please try again later.
Graduate student in physics. Studying Green's functions is slippery. But here is a book that will allow you to trace far enough down the rabbit hole to the very roots of a subject. There are enough examples for sure, and as often as possible the authors discuss information in the context of physical phenomena. Relations of Green's functions to electrostatics is made, and a brief discussion is done on the role they play in the scattering problems of quantum mechanics ie the Born Approximation. Must have. BUY IT!
An indispensable companion to any soul brave enough to venture into the world of graduate-level physics. Which is not to say it is necessarily best at everything. The very inexpensive book by Bryson and Fuller is a very hand adjunct.
7th edition of a classic, with, in the words of the authors, "a substantial and detailed revision".of the 6th edition. Specially improved by the new chapters on vector spaces, Green's functions and angular momentum, and, last but not least, the inclusion of the dilogarithm special function. The on-line material at [...] supplements the printed text. A basic technical reference for all future and present scientists and teachers. | 677.169 | 1 |
Product Description
▼▲
The Developmental Mathematics workbook series covers basic mathematics through early algebra. Workbooks aren't grade-specific, but rather focus on individual skills, making it an ideal curriculum for self-paced learners at any ability level. This is a self-teaching curriculum that was specifically designed for students to read, learn, and complete themselves, cultivating independent learning skills.
Lessons begin with an explanation of the concept and example problems that are solved step-by-step. A number of practice problems are provided on the following "Applications" pages.
Level 6 covers tens and ones, including addition and subtraction with grouping in horizontal and vertical form. | 677.169 | 1 |
When I was a student, I referred to this book as "Abstract Algebra for third-graders." That was because it is so clearly written that I figured anyone could pick it up and read it. (Admittedly maybe that should have been third year undergrads.) I used it often to fill in the blanks for topics covered briefly in other books. The authors make a point of giving complete proofs without "clearly..." or "proof left to the reader." This is the first book I reach for when I need to remember something from Abstract Algebra. ( )
Wikipedia in English
This is a self-contained text on abstract algebra for senior undergraduate and senior graduate students, which gives complete and comprehensive coverage of the topics usually taught at this level. The book is divided into five parts. The first part contains fundamental information such as an informal introduction to sets, number systems, matrices, and determinants. The second part deals with groups. The third part treats rings and modules. The fourth part is concerned with field theory. Much of the material in parts II, III, and IV forms the core syllabus of a course in abstract algebra. The fifth part goes on to treat some additional topics not usually taught at the undergraduate level, such as the Wedderburn-Artin theorem for semisimple artinian rings, Noether-Lasker theorem, the Smith-Normal form over a PID, finitely generated modules over a PID and their applications to rational and Jordan canonical forms and the tensor products of modules. Throughout, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises. In addition, the book contains many examples fully worked out and a variety of problems for practice and challenge. Solution to the odd-numbered problems are provided at the end of the book to encourage the student in problem solving. This new edition contains an introduction to categories and functors, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Noether-Lasker theorem. In addition, there are over 150 new problems and examples. | 677.169 | 1 |
Q - how you mcgraw hill 7th grade math textbook answers be administrator of DeviantArt what you would get Q - Frist your game. A - Bowser the Warrior Q - Where your Still Fandoms. Each process has three independent interval timers available: A real-time timer that counts elapsed time. This timer sends a SIGALRM signal to the process when marh expires. A virtual timer that counts processor holiday graph art christmas used by the process. This timer sends a SIGVTALRM signal to the process when it expires.
The benefits of being Becoming an independent contractor can give you better working conditions, more textbolk hours and aanswers higher income. But there are disadvantages, too. Xero Xero A different way of working Businesses around the world are moving to more flexible employment methods in order to cut costs. Contractors are people who work on a contract basis, not as regular employees. Each contract might span a few months, a few weeks mcgraw hill 7th grade math textbook answers even a one-off piece of work.
For mcgraa this would be ok by me: To provide vrade context. I have a drop down box that shows info very similar to this. Right now my code to create the item in the drop down looks like this: The HtmlDecode changes the to a space that can withstand the space removing formatting of the dropdown list. Seriously, we do. If you find yourself opening more stories than you can possibly read in one sitting, Pocket is definitely worth checking out. Not a big deal, but mcgraw hill 7th grade math textbook answers mentioning as it felt like a bit of a bait-and-switch to me. | 677.169 | 1 |
Infinite Geometric Series Guided Notes
Be sure that you have an application to open
this file type before downloading and/or purchasing.
168 KB|4 pages
Share
Product Description
These attractive and organized guided notes begin with an example and the formula for finding the sum of an infinite geometric series. Then, students look at how the formula is developed and use it to solve problems. The second page of the notes teaches how to write recurring decimals as fractions by writing the decimals as the sum of an infinite series. Very fun IB Math topic! Created by Math with Mrs. Holst. | 677.169 | 1 |
Department of Mathematics
Overview In a world of increasing technological complexity,
knowledge of mathematics is the gateway to the pursuit of many
fields. Mathematics has long been the language of choice for expressing
complex relationships and describing complicated patterns and
processes. It is now true that many fields, in addition to mathematics
and the sciences, rely on this in a fundamental way.
What was formerly "abstract" mathematics to many has
become the concrete stuff of everyday life. "The unreasonable
effectiveness of mathematics" manifests itself today in such
familiar things as CAT scans, compact discs, satellite communications,
and computer animation. These were all rendered possible by new
discoveries made by mathematicians, within the last fifty years.
Even the efficient operation of our financial markets is based,
in part, on relatively recent theorems of mathematical analysis
and probability theory.
Mathematics research today is a vibrant and dynamic enterprise.
Thousands of mathematicians world-wide are at work on an unimaginably
broad range of questions. Exciting recent advances include the
solution of Fermat's Last Theorem, the classification of the finite
simple groups, the solution of the Bieberbach conjecture, and
the (controversial) proof of the four-color theorem.
The discipline and creativity required by the study of mathematics
can be a formidable preparation for later life. Past students
of mathematics have had successful careers in almost every sphere,
including all the professions. (In fact, presidents
of Peru, Ireland and France have been mathematicians.)
The scope of mathematics courses offered at the University of
Virginia allows each of its majors to tailor a program of study
to meet their needs. Students electing to major in mathematics
should consult carefully with a faculty advisor to ensure the
selection of a program of courses which provides a solid grounding
in the fundamentals of higher mathematics and is appropriate to
future goals.
Faculty The faculty of the Department of Mathematics is
committed to excellence in teaching and research. Its members
have been widely published in prestigious research jour-nals and
are internationally recognized scholars. The faculty have held
Sloan fellowships, Humboldt fellowships, and other scholarly honors,
as well as, numerous research grants. Many are currently supported
by grants from the National Science Foundation and other federal
agencies. Most have held visiting professorships abroad. In addition,
the department offerings and ambiance are enhanced each year by
the presence of several internationally recognized visiting faculty.
Students There are currently about 75 students majoring
in mathematics. Class sizes vary from a few large introductory
classes to an average class size of twenty students for upper
level courses. This small class size affords students the opportunity
to get individual attention.
Students who graduate with degrees in mathematics successfully
pursue a variety of different careers. Many go directly into jobs
in industry, insurance (as actuaries), government, finance, and
other fields. Employers in the past have included Morgan Stanley,
General Motors, MITRE Corp., the Census Bureau, the National Security
Agency, and various consulting firms. Many find themselves well-equipped
to go on to professional schools in law, medicine, and business.
Some go directly into teaching. Others have gone on to graduate
programs in mathematics, applied mathematics, statistics, engineering,
systems engineering, economics, and computer science. Students
who have combined the mathematics major with courses in computer
programming, economics, and business have done exceptionally well
in the job market.
Requirements for Major Normally, the calculus sequence
MATH 131, 132, and 221 or its equivalent must be completed before a student can declare a major in
mathematics. At least a 2.2 average in the calculus sequence and
a minimum grade of C in MATH 221 or its equivalent are required.
However, the department may grant special permission to declare
a major to a student who has only completed MATH 131 and 132,
and at least one mathematics course (other than MATH 221 or its
equivalent) which could be counted towards the major in mathematics,
provided the student completes MATH 221 or its equivalent in the
semester following the declaration of a mathematics major.
To graduate with a major in mathematics the student must show
computer proficiency by completing CS120, CS101, or CS182, or
an approved equivalent course. This should be done as early as
possible.
To help guide the student through the major, the mathematics department
offers five options. Completion of one of these options is required.
Each option contains a set of nine required courses (approximately
28 credit hours). To graduate, a student must obtain minimum grades
of C in seven of these courses and C- in the other two.
Certain substitutions are allowed in all options, for example,
MATH 531 for MATH 331 and MATH 551 for MATH 354.
A. The basic option
This is a traditional program for the mathematics major which
provides an overview of key areas of mathematics:
Students fulfilling the requirements for this option will have
a wide range of career opportunities, from law to business to
any field which requires deductive, logical reasoning skills.
B. The graduate preparatory option
This option is for the student who plans to attend graduate school
in mathematics or an allied field. The program emphasizes the
fundamental ideas of mathematics with substantial work in proving
and understanding the basic theorems. It consists of:
Three electives at the 300 level or higher (Students may wish
to take MATH 354 and/or MATH 331 in preparation for MATH 551 and/or
MATH 531.)
This constitutes a minimum expected of an incoming graduate student
in most programs nationwide. We strongly recommend MATH 532 (Real
Analysis in Several Variables), as well as courses in differential
geometry and topology (MATH 572 and MATH 577). Many of our graduate
school bound students take additional courses including 700-level
graduate courses.
C. The probability and statistics option
This option is designed to give the student a good theoretical
underpinning in probability and statistics, as well as the opportunity
to go deeper in these fields. The program can lead to a Master
of Science in Statistics with one additional year of course work,
if additional courses in Statistics are taken in the fourth year.
(Those interested in the M.S. in Statistics should contact the
graduate advisor in the Statistics Division prior to the beginning
of their fourth year.) The requirements for the option are the
following:
This program provides the student with a broad background of basic
mathematics which is essential for an understanding of the mathematical
models used in the financial markets. The mathematics of modern
finance includes, but is not limited to, probability, statistics,
regression, time series, partial differential equations, stochastic
processes, stochastic calculus, numerical methods, and analysis.
Probability and statistics and some acquaintance with numerical
methods are essential as is some knowledge of economics/accounting
and some computing experience. Additional background in statistics,
optimization, and stochastic processes is also desirable. The
program consists of:
Other requirements: Two courses chosen from ECON 201, ECON 202,
COMM 201, or COMM 202. It is recommended, however, that the student
complete all four of these courses.
E. Five-year teacher education program
This option leads to both Bachelor of Arts and Master of Teaching
degrees after five years. The program is for both elementary and
secondary teachers; it is administered by the Curry School of
Education. The requirements are:
Distinguished Majors Program The department offers a Distinguished
Majors Program to qualified majors in mathematics. Admission to
the program is granted by the departmental committee for the DMP,
usually at the end of the student's fourth semester. Criteria
for acceptance into the program are based on the GPA in mathematics,
letters of recommendation from mathematics instructors, and the
cumulative GPA in the College (which should be near 3.4 or higher).
The distinguished majors program is the same as the Graduate School
preparatory option, except that in the fourth year the student
will also take the seminar course MATH 583 in which the student
will give an hour lecture and prepare a written exposition of
his or her work in the seminar, under faculty guidance. Note that
MATH 531 and 551 are prerequisites for the seminar). As with the
options, the DMP must consist of at least nine courses.
Three levels of distinction are possible: distinction, high distinction,
or highest distinction. The departmental recommendation for the
level of distinction to be awarded is based on the quality of
the student's seminar presentations, the overall work in the DMP
and the entire major program of the student, as well as the student's
College GPA.
Requirements for Minor in Mathematics Students not majoring
in mathematics but who wish to declare a minor in mathematics
must complete the calculus sequence through MATH 221 or its equivalent
with at least a 2.0 average.
To graduate with a minor in mathematics a student must complete
five courses approved by the department of mathematics with minimum
grades of C in three of these courses and minimum grades of C-
in the other two. An approved course must carry at least three
credits. Currently, the approved courses are those from the College
department of mathematics with the MATH mnemonic numbered 225
or higher. Courses with the STAT mnemonic or from other departments
or institutions can be offered if approved by the undergraduate
committee.
Courses which are being counted for a major or another minor cannot
also be counted for the minor in mathematics.
Requirements for Minor in Statistics Before declaring a minor in statistics, students must complete the calculus sequence through MATH 221 or its equivalent with at least a 2.0 average. To graduate with a minor in statistics, students must complete five courses approved by the Department of Mathematics with minimum grades of C in three of these courses and minimum grades of C- in the other two.
The program must include the following two core courses: MATH 311-312 or their equivalents. The other three courses must be chosen from MATH 511, MATH 531, STAT 512, STAT 513, STAT 514, STAT 516, STAT 517, STAT 518, or STAT 519. Students wishing approval for courses not listed above may file a petition with the undergraduate committee. Courses are ordinarily approved if they compare favorably in content and quality with the listed courses.
Except for MATH 531, courses which are being counted for the mathematics major or minor may not also be counted for the minor in statistics.
Echols Mathematics Club is an undergraduate club for mathematics
students which sponsors lectures, mathematics films, problem solving
sessions for the Putnam Mathematical Competition and other similar
activities.
Courses
Mathematics
The entering College student has a variety of courses in mathematics
from which to choose. Among those which may be counted towards
the College area requirement in natural science and/or mathematics,
there are several options in calculus, elementary (non-calculus
based) courses in probability and in statistics, and courses dealing
with computer techniques in mathematics.
The courses MATH 100 (algebra and trigonometry) and MATH 103 (precalculus)
are available for students who need to improve basic skills that
are required in other courses such as calculus, chemistry, psychology,
economics, and statistics. However, neither MATH 100 nor MATH
103 may be counted toward the area requirement in natural science
and/or mathematics. Courses equivalent to MATH 100 may not be
transferred for College credit.
Students planning to major in the social sciences, arts, or humanities
who wish to take a mathematics course but omit the study of calculus
may choose from elementary probability theory, MATH 111, or elementary
statistics, MATH 112. Even though it is not a prerequisite for
MATH 112, MATH 111 is frequently taken prior to MATH 112. Note
that both MATH 111 and MATH 112 may be counted toward the area
requirement in natural science and/or mathematics.
The study of calculus is the foundation of college mathematics
for students planning to major in mathematics or the physical
sciences or anticipating a career or graduate study in any of
the natural sciences, engineering, or applied social sciences
(such as economics). There are essentially three programs of study
available in the calculus:
1. MATH 121, 122 is a terminal one-year sequence intended for
business and social science majors;
2. MATH 131, 132, 221 is the traditional calculus sequence intended
for students of mathematics and the natural sciences, as well
as for students intending to pursue graduate work in the applied
social sciences;
The MATH 121, 122 sequence is unacceptable as a prerequisite for
mathematics coursed numbered 221 and above. Students anticipating
the need for higher mathematics courses such as MATH 225 (Differential
Equations) or MATH 311, 312 (Probability and Statistics) should
instead elect the MATH 131, 132, 221 sequence. Note that credit
is not allowed for both MATH 121 and MATH 131 (or its equivalent).
Students who begin their calculus with MATH 121 but wish to transfer
into the traditional calculus sequence may follow MATH 121 with
MATH 132A, which is equivalent to MATH 132.
Students with no previous calculus may elect MATH 121, MATH 131.
An alternative to MATH 121 is MATH 121S, which places greater
emphasis on problem solving and giving more individual attention
to the student.
Students who have previously passed a calculus course in high
school may elect MATH 122, 131, 132, or 221 as their first course,
depending upon placement, preparation, and interest. A strong
high school calculus course is generally adequate preparation
for MATH 132 as a first calculus course, even if advanced placement
credit has not been awarded for MATH 131. Students planning to
take any advanced course in mathematics should not take MATH 122,
because credit in MATH 122 must be forfeited if one takes MATH
132 (or its equivalent).
Exceptionally well prepared students (who place out of both MATH
131 and 132) may choose either MATH 221 or MATH 225 (Differential
Equations) as their first course.
Advanced placement credit in the calculus sequence is granted
on the basis of the College Entrance Examination Board Advanced
Placement Test (either AB or BC). A score of 4 or 5 on the AB
test or a score of 3 on the BC test will give the student credit
for MATH 131. A score of 4 or 5 on the BC test will give the student
credit for both MATH 131 and 132. Students who wish to enter the
calculus sequence but who have not received advanced placement
credit should consult the First-Year Handbook for placement guidelines
based on grades and achievement test scores. The Department of
Mathematics offers short advisory placement tests during fall
orientation.
Pre-commerce students are required to take a statistics course,
usually MATH 112, and one other mathematics course, usually MATH
111, 121, 122, or MATH 131.
Warning: There are numerous instances of equivalent courses
offered by the Department of Mathematics as well as by the Department
of Applied Mathematics in the School of Engineering. A student
may not offer for degree credit two equivalent courses, e.g.,
MATH 131 and APMA 101, or MATH 131 and MATH 121.
MATH 108 - (3) (Y) Modes of Mathematical Thinking
Logic, number systems, functions, analytic geometry, equations,
matrices, enumeration, computer algebra systems. Intended for
liberal arts students and emphasizes the connection between analytic-algebraic
and geometric reasoning in the understanding of mathematics. Facilitated
by the use of a modern computer algebra system, such as Maple.
MATH 110 - (3) (Y) Mathematics for Elementary Teachers
Prerequisite or corequisite: EDHS 201
A study of numbers, operations, and their properties; geometric
figures and their properties; and introductory probability and
statistics.
MATH 121 - (4) (S) Applied Calculus I
Limits and continuity. Differentiation and integration of algebraic
and elementary transcendental functions. Applications to maximum-minimum
problems, curve sketching and exponential growth. Credit is not
given for both MATH 121 and MATH 131.
MATH 122 - (3) (Y) Applied Calculus II
Prerequisite: MATH 121 or equivalent
A second calculus course for business, biology, and social science
students. Functions of several variables, their graphs, partial
derivatives and optimization; multiple integrals. Includes a review
of basic single variable calculus (MATH 121 or equivalent) and
an introduction to differential equations and infinite series.
Credit is not given for both MATH 122 and MATH 132.
MATH 131 - (4) (S) Calculus I
Prerequisite: Background in algebra, trigonometry, exponentials,
logarithms, and analytic geometry
Introductory calculus with emphasis on techniques and applications.
Recommended for natural science majors and students planning additional
work in mathematics. The differential and integral calculus for
functions of a single variable is developed through the fundamental
theorem of calculus. Credit is not given for both MATH 121 and
MATH 131.
MATH 132 - (4) (S) Calculus II
Prerequisite: MATH 131 or permission of instructor
Continuation of 131. Applications of the integral, techniques
of integration, infinite series, vectors. Credit is not given
for both MATH 122 and MATH 132.
MATH 132A - (5) (Y) Calculus II
Prerequisite: MATH 121 or permission of instructor
Continuation of MATH 121 for students who wish to cover the material
of MATH 132.
MATH 221 - (4) (S) Calculus III
Prerequisite: MATH 132 or its equivalent
A study of functions of several variables including lines and
planes in space, differentiation of functions of several variables,
maxima and minima, multiple integration, line integrals, and volume.
MATH 225 - (4) (S) Ordinary Differential Equations
Prerequisite: MATH 132 or its equivalent
Introduction to the methods, theory and applications of differential
equations. Topics: first-order, second and higher-order linear
equations, series solutions, linear systems of first-order differential
equations and the associated matrix theory. Additional topics
may include numerical methods, non-linear systems, boundary value
problems, and additional applications. Section 1 of MATH 225 in
the spring covers, in addition to the basic topics, an introduction
to Sturm-Liouville theory, Fourier series and boundary value problems
and their connection with partial differential equations. Physics
majors should enroll in section one. No knowledge of physics is
assumed in MATH 225, section 1.
MATH 300 - (3) (Y) Foundations of Analysis
Prerequisite: MATH 132 or equivalent
Selection of topics from logic and the construction of mathematical
proofs, basic set theory, number systems, continuity of functions
and foundations of analysis. Introduces at an intermediate level
the standards of mathematical rigor and abstraction that are encountered
in advanced mathematics, based on the material of the calculus
and other basic mathematics.
MATH 331 - (3) (S) Basic Real Analysis
Prerequisite: MATH 132
Concentrates on proving the basic theorems of calculus, with due
attention to the beginner with little or no experience in the
techniques of proof. Topics include limits, continuity, differentiability,
the Bolzano-Weierstrass theorem, Taylor's theorem, integrability
of continuous functions, and uniform convergence.
MATH 354 - (3) (Y) Survey of Algebra
Prerequisite: MATH 132 or equivalent
An introductory survey of the major topics of modern algebra:
groups, rings, and fields. Applications to other areas of mathematics,
such as geometry and number theory are presented. The rational,
real, and complex number systems are developed, and the algebra
of polynomials explored.
MATH 355 - (3) (IR) Algebraic Automata Theory
Prerequisite: MATH 351
An introduction to the theory of sequential machines, including
an introduction to the theory of finite permutation groups and
transformation semigroups. Examples from biological and electronic
systems as well as computer science. The Krohn-Rhodes decomposition
of a state machine. Mealy machines.
MATH 452 - (3) (IR) Algebraic Coding Theory
Prerequisite: MATH 351 and MATH 354 or permission of instructor
An introduction to the use of algebraic techniques for communicating
information in the presence of noise. Topics include the basic
concepts of coding theory: linear codes, bounds for codes, BCH
codes and their decoding algorithms. Other topics may include
quadratic residue codes, Reed-Muller codes, algebraic geometry
codes, and connections with groups, designs, and lattices.
MATH 493 - (3) (IR) Independent Study
Reading and study programs in areas of interest to the individual
student. This course is primarily for juniors and seniors who
have developed an interest in a branch of mathematics not covered
in a regular course. It is the responsibility of the student to
obtain a faculty advisor to approve and direct the program.
MATH 501 - (3) (Y)
Prerequisite: MATH 221 and MATH 351 or permission of instructor
Evolution of the various mathematical ideas leading up to the
development of the calculus in the seventeenth century, and how
those ideas were perfected and extended by succeeding generations
of mathematicians. Special emphasis placed, wherever possible,
on primary source materials.
MATH 503 - (3) (Y) The History of Mathematics
Prerequisite: MATH 221 and MATH 351 or permission of instructor
The development of mathematics from classical antiquity through
the end of the nineteenth century, focusing on the critical periods
in the evolution of such areas as geometry, number theory, algebra,
probability and set theory. Special emphasis placed, wherever
possible, on primary source materials.
MATH 509 - (3) (Y) Mathematical Probability
Prerequisite: Three semesters of calculus, and graduate standing.
Students who have received credit for MATH 311 may not take MATH
509 for credit.
The development and analysis of probability models through the
basic concepts of sample spaces, random variables, probability
distributions, expectations, and conditional probability. Additional
topics covered include distributions of transformed variables,
moment generating functions, and the central limit theorem.
MATH 510 - (3) (Y) Mathematical Statistics
Prerequisite: MATH 509 and graduate standing. Students who have
received credit for MATH 312 may not take MATH 510 for credit.
Methods of estimation, general concepts of hypothesis testing,
linear models and estimation by least squares, categorical data,
nonparametric statistics.
MATH 596 - (3) (S) Supervised Study in Mathematics
Prerequisite: Permission of instructor and graduate standing
In exceptional circumstances, a student may undertake a rigorous
program of supervised study designed to expose the student to
a particular area of mathematics. Regular homework assignments
and scheduled examinations are required.
Courses in Statistics Courses at the 300-500 levels offered
by the Division of Statistics may be found later in this chapter. | 677.169 | 1 |
Collins New GCSE Maths Student Books are the perfect way to help students working at Grades C to A* tackle the AQA GCSE Mathematics Higher Linear specification. Packed with functional skills, problem solving and graded maths practice, they give your students confidence to take on all aspects of the new curriculum and succeed. Collins New GCSE Maths AQA Linear Student Book Higher 2, written by experienced teachers and examiners, is organised exactly according to the AQA GCSE Mathematics Linear specification. It is the ideal resource to help students get the best results: * Enable students to monitor their own progress through the GCSE Maths course with Collins' colour-coded grades on every page and a grade booster at the end of every chapter* Be confident that students are practising the key elements of the new curriculum in every lesson with functional skills, problem solving and AO2/AO3 new exam-style questions within every exercise* Use the colourful functional skills and problem-solving pages at the end of every chapter to engage students with rich tasks that will develop their process skills and allow them to apply maths in stimulating real-life contexts* Show students exactly why each chapter matters to them with new chapter openers that develop the cross-curricular nature of maths* Give students the opportunity for self-assessment and guidance for their exam technique by using the comprehensive exam practice and worked exam questions with examiner notes at the end of every chapter Suitable for the second year of the AQA Linear course. For the first year, you will need Collins AQA Linear Student Book Higher 1 (9780007489329)
Book Description Collins Educational, 2012. Paperback. Book Condition: Very Good. New GCSE Maths â€" AQA Linear Higher 2 Student7489336
Book Description Collins Educational6661769 | 677.169 | 1 |
9 - 12), or Technology Education (9 - 12) Title: Predict the Future? Description: Students will use data collected and a "best-fit line" to make predictions for the future. The example the students will be working on for this lesson will demonstrate an exponential regression.
Subject: Mathematics (6 - 12) Title: Swimming Pool Math Description: Students will use a swimming pool example to practice finding perimeter and area of different rectangles.
Subject: English Language Arts (9 - 10), or Mathematics (9 - 12) Title: Rags to Riches or Riches to Rags? Description: In this lesson students will apply the interest formulas to make the best investment decision for our fictional character Algebrea Calculique. The students will be provided with a story explaining the situation that Algebrea faces and the investment choices that she has. To present their findings and conclusion, the students will write an expository essay. This lesson integrates mathematics into the writing process.
Thinkfinity Lesson Plans
Subject: Mathematics,Science Title: Symbolic AnalysisAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students use iteration, recursion, and algebra to model and analyze a changing fish population. They work to find additional equations and formulas to represent the data. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics,Science Title: Analyzing the DataAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students explore the development of a mathematical model for the decay of light passing through water, in a rich exploration of exponential models in context. They use an interactive Java applet to explore related algebraic functions Counting Embedded FiguresAdd Bookmark Description: In this Illuminations lesson, students look for patterns in an embedded-square problem. After looking at the patterns, students form generalizations for the pattern. This activity sharpens students algebraic thinking and visualization skills. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
Subject: Mathematics,Science Title: Shedding the LightAdd BookmarkSubject: Mathematics Title: Investigating Pick's TheoremAdd Bookmark Description: In this unit of three lessons, from Illuminations, students rediscover Pick's Theorem, which they were likely introduced to in middle school, and use algebra to determine the coefficients of the equation. They explore the concept of change as a mechanism for finding the coefficients of Pick's Theorem. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 | 677.169 | 1 |
Does someone here know anything concerning how to program calculate online examination with the java statistic? I'm a little lost and I don't know how to finish my math homework about this topic. I tried reading all tutorials about it that could help me figure things out but I still can't finish it. I'm having a hard time answering it especially the topics rational expressions, conversion of units and quadratic inequalities. It will take me forever to finish my math homework if I can't get any help . It would really help me if someone would recommend anything that can help me with my algebra homework.
Algebrator is a useful program to solve how to program calculate online examination with the java statistic problems . It gives you step by step answers along with explanations. I however would warn you not to just copy the answers from the software. It will not aid you in understanding the subject. Use it as a guide and solve the questions yourself as well.
I remember I faced similar problems with monomials, matrices and mixed numbers. This Algebrator is truly a great piece of algebra software program. This would simply give step by step solution to any algebra problem that I copied from workbook on clicking on Solve. I have been able to use the program through several Algebra 1, Algebra 1 and Algebra 1. I seriously recommend the program. | 677.169 | 1 |
Mathematics
The department of mathematics, started since the inception of the college, has the credit of successful degree holders with flying colors. As an integral part of the institute, the department arranges various activities to inculcate interest towards the subject and the prime focus of the department is to provide a conceptual base in mathematics which helps the students continue their studies. Apart from this, the department provides a right sense of direction for living as a responsible citizen.
Courses
B.Sc. in Mathematics
Facilities
Books in the Central Library:
Number of Mathematical books available in College library is 290.
Books in the Departmental Library:
Number of Mathematical books available in Departmental library is
93.
Number of Computer Labs: 03
Activities
1. Expert Lecture:
Dr. Nirav Vyas, Assistant Professor, Atmiya Institute of Technology and Science, Rajkot delivered a talk on "Career Opportunities with Mathematics " on 25th July, 2015. More than 150 Students of SY BSc (A) and TY BSc Maths attended the talk.
2. Problem solving Contest:
Department of Mathematics had organized problem solving contest-2015 for the students of Christ College, Rajkot. | 677.169 | 1 |
About this product
Description
After a brief introduction reviewing the concepts of principal ideal domains and commutative fields, the book discusses residue classes; quadratic residues; algebraic integers, their discriminant; decomposition, rm, and classes of ideals; the ramification index; and the fundamental theorem of Abelian extensions. | 677.169 | 1 |
Students in my courses will be given the opportunity to challenge themselves within the realm of mathematics. These challenges can be met with hard work and practice.
In my classroom, I believe there are three steps to being successful: be prepared, be respectful, and be your best. I expect all students to follow these steps to allow for a better learning environment.
The syllabus outlines the classroom expectations, the components of the course grade (homework, test/quizzes, and projects), and how to be successful in class. Overall, communication is going to be essential between the student, teacher, and parents/guardians.
I try to update Skyward daily, with homework scores. Feel free to contact me with questions, comments, or concerns.
I look forward to a great semester full of mathematics with great students at Spring Hill High! | 677.169 | 1 |
Survey of Mathematical Problems
Preface
Everybody talks about the weather, but nobody does anything about it. Mark Twain College mathematics instructors commonly complain that their students are poorly prepared. It is often suggested that this is a corollary of the students' high school teachers being poorly prepared. International studies lend credence to the notion that our hard-working American school teachers would be more effective if their mathematical understanding and appreciation were enhanced and if they were empowered with creative teaching tools. At Texas A&M University, we decided to stop talking about the problem and to start doing something about it. We have been developing a Master's program targeted at current and prospective teachers of mathematics at the secondary school level or higher. This course is a core part of the program. Our aim in the course is not to impart any specific body of knowledge, but rather to foster the students' understanding of what mathematics is all about. The goals are: • to increase students' mathematical knowledge and skills; • to expose students to the breadth of mathematics and to many of its interesting problems and applications; • to encourage students to have fun with mathematics; • to exhibit the unity of diverse mathematical fields; • to promote students' creativity; • to increase students' competence with open-ended questions, with questions whose answers are not known, and with ill-posed questions; iii
The material evolves each time we teach the course.
We hope that after completing this course. and • to give students confidence that.tamu. proving theorems. when their own students ask them questions. Suggestions. We deliberately have not flagged the "difficult" exercises. in the sense of solving problems. superficial level or at a deeper. Many of the exercises can be answered either at a naive. struggling with difficult concepts. depending on the background and preparation of the students. corrections. because we believe that it is salutary for students to learn for themselves whether a solution is within their grasp or whether they need hints.iv
PREFACE • to teach students how to read and understand mathematics. Please email the authors at boas@math.tamu.edu. they will either know an answer or know where to look for an answer.
. and comments are welcome. The exercises range in difficulty from those that are easy for all students to those that are challenging for the instructors. We try to teach in a hands-on discovery style. We distribute the main body of this Guide to the students. students will have an expanded perspective on the mathematical endeavor and a renewed enthusiasm for mathematics that they can convey to their own students in the future. reserving the appendices for the use of the instructor. making conjectures. searching for understanding. more sophisticated level. typically by having the students work on exercises in groups under our loose supervision.edu and geller@math. We emphasize to our students that learning mathematics is synonymous with doing mathematics.
available from gopher://wiretap.2
Reading
1. 1
. contrapositive. • Be aware that foundational problems (paradoxes) exist. and Seeking Truth in the Sciences. Indianapolis.Chapter 1 Logical Reasoning
1. Lafleur. Ren´ e Descartes. excerpt from Part II.
1. and so on. if and only if. • Be able to construct valid logical arguments and to solve logic problems.1 Goals
• Know the meanings of the standard terms of logic: converse. • Be able to recognize valid and invalid logic. implication. necessary and sufficient conditions. txt. translated by Laurence J. 1964.area. 2.com:70/00/Library/Classic/reason. pages 156–162. Philosophical Essays. Ren´ e Descartes. Discourse on the Method of Rightly Conducting the Reason. • Be able to identify and to construct valid proofs by the method of mathematical induction. Bobbs-Merrill Company.
2
The Liar Paradox
Epimenides of Crete (attributed)
All Cretans are liars. pages 20–29. 6. 1984. no. Scribner's.ab. Stephen B. no. 5.3.1. 2 Goldbach's Conjecture: every even integer greater than 2 can be expressed as the sum of two prime numbers.ca/bottleimp. (Compare the surprise examination paradox. 1893. 91–96. in the public domain and available on the world-wide web at The paradox has a number of different guises. Robert Louis Stevenson. for example: Please ignore this sentence.
1
.1 The Captain wishes to determine the truth of a rumor that one of the miners has recently proved Goldbach's Conjecture. The recursive paradigm: suppose we already knew. "The Bottle Imp". Excerpt from Alice in Puzzleland by Raymond M.)
1. Penguin.
How are we to understand this statement? Apparently.mtroyal. 2 (February).
Raymond Smullyan refers to such scenarios as "knights and knaves" and discusses many such in his puzzle books. Smullyan. The Starship Enterprise puts in for refueling at the Ether Ore mines in the asteroid belt. There are two physically indistinguishable species of miners of impeccable reasoning but of dubious veracity: one species tells only truths. 62– 63. Maurer. Proofs without words from Mathematics Magazine 69 (1996). LOGICAL REASONING 3.2 What single question—answerable by "Yes" or "No"—can the Captain ask of an arbitrary miner in order to determine the truth?
1. 1. while the other tells only falsehoods. School Science and Mathematics 95 (1995).3.2
CHAPTER 1.3
1. 4. in Island Nights' Entertainments. it is true if and only if it is false.1
Classroom Discussion
Warm up
Exercise 1.htm.
Exactly one statement on this list is false. 2." A version due to Bertrand Russell is: The village barber shaves those and only those villagers who do not shave themselves. Let n be an integer greater than 1. n. Is S an element of itself? G. Exactly n statements on this list are false. Kurt Grelling asked if the adjective "heterological" is heterological (that is. Berry asked for a determination of "the least integer not nameable in fewer than nineteen syllables" (the quoted phrase consisting of eighteen syllables). CLASSROOM DISCUSSION An even sharper form is: This sentence is false. Exactly two statements on this list are false. . (This problem was published by David L. cited in Martin Gardner. 1986.2. Silverman in the Journal of Recreational Mathematics (1969). not describing itself). Freeman.
3
Exercise 1. Who shaves the barber? The mathematical formulation of Russell's paradox is: Let S be the set whose elements are those sets that are not elements of themselves. page 29. 1. Jourdain is a piece of paper that says on one side "The sentence on the other side is true" and on the other side "The sentence on the other side is false. Knotted Doughnuts and Other Mathematical Entertainments.) A version of the liar paradox attributed to P. Self-referential paradoxes have appeared in popular literature. Determine the truth or falsity of each statement on the list. G. E. B.1.3. Chapter Six. and consider the following list of sentences. . For example:
. There is a non-paradoxical generalization of the preceding example to more than one sentence. .
If he flew them he was crazy and didn't have to. It happened. and the judges let them pass free. and if he swears truly. when they came to take his declaration." Though the law and its severe penalty were known." Miguel de Cervantes
. LOGICAL REASONING "Can't you ground someone who's crazy?" "Oh. and nothing else. Orr would be crazy to fly more missions and sane if he didn't. but in their declarations it was easy to see at once they were telling the truth. and a sort of tribunal. he would no longer be crazy and would have to fly more missions. The judges held a consultation over the oath. that one man. Orr was crazy and could be grounded. and which was to this effect. however. but if falsely. sure. many persons crossed.. I have to. and therefore swore the truth. by the same law he ought to go free. without any remission. All he had to do was ask. "If we let this man pass free he has sworn falsely. he shall be put to death for it by hanging on the gallows erected there.] There was only one catch and that was Catch-22. and at one end of it a gallows. but if he didn't want to he was sane and had to. bridge and the lordship had enacted. There's a rule saying I have to ground anyone who's crazy. Joseph Heller Catch-22 Here is another example: Well then. swore and said that by the oath he took he was going to die upon that gallows that stood there. and they said." [ . as he swore he was going to die on that gallows.. where four judges commonly sat to administer the law which the lord of river. "If anyone crosses by this bridge from one side to the other he shall declare on oath where he is going to and with what object. and as soon as he did. which specified that a concern for one's own safety in the face of dangers that were real and immediate was the process of a rational mind.4
CHAPTER 1. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle. on this river there was a bridge. he shall be allowed to pass. but if he was sane he had to fly them. and by the law he ought to die. but if we hang him.
Q. we will be content with reviewing some basic procedures of logical analysis. let R denote the statement "We are studying". We will return later to the axiomatic method.3. let Q denote the statement "It is Saturday". or we need to accept that some apparently wellformed sentences are meaningless: neither true nor false.3. Intuitively. For a complete resolution of Russell's paradox. we should have to go into set theory rather deeply. and ¬ (negation). are we to make of the following grammatical English sentence? Colorless green ideas sleep furiously. and S ." if only if is sufficient for is necessary for if and only if ⇒ ⇐ ⇔
. What. we need to rule self-referential statements out of bounds. but we have a picnic on Saturdays when it is not raining. CLASSROOM DISCUSSION Don Quixote (John Ormsby. and let S denote the statement "We are having a picnic. translator)
5
A common element of these paradoxes is their self-referential or circular property.3
The Formalism of Logic
Exercise 1. ∧ (conjunction)." Using the letters P . for example.3. then we had better have some rules about determining truth. and the symbols ∨ (disjunction).1. Let P denote the statement "It is raining". fill in as many lines as possible of the following table to be compatible with the sentence: "We always study when it rains.
1. Noam Chomsky
The lesson of the above discussion is that if we are not to build our mathematics on a foundation of quicksand. For the moment. R.
Solve the problems from the "Who Is Mad?" chapter in Raymond Smullyan's Alice in Puzzle-Land. This can be accomplished by completing the following truth table and observing that the last two columns are identical.4. Exercise 1. Everything in the world is good for something. 5. y )). 1. Oscar Wilde 7. The House of Peers. There is only one thing in the world worse than being talked about. LOGICAL REASONING
Exercise 1. it would be necessary to invent Him. iii.) William Shakespeare Hamlet I. 4. il faudrait l'inventer.
W. Gilbert
. 3. Demonstrate that the biconditional "⇔" is associative. S. For all positive x and y . If God did not exist.5. Did nothing in particular. y ) ⇒ Q(x. Neither a borrower nor a lender be. Abraham Lincoln (attributed) 8.7. throughout the war.6. and that is not being talked about. A B C A⇔B B⇔C (A ⇔ B ) ⇔ C A ⇔ (B ⇔ C )
Exercise 1. (Si Dieu n'existait pas. Negate the following statements. 75 John Dryden Voltaire
6. 2. Exercise 1. and some of the people all the time. but you can not fool all the people all of the time. Create your own examples similar to the one above. either x2 ≥ y or y 2 ≥ x. You can fool all the people some of the time. And did it very well. ∀x ∃y (P (x.6
CHAPTER 1.
I think that everybody does the same. and each time the doorbell rings it's because there is somebody there. . Martin: But yes.4
Mathematical Induction
Exercise 1. Mr. Read aloud the following excerpt from Scenes VII–VIII of Eug` ene Ionesco's "anti-play" The Bald Soprano.8.3. (She sits down again. Martin: Not always. (The doorbell rings. first performed May 11. I ring the bell to get in. when I go to someone's house. Mrs. .) Nobody. (She goes to see. Smith: There must be somebody there. Martin (who has forgotten where he was ): Uh . Martin: You were saying that you were going to give another example. Martin: I'm going to give you another example . Smith: Well. as you have just seen! Mr. Martin. Martin. Mrs. the doorbell is ringing. the doorbell is ringing. and the Fire Marshall. .) Mr.3. I'll go see. Smith: Say. Mrs. Why do you think there will be somebody now? Mr. (She goes to see. Smith: The first time. Mrs. Martin: Oh. there was nobody. Mr. again nobody.) Mr. Smith. yes .) Mr. . She opens the door and returns. Smith: Say. .) Nobody. (She returns to her seat. She opens the door and returns. Mrs.3 There are five actors involved: Mr.
3
La cantatrice chauve.1. Smith: I am not going to open the door again. Mr. Mrs. Martin: That's not a reason. (The doorbell rings. Smith: Myself. . Smith: Because the doorbell rang! Mrs. but there must be somebody there! Mrs. it's because there is somebody at the door who is ringing to have the door opened. Smith. (The doorbell rings. Mrs. the doorbell is ringing.) Mr. I'll go see. Mr.
. most of the time. Smith: Say.) Mr. Smith: It must be somebody. CLASSROOM DISCUSSION
7
1. Mr. The second time. Martin: What? When one hears the doorbell ring. 1950.
You have just seen so.) Mr. Mrs. Mrs. ladies and gentlemen. Smith: He won't admit he's wrong. She opens the door and shuts it again. I'll go see. Mr. You can't say that I am stubborn. Most of the time. Mrs. Smith: I tell you no. there is nobody. Mrs. of course. Mrs. Smith: I'm going. There must be somebody there. Mrs. Smith: Say. You have seen that it is useless.) Hello. (She returns to her seat. but you will see that there is nobody there! (She goes to see. Mrs. Mrs. Mr. these men who always want to be right and who are always wrong! (The doorbell rings again. Smith: There is somebody there. averts her head and does not respond. Smith shrugs her shoulders.) It's the Fire Marshall! Fire Marshall (he has. But in reality things happen differently. You seem to be angry. it's because there is somebody there. Martin tosses her head.) Mr. Mr. who are all surprised. Smith (to her husband ): No.) Mrs. Mrs. If you want to go see. Martin: Your wife is right. Smith. Smith (opening the door ): Oh! Comment allez vous? (He glances at Mrs. Martin: Oh. Martin: It's not impossible.) You see. Smith: Yes. Mrs. Smith: It's even false. Mr. (Mrs. you women! Always defending each other. (They are still a bit stupefied. Smith: Oh.8
CHAPTER 1. Mr. Martin: That's not certain. Smith: That is true in theory. Mrs. LOGICAL REASONING
Mrs. Mrs. Martin: My husband too is very stubborn. At any rate. when the bell rings. Smith and at the Martins. Smith: All right. Mr. Smith (in a fit of anger ): Don't send me to open the door again. it's because there is never anybody there. the doorbell is ringing. angry. Smith: Oh!
. Experience shows us that when the bell rings. Martin: Never. go yourself! Mr. an enormous shiny helmet and a uniform ): Good day. Smith. you are not going to bother me again for nothing.
Mrs. because the Fire Marshal is there. Martin: Yes. not by theoretical demonstrations. Martin: Always. It troubles me to speak openly to you. Martin: It might seem strange. Smith: It's like this. but it was only after the bell rang a fourth time that someone was there. what's it all about? Mrs. Smith: We were arguing because my husband said that when the doorbell rings. but a Fire Marshall is also a confessor. The Fire Marshall is an old family friend. He rang. Mr. darling. Tell me about it. Mr. Martin: There has been. but by facts. Mrs. And the fourth time doesn't count. . Mrs. Fire Marshall: Well? Mrs. Mrs. Mr. Smith: My husband was claiming . Only the first three times count. he was there.1. Martin: When? Mr. it was him. Smith. Mrs. Mr. you see . He died waiting. I opened the door. Mr. it was you who was claiming. Mr. Smith: All right. Fire Marshall: Go ahead. Martin: It's plausible. Mr. it's not so serious. and Mr. Mr. Martin: It's neither his fault nor yours. Smith. Smith: No. He asked to marry my daughter if I ever had one.3. Fire Marshall: Don't get excited. Smith (to Mr. Smith: Oh. and I knew his father. Martin ): It's none of your business! (To Mr. Mrs. Smith: It's false. Smith: And I said that each time the doorbell rings. Martin: No. CLASSROOM DISCUSSION Mr. Mrs. Smith: But it is proven. it is because nobody is there. Smith: Fire Marshall. then. my wife is a bit humiliated to have been wrong. Mrs. Smith: Yes. Mrs. an argument between Mrs. Martin: Why. Smith ): I beg you not to bring strangers into our family quarrels. His mother courted me. Fire Marshall. there is always somebody there. . . Mrs. it was her. Fire Marshall: So. just now. let me ask you a few questions. .
9
.
Martin: Did you hear the bell ring the second time? Fire Marshall: Yes. Mrs. it was me. Smith: Were you at the door a long time? Fire Marshall: Three quarters of an hour. I was standing there. whether there is someone there or not! Mrs.10
CHAPTER 1. Smith: And you saw nobody? Fire Marshall: Nobody. Mrs. You can't say that the Fire Marshall is not somebody. Fire Marshall. was it you? Fire Marshall: No. Mrs. . because the fourth time does not count. .
. Mrs. Smith (to his wife. as a joke. . And there was still nobody. Mrs. I repeat that I speak only of the first three times. Smith: Always somebody. we did not see you. Martin: Perhaps it was someone else? Mr. Martin (to the Fire Marshall ): But the third time . Mr. LOGICAL REASONING
Mr. was it indeed you who had rung? Fire Marshall: Yes. I'm certain. it is because somebody is there. Mr. Mr. it was me. Smith: But when we opened the door. Mr. Mr. Smith: When I opened the door and saw you there. victoriously ): You see? I'm right. Mr. Smith: Certainly not. Smith (to his wife ): Not so fast. . it was not me. Mr. When the bell rings. Martin: You were at the door? You rang to be let in? Fire Marshall: I don't deny it. Fire Marshall: That's because I hid myself . Martin: You see? The bell rang. Smith: Victory! I was right. we still don't know. when the doorbell rings. Martin: To sum up. Mr. Mrs. Mr. Martin: When the bell rang the first time. The situation is too sad. (To the Fire Marshall ) And what were you doing at the door? Fire Marshall: Nothing. and again it was not me. I was thinking about a lot of things. Smith: Never anybody. and nobody was there. it was not you who rang? Fire Marshall: Yes. Smith: Don't joke.
Mrs. Martin: I think so too. ? The principle of induction says. because it is not obvious how to relate statement n to statement (n + 1). Exercise 1.10. When the doorbell rings. 3. Chinese Proverb Who has begun. and if you can always take another step—no matter how far you have gone already—then you can travel an arbitrary distance. Exercise 1. A journey of a thousand miles begins with a single step. You are both partly right. Jesus says: Let he who is without sin among you cast the first stone. Mr. John 8:7 The most familiar type of induction problem involves proving an equality. .3.11. 9. What is the next term in the sequence 2. has half the job done. . Proving an inequality can be trickier. but it is nonetheless a crucial part of the argument. The two common mistakes in creating an induction proof are (i) neglecting to check the basis step.)
Horace Epistles I. CLASSROOM DISCUSSION Fire Marshall: I will reconcile you. i. and other times there is nobody. 5. 40
An example of an inductive situation that founders for want of an initial step is the parable in the New Testament of the woman who is to be stoned for transgressing the Mosaic law. sometimes there is somebody. that if you can take a first step. intuitively. . Fire Marshall: Things are simple. and (ii) failing to make a completely general argument for the induction step.
11
Exercise 1.9.1. Prove that 2n > 2n + 1 when n is an integer greater than 2.
. Martin: That seems logical to me. (Dimidium facti qui coepit habet. really. The "domino effect" is the same phenomenon. The basis step is usually easy to confirm. Comment on the implications of the above scene for mathematical logic.
All horses are the same color. all horses are the same color. it must be that all n + 1 horses are the same color. page 145. P (n) is true for every positive integer n. 1991. In a round-robin tournament.12
CHAPTER 1. Here is a false proof of a true statement.12. Maurer and Anthony Ralston.
Exercise 1. The sum of the angles of a regular n-gon is 180(n − 2)◦ . LOGICAL REASONING In the next example. Since it does not matter which horse is removed. Maurer and Anthony Ralston. "Proof " by induction. 1991. and team 3 beat team 4. Addison-Wesley. there will be some way to number the teams so that team 1 beat team 2.14. Suppose that P (n) holds. Discrete Algorithmic Mathematics.4 Here is an example of a formula with cases. Discrete Algorithmic Mathematics." Theorem. The previous example was a false "proof" of a false statement. Addison-Wesley. Trivially P (1) is true. 2 Exercise 1. 2.15. By induction.
4
. Show that if no games end in ties. Show that the sum of the first n positive integers that are divisible by neither 2 nor 3 is 3 2 n −1 if n is odd. 1. If S is an arbitrary set of n + 1 horses. Find the mistake in the following "proof. and so on. where is the mistake?5 Theorem. 2 2 3 2 n if n is even. then no matter what the outcomes of the games. Exercise 1. the remaining n horses are the same color by the induction hypothesis. 5 Example paraphrased from Stephen B. Find a formula for the nth positive integer that is divisible by neither 2 nor 3. Exercise 1.13. each team plays every other team exactly once. page 171. the induction statement is not a formula. and team 2 beat team 3. that is.
Problem paraphrased from Stephen B. Let P (n) be the proposition that all members of an arbitrary set of n horses are the same color. and one is removed.
For instance. Exercise 1. Now suppose that the statement has been proved for a certain value of n (where n ≥ 3). 1986. 1984.4. 8 ?emit dnoces eht rof tnemetats siht ekam thgink a ro evank a naC
6
. 1992. Given a regular (n + 1)-gon. 1985. Knopf. What Is the Name of This Book?. 1981. Prentice Hall. Knopf. page 44. Much of modern science is an effort to find patterns in nature. of art. 7 Raymond Smullyan. of mathematics. where knights speak only truths. it may not be obvious what statement n should be. Invent your own scenario of an encounter with the residents of the island of knights and knaves. The Lady or the Tiger?. reprinted by Times Books. Knopf. Since the knights are physically indistinguishable from the knaves. 1.1. 1978. Children of all ages enjoy guessing patterns. reprinted by Penguin. the visitor must exercise ingenuity to extract information from the inhabitants' statements. 1982. Alice in Puzzle-Land: A Carrollian Tale for Children Under Eighty. Knopf. is a native who states. 1993. Knopf. 1982.4
Problems
Problem 1. 1992. reprinted by Oxford University Press. reprinted by Simon & Schuster. 1980. Pattern recognition is an element of games.7 "This is not the first time I have said what I am now saying" a knight or a knave? See the footnote for a hint. 1987.
1.2. Forever Undecided.8 2. reprinted by Penguin. Problem 1. Adding back the triangle increases the angle sum by 180◦ for a final total of 180(n − 1)◦ = 180((n + 1) − 2)◦ . PROBLEMS
13
"Proof " by induction. To Mock a Mockingbird. Morrow. Cantor and Infinity.16. take three consecutive vertices and cut off the triangle they determine. The philosopher Raymond Smullyan has written several puzzle books6 featuring the island of knights and knaves. In the next exercise. The remaining n-gon has angle sum of 180(n − 2)◦ by the induction hypothesis. To Mock a Mockingbird.
Some of his books featuring logic puzzles are Satan. This Book Needs No Title. The basis step (n = 3) is the well-known fact that the angles of a triangle sum to 180◦ .1. 1985. Construct an induction proof of the proposition that every set (possibly infinite) of positive integers has a least element. and knaves speak only falsehoods. Prentice Hall.
page 137. Problem 1. 1995 (available online at Problem 1. T. 14. T. Discrete Algorithmic Mathematics. T. T. . 31.eerht esab ni si yrtne gnissim ehT 12 Example paraphrased from Stephen B. page 172. Academic Press. revised edition published by the Mathematical Association of America.3. 2.4. 16.hcnerF ni tnereffid eb dluow elzzup sihT Martin Gardner. S. 1977. See the footnotes for a hint. New York. 15. 22. and prove it by induction. The Encyclopedia of Integer Sequences. Find a formula for the sum of the first n positive integers that are not divisible by 4. 100. 11. .research. .9 The element of surprise often makes an amusing puzzle. 13. 10000
Hint: the sequence terminates—there are no more terms. See the footnote for a further hint. E. E. J. F. F. Prove by induction that 2+ 2+
n
√
2 + · · · is irra-
tional for each positive integer n. Maurer and Anthony Ralston.5. 17. 24.
2
2+
3
3+
n
√ 4
4 + · · · is irrational for each pos-
Problem 1. att. Can you figure out the missing entry in the following sequence?10 10.11 A great many special sequences of counting numbers may be found in Simon Plouffe and N.com/~njas/sequences/). Knopf. Sloane. S. 1989. LOGICAL REASONING
You can amuse any class by asking for the rule generating the following sequence of letters: O. San Diego. 20. 12. Prove by induction that itive integer n. .
10 9
. Invent your own pattern recognition problem. A. Addison-Wesley. 1. . N. What is wrong with the following "proof"?12
. 1991.14
CHAPTER 1. Mathematical Magic Show. 11 .
in the case of appending a 1. Hence there twice as many sequences of length (n + 1) as there are of length n. New York. Mackie.S66 1995)
. Vicious Circles and Infinity: A Panoply of Paradoxes. Garden City. 1989. New York. Oxford University Press. Van Nostrand. We can create sequences of length (n + 1) by appending either a 0 or a 1 to the right-hand end of a sequence of length n. Knopf. The Liar: An Essay on Truth and Circularity.
1.P2 B37 1987) • Bryan H. so the theorem is proved because 2 · 2n = 2n+1 .5. where n ≥ 1. 1975. NY. L.1. updated and revised edition published by the Mathematical Association of America. J. Sloane. Bunch. (QA9 B847) • Martin Gardner. Catch-22. New York. • Joseph Heller.5. • Simon Plouffe and N. Doubleday. 1973. there two such sequences. Academic Press. we might produce a 11 at the right-hand end. Mathematical Magic Show. 1987. Mathematical Fallacies and Paradoxes. Maurer and Anthony Ralston. The Encyclopedia of Integer Sequences. (BC171. 1977. (BC199.5
Additional Literature
• Jon Barwise and John Etchemendy. so the theorem holds in the base case. There are 2n sequences of 0's and 1's of length n with the property that 1's do not appear consecutively except possibly in the two rightmost positions. Probability and Paradox: Studies in Philosophical Logic. (QA246. Addison-Wesley. "Proof ". 1991. 1961. Truth. Suppose the theorem holds for a certain integer n.M24) • Stephen B. but that is allowed. Simon and Schuster. 1982. ADDITIONAL LITERATURE
15
Theorem (false).P2 H83) • J. (BC199. (PZ4 H47665 Cat) • Patrick Hughes and George Brecht. Oxford University Press. San Diego. 1995. A. When n = 1. Discrete Algorithmic Mathematics.
scene 4
2. William Shakespeare Twelfth Night Act III. Chapter VII.Chapter 2 Probability
If this were played upon a stage now.1
Goals
• Understand the notion of discrete probability. • Be able to apply your knowledge of probability to unfamiliar situations. "Chance and Chanceability". Simon and Schuster.
17
. pages 223–264. 1940.2
Reading
1. • Be able to count cases using permutations and combinations.
2. I could condemn it as an improbable fiction. • Be able to calculate discrete probabilities. of Mathematics and the Imagination by Edward Kasner and James Newman. This selection is an introduction to probability.
two. The weather tomorrow is not a repeatable experiment.
n k
=
n! k!(n−k)!
is the number of combinations of n things taken k at a
Exercise 2. Chap.3.1. Keep probability in view. John Gay 1688–1732 The Painter who pleased Nobody and Everybody
2. . sir.
1
. you could compute this probability as the fraction
26 2 52 2
=
25 ≈ 0. . PROBABILITY
2. queen. jack.3.2.3
2. Samuel Johnson Life of Boswell. hearts ♥. why. and clubs ♣). so what does it mean when the weather forecast is "30% chance of rain tomorrow"? Lest men suspect your tale untrue.1
Classroom Discussion
Warm up
Exercise 2. and each suit has thirteen cards (ace.2
Cards and coins
A typical sort of question in discrete probability is: "If two cards are dealt from a standard deck. king). ten.245 102
(where time). diamonds ♦.1 what is the probability that both are red?" Using the principle that probability is computed as the number of favorable situations divided by the number of all possible situations (assuming that all situations are equally probable). three. . Therefore the probability that it will rain tomorrow is 1/2. ii. the spades and clubs being black and the hearts and diamonds being red. 1763
You need to know that a standard deck of playing cards consists of four suits (spades ♠. Why—since half the cards are red—is the answer not just the product 1 ·1=1 ? 2 2 4 If he does really think that there is no distinction between virtue and vice. What is wrong with this argument? Exercise 2. . v. Either it will rain tomorrow.3. when he leaves our houses let us count our spoons. or it will not rain tomorrow. Vol.18
CHAPTER 2.
Three of a kind means three cards of the same rank and two extra cards (but excluding all of the previous cases). One pair means two cards of the same rank and three other cards (but excluding all of the previous cases). Examples are ♥K ♥Q ♥J ♥10 ♥9 and ♣10 ♣9 ♣8 ♣7 ♣6. it is historically appropriate to analyze a popular modern gambling game: poker. Examples are ♠A ♥K ♥Q ♠J ♣10 and ♠5 ♥4 ♥3 ♣2 ♥A. A flush is five cards all of the same suit (but excluding all of the previous cases). Examples are ♣A ♣7 ♣5 ♣3 ♣2 and ♠K ♠Q ♠10 ♠9 ♠7. Four of a kind is all four cards of the same rank together with any other card.
. five cards are dealt from a standard deck. 7. (Keep in mind that aces can count as either high or low. A straight flush consists of five consecutive cards in the same suit (but excluding a royal flush). Examples are ♠A ♥A ♠6 ♦5 ♣3 and ♦9 ♣9 ♣10 ♦8 ♣3.) You will finish this exercise for homework. Examples are ♠K ♦K ♣K ♦3 ♣3 and ♥5 ♦5 ♣5 ♥10 ♣10.2. Nothing is any other hand not previously enumerated.3. Examples are ♠A ♥A ♦A ♥9 ♣7 and ♠3 ♥3 ♦3 ♣K ♦10. 4. Examples are ♠A ♠K ♠Q ♠J ♠10 and ♦A ♦K ♦Q ♦J ♦10. A full house consists of three of a kind together with a pair. 3. 6. A straight is a sequence of five cards in order in mixed suits. 5. In the simplest version of poker. Examples are ♠J ♥J ♦J ♣J ♥3 and ♠7 ♥7 ♦7 ♣7 ♣10. Two pairs means two separate pairs and an extra card (but excluding all of the previous cases).4. 1. 8. 9. 10. Begin determining the probabilities of being dealt the following poker hands. A royal flush consists of the five highest cards in one suit. Examples are ♦K ♣K ♥7 ♣7 ♠9 and ♠10 ♣10 ♥6 ♦6 ♣4. Exercise 2. 2. CLASSROOM DISCUSSION
19
Since the mathematical theory of probability had its beginnings in gambling games.
exactly one set of weights will work. Three solutions are conceivable: more than one set of weights will work. Chap. Miguel de Cervantes 1547–1616 Don Quixote. Mathematically more challenging is the inverse problem of determining the weights needed to produce specified probabilities.1. v
2. (This effect is noticeable in the third decimal place.
Exercise 2. Actual United States coins are not precisely fair.20
CHAPTER 2. the three possible outcomes (both heads. so that the probabilities are different from the ordinary uniform distribution. and a second coin has been weighted so that it comes up heads 3 . because one side is slightly hollowed out to form a relief image. and If a "fair coin" is tossed. Instead.4
Problems
Problem 2. In such cases.) There are many probability problems dealing with coins (or dice) that have been weighted. The problem of computing probabilities of results of coin tosses for coins weighted in a specified way can be difficult. or one head and one 1 tail) all have probability 3 ? The two coins do not have to be weighted the same as each other.5. If one coin has been weighted so that it comes up heads with probability 1 . but it is routine in the sense that all such problems use the same principle. computing the number of favorable outcomes divided by the number of possible outcomes is no longer a valid way to find the probability of an event (because the outcomes are not equally likely). or no set of weights will work. Can you weight two coins in such a way that if the two coins are tossed. Part i.
. 1. what is the probability that when the two coins are tossed. PROBABILITY
1 of landing heads up. The lighter side with the head is slightly more likely to land facing up. Similar considerations apply to the dice accompanying children's games: the side with six pips hollowed out is more likely to land up than the opposite heavier side with only one pip hollowed out. Book iii. with probability 1 4 one of them comes up heads and the other one comes up tails? Let the worst come to the worst. one has to add the probabilities of the individual outcomes. it has probability 2 1 probability 2 of landing tails up. both tails.
At any point. and A affirms that B denies that C declares that D is a liar. or 10 2 .4. but his solution is disputed by other authors. pull out slips of paper one at a time.
2
. you stop automatically. but they need not be integers: they could √ be −17π . page 121. The numbers are all different. PROBLEMS
21
2. It is rather difficult to find an exact solution to the problem. and look at each number. This problem sometimes goes by the name of "the secretary problem. then you win $10. Otherwise you win nothing. both the numbers you have drawn and the numbers that are left in the hat). 1935. MacMillan. What about three coins? Can you weight them so that the four possible outcomes (all heads. You may want to get started by considering a similar problem in which there are only a small number of slips of paper.3. 1. or −22/7. C . New York.2. What is a reasonable amount to pay for the privilege of playing this game? Ten cents? Fifty cents? One dollar? Two dollars? Remarks 1. you may choose to stop. B .
New Pathways in Science. what is the probability that D was speaking the truth? Problem 2. You reach into the hat.2. What is a reasonable strategy to use for playing this game? How should you decide when to stop? 2. This problem was discussed by Sir Arthur Eddington:2 If A. but you should be able to make some estimate of the expected value that is in the right ballpark.) If the last slip you draw has the largest number on it (largest of all 100 numbers. two heads and one tail. Can you generalize to an arbitrary number of coins? Problem 2. D each speak the truth once in three times (independently). all tails) are equally likely? 3. say four. (If you get to the last slip. 2. one head and two tails." A hat has 100 slips of paper in it with different real numbers written on them.
New York. Mathematics and the Imagination. 1940. • Sir Arthur Eddington. 1994.
. H."
2. • T. There is an extensive literature about this problem and its relatives. 1965. the problem concerns a business executive who is interviewing 100 secretaries for a job and wants to hire the best one.5
Additional Literature
• Richard Durrett. New Pathways in Science. In its traditional formulation. This is the source for "Eddington's problem. PROBABILITY 3. chapter VII. This is an undergraduate textbook. New York. 1935. MacMillan. Puzzles and Paradoxes. Belmont." • Edward Kasner and James Newman.22
CHAPTER 2. Oxford University Press. CA. O'Beirne. The Essentials of Probability. Hence the name of "the secretary problem. Duxbury Press. Simon and Schuster.
Robert Browning 1812–1890 Paracelsus. his good time. but unless God send his hail Or blinding fire-balls. I shall arrive. such as the traveling salesman problem. and its consequences. 4.1
Goals
1. Learn about Euler's formula. Part i
3. I ask not. 3.
23
. its proof. sleet or stifling snow.—what time. Experience the fun of discovering and creating mathematics. what circuit first. and the four-color problem. planar graph. Become familiar with some famous problems of mathematics. 2. and dual graph. I shall arrive: He guides me and the bird. In some time. the K¨ onigsberg bridge problem. Hamiltonian graph. In his good time. Understand the notions of Eulerian graph.Chapter 3 Graph Theory
I see my way as birds their trackless way.
1
Examples of graphs
cyclic The cyclic (or circuit) graph Cn is a connected graph having n vertices each incident to exactly two edges. for the gain from it is better than gain from silver and its profit better than gold.3
Classroom Discussion
How sweet a thing it is to wear a crown. the cube (hexahedron).3. bipartite A bipartite graph has two sets of vertices.3. Platonic The Platonic graphs are formed by the vertices and edges of the five regular Platonic solids: the tetrahedron.
. The graph Kn has n(n − 1)/2 edges. Act i. Proverbs 3:13-17 A graph is called Eulerian if there is a closed path in the graph that traverses each edge once and only once. and the person who gets understanding. in her left hand are riches and honor. The complete bipartite graph Km. utilities The utilities graph is the complete bipartite graph K3. CLASSROOM DISCUSSION
25
3. n blue vertices.n has m red vertices.3 . It is also the problem of tracing a graph without backtracking and without lifting the pencil from the paper. Within whose circuit is Elysium And all that poets feign of bliss and joy! William Shakespeare King Henry VI Part III. and all her paths are peace. Long life is in her right had.3. and an edge for each of the mn red-blue pairs of vertices. the octahedron. the dodecahedron.
3. say red and blue. She is more precious than jewels. scene 2
3.3.2
Eulerian graphs
Happy is the person who finds wisdom. and nothing you desire can compare with her. Finding such a path is sometimes called the highway inspector problem. and the icosahedron. and every edge of the graph has one red vertex and one blue vertex. Her ways are ways of pleasantness. complete The complete graph Kn has n vertices each of which is adjacent to all the other vertices.
The faces of a map are the connected components of its complement.
While fancy.
.3. a map cannot have a dangling edge.) In particular. Which of the examples of graphs in section 3. Show that Euler's condition is sufficient by finding an algorithm for constructing Eulerian paths. The surrounding "ocean" (the unbounded component of the complement) is ordinarily counted as a face. GRAPH THEORY
Exercise 3.1 are Hamiltonian? For those that are. The famous four-color theorem states that the faces of an arbitrary map can be colored with (at most) four colors in such a way that no two faces sharing an edge have the same color. that the condition is also sufficient.1. Hint: if you start anywhere and begin traversing edges at random. Runs the great circuit. Evidently there are no interesting maps that can be colored with one color (for such maps are all ocean). It is an interesting problem to determine which maps can be colored with fewer than four colors.1 are Eulerian? For those that are. Which maps can be colored with two colors?
3. find Hamiltonian paths.4. Exercise 3. Book iv The Winter Evening.3
Hamiltonian graphs
William Cowper 1731–1800 The Task.2. Exercise 3. line 118.30
CHAPTER 3. Euler implies.
A graph is called Hamiltonian if there is a closed path in the graph that includes each vertex (other than the vertex that is the common start and end of the path) once and only once. find Eulerian paths. Euler's seminal paper on the problem of the K¨ onigsberg bridges gives a necessary condition for a graph to be Eulerian: each vertex should have even degree. Which of the examples of graphs in section 3. and is still at home. Exercise 3. what could go wrong? A map is a special kind of connected planar graph: one which cannot be broken into two pieces by removal of a single edge. but does not prove. (Such an edge is a bridge.3. like the finger of a clock.3.3.
It is known that the problem of determining whether or not a given graph contains a Hamiltonian cycle is an NP-complete problem.8. Hint: Proof by contradiction. then it is Hamiltonian. (It is assumed that there are at least three vertices.
1
. If the graph is not Hamiltonian.7. Exercise 3. Note that is often more convenient to draw them with edges crossing.4
Euler's formula
A graph is said to be planar is it can be drawn in the plane with no edges crossing. roughly speaking.
3. Exercise 3. then the resulting graph is still Hamiltonian. the number f of faces. Prove Dirac's theorem : if the degree of each vertex in a graph is at least half the total number of vertices.6. that if a graph has "enough" edges. Find a proof of Euler's formula by induction on the number of edges.1 that happen to be planar. Euler's formula says that v − e + f = 2 for every connected planar graph. and that there are no multiple edges or loop edges).9.3. for n ≥ 3. Exercise 3. Find a proof of Euler's formula by induction on the number of faces. count the number v of vertices. and then compute the quantity v − e + f . 1. For those examples of graphs in section 3. If new edges are added to a Hamiltonian graph (without changing the set of vertices).3.3. Exercise 3. A graph is said to be simple if it has no multiple edges nor loops. Take a non-closed path through all the vertices and examine the collections of vertices adjacent to the beginning and the end of the path. the number e of edges. Show that a complete graph is Hamiltonian. add edges until it is "just barely" non-Hamiltonian. Exercise 3. CLASSROOM DISCUSSION
31
It is noteworthy that finding a characterization of Hamiltonian graphs (analogous to Euler's theorem for Eulerian graphs) is an unsolved problem. then the graph is Hamiltonian. 2. so just looking a a graph with crossings doesn't determine whether or not it is planar.3.1 Many theorems about Hamiltonian graphs say.5.
3.12. also determine the number of colors needed to color its vertices so that adjacent vertices are different colors. Exercise 3. For bridge edges of G. and for each edge of G that is incident to two faces. Prove by induction on the number of vertices that every planar graph is (vertex) six-colorable. Every simple planar graph has a vertex of degree at most five.11. so it is enough to study vertex coloring problems (which are technically simpler than face coloring problems).13.32
CHAPTER 3. connected. Exercise 3. Exercise 3. crowd into the edges of their maps parts of the world which they do not know about. we get a loop edge in the dual graph.5
Coloring graphs
As geographers. the dual of a map is particularly simple to write down. 1. planar graph with at least three vertices. e ≤ 3v − 6 for every simple. Determine the duals of the five Platonic graphs.
3. adding notes in the margin to the effect that beyond this lies nothing but sandy deserts full of wild beasts. 3.) Since maps have no bridge edges (by definition). The complete graph K5 on five vertices is not planar. 2. For each Platonic graph. It is clear that face colorings of maps correspond to vertex colorings of their dual graphs. Plutarch Life of Theseus If G is a planar graph.
. (They turn out to be Platonic graphs again. determine the number of colors needed to color its faces so that adjacent faces are different colors.10. Sosius. we form the dual graph by placing a vertex in each face of G. and unapproachable bogs. GRAPH THEORY
Exercise 3. we make an edge in the dual graph that joins the vertices inside the two incident faces. Deduce the following facts from Euler's formula.
and show by example that this upper bound is attained. Is the converse true? Prove or give a counterexample. 1. Define the line graph L(G) to be the graph having one vertex for each edge of G. with an edge joining two vertices if and only if the vertices lie in different sets.s. two vertices of L(G) being joined by an edge if and only if the corresponding edges of G have a common vertex. Characterize the graphs that are isomorphic to their line graphs. Give a proof by induction. scene 1
Problem 3.4. Problem 3. Show that the line graph of an Eulerian graph is Hamiltonian. there must be either three people who all know each other.1. 2. Under what conditions on r. and indeed the sundry contemplation of my travels.s.4. then the graph has at most k 2 edges. 1.6. s. In any gathering of six people. Problem 3.5. compounded of many simples.
. or three people who are all strangers to each other. The mail carrier problem asks for necessary and sufficient conditions on a graph for the existence of a closed path that includes each edge of the graph exactly twice. and t). How many edges does Kr. Problem 3. but contains no triangles.t have? 2. 3.t an Eulerian graph? Problem 3.s.4
Problems
It is a melancholy of mine own.) Solve the problem. and t is Kr. Act iv. Problem 3. Prove that if G is Eulerian. The complete tripartite graph Kr. Let G be a graph (with no loops or multiple edges). (A mail carrier must traverse both sides of each street. and reformulate it as a statement about graphs.t consists of three sets of vertices (of sizes r.2.3. PROBLEMS
33
3. then so is L(G). William Shakespeare As You Like It. s. extracted from many objects.3. Prove this. in which my often rumination wraps me in a most humorous sadness. Tur´ an's extremal theorem: If a graph (with no loops and no multiple edges) has 2k vertices.
Applications of subgraph enumeration. on the other hand. Problem 3. Rosen.) Problem 3.
. GRAPH THEORY
Problem 3. and the icosahedron. pages 241–262. Each guest knows n of the other guests. 1.8. Show that the (nonplanar) complete graph K5 on five vertices can be drawn on the surface of a torus in such a way that no edges cross. the cube. 2.3 can be drawn on the surface of a torus (donut) in such a way that no edges cross. Exhibit. At a dinner party. in Applications of Discrete Mathematics. Michaels and Kenneth H. and all of its vertex angles are equal. Characterize the graphs admitting a path that is simultaneously Eulerian and Hamiltonian. the dodecahedron. the octahedron. Tur´ an's graph theorem. or by using the fact (Exercise 3. edited by John G. 808–816.10) that a planar graph cannot contain K5 . (A regular polyhedron has congruent regular polygons for faces. 2. Prove by induction on the number of vertices that every planar graph is (vertex) five-colorable. McGraw-Hill.9.34
CHAPTER 3. Rispoli. Problem 3. 1991. chapter 14.10. American Mathematical Monthly 102 (1995). a graph not of this type that is nonetheless simultaneously Eulerian and Hamiltonian. This can be done by adapting Kempe's false proof of the four-color theorem.7.5
Additional Literature
1. Apply Euler's formula v − e + f = 2 to prove that the only regular polyhedra are the tetrahedron.
3. Show that the (nonplanar) utilities graph K3. Problem 3. Show that it is possible to seat the guests so that each is between two acquaintances.11. there are 2n guests to be seated at a round table. Martin Aigner. Fred J.
4. Euler's φ function.1
Goals
1. Learn about and know how to use the Chinese Remainder Theorem to solve systems of equations exactly. that voluntarie move Harmonious numbers. unsolved problems that exist in number theory. William Collins 1720–1756 Ode to Simplicity
4. Appreciate the breadth of easily stated.
4. 2. John Milton 1608-1674 Paradise Lost.Chapter 4 Number Theory
In numbers warmly pure and sweetly strong. especially problems about the integers. lines 37–38 35
. Book III. 3. and Euler's generalization of Fermat's Little Theorem.2
Reading
Then feed on thoughts. Learn about Fermat's Little Theorem. Recall or learn basic facts about the integers.
A (complex) number is called transcendental if it is not algebraic. Definition. Notice that if a number α is a root of a polynomial p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 with rational coefficients. and x2 + 1 respectively. one uses a proof by contradiction to show that a number α is transcendental. x2 − 2. You are familiar with various classes of numbers: the integers Z. Suppose x is an integer larger than 1. . a n .3. Prove the Fundamental Theorem of Arithmetic by induction. Definition. but there are two other common designations for real (and complex) numbers: algebraic and transcendental.4. Arbuthnot. and the complex numbers C.
Theorem (Fundamental Theorem of Arithmetic). . line 127 As yet a child.3. Exercise 4. CLASSROOM DISCUSSION
37
4.1. 2. and the ni are positive integers. Then x can be written uniquely in the form
nm 1 n2 x = pn 1 p2 . . Often we separate the real numbers into the rational numbers and the irrational numbers. 2 and i = −1 are algebraic since they are roots of the polynomials x − 2. Showing that a number is transcendental is the same as showing that it is not the root of any polynomial with integer coefficients. Prologue to the Satires. . √ √ For example.3
4.1
Classroom Discussion
Basic Number Theory
Alexander Pope 1688-1744 Epistle to Dr. for the numbers came. a1 . nor yet a fool to fame.
where the pi are prime numbers ordered so that p1 < p2 < · · · < pm . Suppose
. A (complex) number is called algebraic if it is a root of some polynomial with integer coefficients. I lisp'd in numbers. then α is also the root of a polynomial with integer coefficients: namely d · p(x). . pm . where d is the least common multiple of the denominators of a0 . Since it is time consuming (not to say impossibly long!) to check every polynomial. We know that Z ⊂ Q ⊂ R ⊂ C. . the real numbers R. the rational numbers Q.
and e represent?
4. 23.
This is strange mathematics! The value x = 5 seems legitimate enough. Modern saying
Exercise 4. The first expedition to Mars found only the ruins of a civilization. You are probably aware that e and π are transcendental numbers. The god delights in odd numbers. . then what digits do the letters A.)
Virgil 70–19 b. Natural History. and z such that xn + y n = z n . Eclogue VIII. then there are no positive integers x. n.38
CHAPTER 4. The explorers were able to translate a Martian equation as follows: 5x2 − 50x + 125 = 0 ∴ x = 5 or x = 8.3. It is somewhat easier to prove the weaker statement that e and π are irrational numbers. Book xxviii.d.2
Unsolved Problems
Why is it that we entertain the belief that for every purpose odd numbers are the most effectual? Pliny the Elder 23–79 a.75
Pierre de Fermat (1601–1665) wrote the following claim in the margin of a book by Diophantus and added that the margin was too small to contain the proof. y . but x = 8 requires explanation. ? The trick is to find a contradiction. Take care of Number One. If n is an integer greater than 2. If the Martian number system developed in a manner similar to ours.2. Theorem (Fermat's Last Theorem). (Numero deus impare gaudet. If Annebase 8 − Annebase 5 = Annebase 7 . You will explore this in the reading.
.c.3. sec. NUMBER THEORY
that α is algebraic: then . These proofs are generally not easy. how many fingers did the Martians have? Exercise 4. .
Unsolved Problem 4. Doug Hensley. Annals of Mathematics (2). c." and if so it would have interesting consequences for Diophantine equations and formal logic. Actually the following mathematics was developed in the attempts to prove Fermat's Last Theorem. Modular elliptic curves and Fermat's last theorem. 2). c.1.) The betting is that the answer is "yes. 16.4. The difference of the difference is the second difference: (c − 2b + a. The following is his reply. e = (n + 4)2 ? (The difference of a sequence in ascending order (a. and recently Wiles and Taylor succeeded in completing a proof. does it follow that there exists an n so that a = n2 . b.1 The claim. Let π (x) denote the number of primes from 2 to x. 25. d. Do there exist positive integers x and y (both larger than 2 to avoid trivial counterexamples) so that π (x + y ) > π (x) + π (y )?
Andrew Wiles. 36). Annals of Mathematics (2). number 3. CLASSROOM DISCUSSION
39
For more than three centuries mathematicians tried to prove this claim. An example of such a sequence of squares is (4. Andrew Wiles and Richard Taylor. counting 2. e) of squares of positive integers in ascending order is (2. d = (n + 3)2 . 2. e − d). 553–572. If the second difference of a sequence (a. c = (n + 2)2 . 141 (1995). c − b. • Noetherian Rings • Elliptic Curves • Projective Plane • Cyclotomic Extensions • Modular forms • Shimura and Taniyama-Weil Conjectures One of our resident number theorists. number 3. doesn't seem to have any use other than being pretty (excuse enough). 9.
1
. and counting x if it is prime. b = (n + 1)2 . d − 2c + b. Unsolved Problem 4. d. e) is the sequence (b − a. e − 2d + c). b. though curious in its own right.2. 443–551. was asked for some unsolved problems that he particularly likes and that are easy to state. who loves problems.3. Ring-theoretic properties of certain Hecke algebras. 141 (1995). d − c.
vol. and the race a life. by Richard K." but the prospects of hitting upon x and y in a random search are slim. one of your readings is taken from the second one. p. Samuel Johnson 1709–1784 Boswell's Life of Johnson. Problem Books in Mathematics.5. • Old and New Unsolved Problems in Plane Geometry and Number Theory. Book i. Guy. Springer-Verlag. For example.40
CHAPTER 4. The set of equivalence classes (mod n) is denoted by Zn . Earl Beaconsfield (1805–1881) Sybil. Prove that addition and multiplication are well defined in Zn . Chapter 13
Knowledge is more than equivalent to force. Round numbers are always false. iii. Dolciani Mathematical Expositions. Prince of Abissinia. No. Mathematical Association of America. Victor Klee and Stan Wagon. 1991. The first example probably involves very large x or y .3
Fermat's Little Theorem and Euler's Generalization
Samuel Johnson 1709–1784 The History of Rasselas. Chapter ii
.4. NUMBER THEORY
Again. Definition. and each moment is a day. and not by calendars. Volume 1.
4. We say a ≡ b (mod n) if and only if b − a is divisible by n. the betting is that the answer is "yes. • Unsolved Problems in Number Theory.
Definition. Prove that congruence (mod n) is an equivalence relation on the integers. What are the units (elements with multiplicative inverses) in Zn ? But what minutes! Count them by sensation.3. Exercise 4. Benjamin Disraeli. Exercise 4. 27 ≡ 13 (mod 7). 11. 226 (30 March 1778)
Here are some books of unsolved problems. 1981.
) Care must be used in dealing with historical dates because different countries adopted the Gregorian calendar at different times. . Monday = 1. 3. . 6). Derive a formula that gives the day of the week of any day of any year in the Gregorian calendar. • dN is the day of the week of March 1 in year N . 1996 = N = 100C + Y where C = 19 and Y = 96). or if Y = 0 and N is divisible by 400. For months. . 2. . • m is the month (1. work modulo 7: Sunday = 0. CLASSROOM DISCUSSION
41
Exercise 4. 12). . 1. Britain and its colonies did not adopt the Gregorian Calendar until 1752. work modulo 12. Saturday = 6. We were not the last to convert: Greece did not change over until 1923. . Next find the first day of month m in year N . . 31).6 (The perpetual calendar). • W is the day of the week (0. . .3. 4. • C is the century. . it is convenient to start in March: March = 1. 99). For example. April = 2. . 2. • Y is the particular year of the century (0. For days of the week. but the year 1900 is not a leap year. since the extra day in leap year is in February. . . Use the following notation. You can do this by finding a function f (m) that matches the shift in the day of the week from
. In 1995. 2. (The calendar in current use is called the Gregorian calendar because Pope Gregory set it up in 1582. 1. • N = 100C + Y is the year (for example. . • k is the day of the month (1. 5. . where ǫ equals 2 in a leap year and 1 otherwise. The year Y is a leap year if Y = 0 and Y is divisible by 4 (notice that Y is divisible by 4 if and only if N is divisible by 4). February 1984 is the 12th month of 1983 in this system.4. you can find dN by counting leap years from some reference date (say 1600). . Find dN . . . . For example. February = 12. 1. January and February are viewed as part of the previous year. March 1 was a Wednesday. Thus. Since dN ≡ dN −1 + ǫ (mod 7). . the years 1996 and 2000 are leap years. Proceed as follows.
But when loud Surges lash the sounding Shore. 3. 11 are relatively prime to 12.
. a luxury that does not always arise. to use Fermat's Little Theorem one must have a prime modulus.
Theorem (Fermat's Little Theorem). Suppose a is a positive integer. The hoarse. rough Verse shou'd like the Torrent roar. Hint: What is the average shift? You can express the function f by using the function ⌊x⌋ that represents the greatest integer less than or equal to x. Soft is the Strain when Zephyr gently blows. For example. Part II Exercise 4. so he made the following definition. Definition (Euler's φ Function). Fermat's Little Theorem is quite useful in computing (mod n). 5.
However. Then ap−1 ≡ 1 (mod p). φ(n) is the number of integers between 1 and n (inclusive) that are relatively prime to n. Here is an example mod 7: (12)53 = ((12)6 )8 · (12)5 ≡ 18 · 55 = (25)2 · 5 ≡ 42 · 5 ≡ 10 ≡3
(mod 7) (mod 7) (mod 7) (mod 7). Adjust for the k th day of the month and gather the final formula for W ≡ dN + f (m) + k − 1 (mod 7).42
CHAPTER 4. When n is a positive integer.7. Alexander Pope (1688–1744) An Essay on Criticism. Use induction to prove Fermat's Little Theorem. And the smooth Stream in smoother Numbers flows. 7. and p is a prime number that does not divide a. φ(12) = 4 because the four numbers 1. NUMBER THEORY March to month m. Euler noticed that p − 1 is the number of units in Zp when p is prime.
10. .
Suppose we want to solve the pair of congruences x ≡ 4 (mod 7) and x ≡ 14 (mod 30) for x. . has a solution. . Every system of congruences x ≡ a1 x ≡ a2 . Furthermore. If x and n > 1 are positive integers that are relatively prime. CLASSROOM DISCUSSION Exercise 4.) Exercise 4. (You may want to start with the case that s and t are primes. then xφ(n) ≡ 1 (mod n). line 125
And wisely tell what hour o' the day The clock does strike. . Use induction to prove Euler's generalization of Fermat's little theorem.
4. Find a formula for φ(n) as follows.9.3. Theorem (Chinese Remainder Theorem).
.8.4. every two solutions are congruent mod M . Theorem (Euler's generalization of Fermat's little theorem). Compute 2586 (mod 21) by applying the previous theorem. Prove that φ(st) = φ(s)φ(t) when s and t are relatively prime.11. Find φ(pr ) when p is a prime.3. Canto i. Exercise 4. Exercise 4. (This could be asking to find a day of the week and a time of the month in terms of the entire year. x ≡ ak (mod m1 ) (mod m2 ) (mod mk ). by algebra.) Is there a solution? What is a good way to find it? The ancient Chinese worked out a method that is still computationally viable. Compute φ(54) by applying the result of the previous exercise. Part I.4
Chinese Remainder Theorem
Samuel Butler (1612–1680) Hudibras.
43
2. where M = m1 m2 . 1.
where the mi are pairwise relatively prime. mk .
and Mi to get a formula for a solution x0 . 4.14. Solve the following system of equations. quarter" quest. Suwarrow stop such sanguinary sounds! Alliteration. round-off errors can be a major problem.3. Use the Chinese Remainder Theorem to solve the following pair of congruences for x.44
CHAPTER 4. 3. Prove the Chinese Remainder Theorem constructively as follows. Exercise 4. If Mi = M/mi .5
Exact Solutions to Systems of Equations
Now noisy. then x1 ≡ x0 (mod M ). or the Siege of Belgrade: a Rondeau In this age of computers. Show that if x1 ≡ ai (mod mi ) for all i. persecution's pest! Quite quiet Quakers "Quarter. first by picking a large enough prime and using the procedure of the reading. it is to our advantage to solve systems of equations using integer arithmetic. religion. redounds. 5x1 − 3x2 = 3 4x1 − x2 = 6
. x ≡ 4 (mod 7) x ≡ 14 (mod 30)
4. NUMBER THEORY
Exercise 4. Exercise 4. A method for doing this was in the reading. Find a way to determine ci so that ci Mi ≡ 1 (mod mi ). Consequently. and then by picking a large enough composite of small primes and using the procedure of the reading and the Chinese Remainder Theorem. Use the ai . then what is gcd(Mi . Of outward obstacles o'ercoming ought.12. ci . noxious numbers notice nought. right. mi )? 2. Poor patriots perish. Reason returns.13. 1.
That is an order!)
4. they do but reckon by them: but they are the money of fools. Part I.4. but this joke came out of the depression.4
Problems
For words are wise men's counters. (Have fun. The following are some fun ones for you to solve. • a divides b or a | b (for integers a and b) • prime number • relatively prime • gcd or greatest common divisor • lcm or least common multiple • division algorithm • Euclidean algorithm • linear combination Number theory has intrigued many people.4.4.1. The different letters represent different digits. Thomas Hobbes (1588–1679) Leviathan. non-mathematicians and mathematicians alike. Chapter IV
Problem 4. please review the following concepts that are often used in number theory. Find them.2. Problems in number theory come in many varieties at various levels of complexities. PROBLEMS
45
4. U SA + F DR = N RA U SA + N RA = T AX
. The dep ression was no joke.1
Cryptoanalysis
Problem 4. So that we may start with a common vocabulary.
and so on. the Erewhon Daily Howler carried the following item: "The famous astrologer and numerologist of Guayazuela.7. find the smallest such positive integer. Away I go. Show that q is composite. 1770. His prediction is based on profound mathematical and historical investigations. 5. p2 − p1 = 2). 000. 027 by hand. the Professor Euclide Paracelso Bombasto Umbugio.2
Other types of number puzzles
Problem 4. Scene i What I tell you three times is true. Not all large numbers are hard to factor.4. if not. without much work. On April 1. show why none exists. Problem 4. William Shakespeare The Merry Wives of Windsor. Now.10. 3.4. and so can be written in the form 2q . 2. the Boston Massacre. Professor Umbugio computed the value of the formula 1492n − 1770n − 1863n + 2141n for n = 0. up to 1945. Problem 4.11. and found that all the numbers which he obtained in many months of laborious computation are divisible by 1946. Show that an integer in an odd base system is odd in the base 10 system if and only if it has an odd number of odd digits. and 1863 represent memorable dates: the Discovery of the New World. 1946. I hope good luck lies in odd numbers.9. If p1 and p2 are consecutive odd primes (that is. 1. and which ends in the digits 11? If so. predicts the end of the world for the year 2141." Deflate the professor! Obtain his result with little computation. This is the third time.)
. Problem 4. PROBLEMS
47
4. either in nativity. then p1 + p2 is even. (For example. They say there is divinity in odd numbers. Find all the prime factors of 1. In which bases b are 35base b and 58base b relatively prime? Problem 4. the numbers 1492. 3. Lewis Carroll The Hunting of the Snark: an Agony in Eight Fits Problem 4. chance. Does there exist a positive integer whose prime factors include at most the primes 2.4.8. 111base 3 is 9 + 3 + 1 = 13 in base 10 and is odd. or death. Act V. obviously. 7. and the Gettysburg Address. What important date may 2141 be? That of the end of the world.12.
Dolciani Mathematical Expositions. 1985. by Maxey Brooke. (200)37 (mod 21). 1. (3100)76 (mod 17). just give a method.16. 2. Dover. Mathematical Association of America.
3x1 + x2 = 1 2x1 + 3x2 = 2
.14 (Perpetual Calendar). New York. No. Mark I.48 The above puzzles were taken from
CHAPTER 4. Use Fermat's Little Theorem and modular arithmetic to compute the following by hand (not computer). 2nd rev. Give an answer between 0 and 20. 2. Larson.. 1993.15. by George T. 3. Conjecture a formula for which rows are all black. Use the Chinese Remainder Theorem and both the techniques (taking M prime or composite) from the reading "Exact Solutions to Systems of Equations" to solve the following system.13 (Fractals with Moduli). New York. 14. ed. NUMBER THEORY
• 150 Puzzles in Crypt-Arithmetic. 1. Trigg. by Charles W. Krusemeyer. Problem 4. • The Wohascum County Problem Book. Prove that n33 − n is divisible by 15 for every positive integer n. Gelbert. True or False: The 13th of the month falls on Friday more often than any other day. Dover. How might you go about justifying your answer? If the method is long or tedious. See if you can prove your formula. and Loren C. Give an answer between 0 and 16. • Mathematical Quickies. 1969. For the first 10 lines of Pascal's Triangle. Problem 4. Problem 4. Problem 4. Is the probability that Christmas falls on a Wednesday equal to 1/7? Prove or disprove. replace the odd numbers by black squares and the even numbers by white squares.
volume x. that the greatest happiness of the greatest number is the foundation of morals and legislation. line 8
49
.4. which procures the greatest happiness for the greatest numbers. PROBLEMS Priestley was the first (unless it was Beccaria) who taught my lips to pronounce this sacred truth. page 142) That action is best. section 3. Concerning Moral Good and Evil. Francis Hutcheson (1694–1746) Treatise II.4. Jeremy Bentham (1748–1832) The Commonplace Book (Works.
.
Shrines to no code or creed confined. not its utility.1
Goals
1. The Meccas of the mind.
Fitz-Greene Halleck (1790–1867) Burns
51
. 3. 2.
5. Alfred. Learn some cryptography. especially what RSA codes are and how to use them. Lord Tennyson (1809–1892) Aylmer's Field
5. Learn about error-correcting codes and their uses. the Palestines.2
Reading
Such graves as his are pilgrim shrines.Chapter 5 Codes
Mastering the lawless science of our law. Gain an appreciation of the usefulness of some mathematics that was originally studied for its beauty.— The Delphian vales.— That codeless myriad of precedent. That wilderness of single instances.
1. CLASSROOM DISCUSSION
53
equivalents p1 .3. decipher the message WBRCSL AZGJMG KMFV. find e. cn .5. . and the block size is 2. Does RSA encryption guarantee that the message is obscured? Suppose that n = 15.2
RSA Code
Exercise 5.) Exercise 5. And dies among his worshippers.6. 2. Given that n = 65 and d = 11 in an RSA code. The Evil Empire thinks it is clever. exercises 17 and 18. Their cryptographers tell the world to send them messages in an RSA code with n = 10573 and e = 2531 and claim that this is a secure method. writhes with pain. . but have forgotten how to factor numbers as large as n. .) Using a Vigen` ere cipher with key SECRET and setting A = 0.3. wounded. encipher the message DO NOT OPEN THIS ENVELOPE. set A = 1. pn to obtain a ciphertext block of letters with numerical equivalents c1 .3. How many of the allowable code blocks are encoded to themselves when e = 3? when e = 5? That is. (Rosen. . . (For the letter/number correspondence. . so that people know theorems such as the Fundamental Theorem of Arithmetic.3. we use a sequence of shift ciphers with ci ≡ pi + ki (mod 26) for each i.5. . How can the Spooks of Goodguyland now decode all the messages that the Evil Empire receives?
5. Exercise 5.— The eternal years of God are hers.4. how many X are there such that 0 ≤ X ≤ 14 and X e ≡ X (mod n) when e = 3? when e = 5? Exercise 5. They know that education in Goodguyland has deteriorated. . William Cullen Bryant (1794–1878) The Battle-Field
. Exercise 5.
5.2. Use the square and multiply method to decode the message 28717160 when n = 77 and d = 13. A clever agent from Goodguyland steals the information (bribery is suspected) that φ(n) = 10368. But Error.3
Error-Correcting Codes
Truth crushed to earth shall rise again. page 243.
Find a parity check matrix for C . 2. Let C be the binary code 1 0 0 0 0 1 1 0 0 0 1 1 1.10. m&m's.9. Calculate the probability of decoding correctly assuming that the probability of correct transmission of a bit is p = 0. u).) whose generator matrix is 1 1 0 1 .54
CHAPTER 5. w) ≤ d(u. 3.9.3. 5. 3. What is the probability of receiving a message correctly if no coding is used? Exercise 5.
. Make a syndrome and coset leader table. 3) code. Prove that the distance function on code words is a metric. Prove that the minimum distance between two code words in a code C is d if and only if C can correct ⌊(d − 1)/2⌋ or fewer errors via maximum-likelihood decoding. (Here q = 3 and r = 2. v ) = 0 if and only if u = v . 1.7. Exercise 5. d(u. 1 0
5. such as matches. d(u. 2. CODES
Exercise 5. that is. tokens. w). 6. Construct the standard array for C . v ) + d(v. the distance function satisfies the following three properties. v ) = d(v. Use the table you constructed to decode 101111 and 111111. Exercise 5. poker chips. Construct a standard array for the ternary Hamming (4. d(u.8. Nim is a two-person game that can be played with any small objects.4
Nim
The game of Nim can be solved by using number theory or by using an area of mathematics called game theory. 4. Determine the syndromes for C . 2. We are going to discuss the game of Nim as a bridge between the two areas.
5.4. PROBLEMS
55
chocolate chips. We will assume the use of matches. To start the game, some piles of matches (it doesn't matter how many piles, but three is typical), each with an arbitrary number of matches, are placed on a flat surface. Each player in turn can take as many matches as desired, but at least one, from any one pile. The person who takes the last match wins. (In an alternate version of the game, the person who takes the last match loses.) For example, suppose there are 2, 7, 6 matches in the initial piles. If Player A chooses to take three matches from the second pile, then there are 2, 4, 6 matches in the piles. If Player B takes all 6 matches of the third pile (leaving 2, 4, 0 matches in the piles), then Player A should take two matches from the second pile (leaving 2, 2, 0). If Player B takes both matches from one pile, Player A can take both matches from the other pile and win. If Player B takes one match from a pile, then Player A should take one match from the other pile so that, whatever pile Player B chooses, Player A takes the last match and wins. Exercise 5.11. Devise a winning strategy for Nim in each of its versions. You might want to try a few games first.
5.4
Problems
Errors, like straws, upon the surface flow; He who would search for pearls must dive below. John Dryden (1631–1701) All for Love. Prologue
Problem 5.1. An affine transformation C ≡ aP + b (mod 26) was used to encipher the message PJXFJ SWJNX JMRTJ FVSUJ OOJWF OVAJR WHEOF JRWJO DJFFZ BJF. Use frequencies of letters to determine a and b and to recover the plaintext. Problem 5.2. Another type of cipher or cryptosystem is a replacement cipher: let τ be a permutation of the alphabet, and apply τ to each letter of the message. Frequency analysis is useful for breaking this type of code, just as it was in the shift cipher. Decode the following, which was encoded using a replacement cipher.
Problem 5.3. Pick n, d, and e to use in your own public key cryptosystem, and encrypt a message. Turn in the answer in two parts: first give the public information and the encrypted message, and then give your decryption key and the original message. Problem 5.4. If the probability of a digit being received correctly is 0.9, what is the probability of having a correct message after decoding the send-it-threetimes code with three information digits? How does this compare with the probability of receiving a three-digit message correctly without any coding? Problem 5.5. Find eight binary vectors of length 6 such that d(u, v ) ≥ 3 for every pair (u, v ) of the vectors. Problem 5.6. Is it possible to find nine binary vectors of length 6 such that d(u, v ) ≥ 3 for every pair (u, v ) of the vectors? Problem 5.7. Give generator and parity check matrices for the binary code consisting of all even weight vectors of length 8. Problem 5.8. If C is an (n, k, d) code with n > 1, prove that any vector of weight ⌊(d − 1)/2⌋ or less is a unique coset leader. Problem 5.9. Show that if d is the minimum weight of a code C , this weight d is even, and t = ⌊(d − 1)/2⌋, then there are two vectors of weight t + 1 in some coset of C . Problem 5.10. A generator matrix code has 0 1 1 0 1 1 A= 1 2 1 2 1 1 G = (I A) for the ternary (12, 6) Golay 1 1 1 1 1 2 2 1 0 1 2 2 . 1 0 1 2 2 1 0 1 2 2 1 0
3. Though a wide compass round be fetched. Understand what the Greeks meant by a number being constructible. My own hope is.) Consolation to Apollonius
1. Plutarch (circa 46–circa 120 a.
Robert Browning (1812–1890) Apparent Failure. That after Last returns the First. 4.Chapter 6 Constructibility
It 's wiser being good than bad.d. Understand what the Greeks meant by a figure being constructible. and not the length. That what began best can't end worst. It 's fitter being sane than mad. 2. Nor what God blessed once prove accurst. 59
. a sun will pierce The thickest cloud earth ever stretched. Learn the algebra of polynomial rings. Learn about extension fields. vii
6. It 's safer being meek than fierce.1
Goals
The measure of a man's life is the well spending of it.
So when the ruler changed. so they couldn't just put the points at the ends of a line segment and copy that segment elsewhere.2
Reading
1. William Shakespeare Macbeth. Take all the rest the sun goes round. but only the kind that govern a country. 1998. Their compasses would not stay open like ours do. Sue Geller. the cubit. Sue Geller.g. so did the length of the cubit and the foot. They did have a straight-edge with which to draw lines and a compass with which to draw circles. one of the "standard" units of measurement. Be able to determine whether or not a given real number is constructible. Factoring Polynomials. we will use the modern compass that can easily copy a line segment. Act I. In fact.
6. Another standard measure was the "foot" which was the length of the ruler's right foot.3
6. Give me but what this riband bound. king's) right arm from the elbow to the end of the middle finger.1
Classroom Discussion
Classical Constructions
There 's no art to find the mind's construction in the face. 2.60
CHAPTER 6. and all that 's fair.3. A narrow compass! and yet there Dwelt all that 's good. But the Greeks still wanted to create lengths and figures in a repeatable way. Algebra for Constructibility. 1997. but they did have a procedure to produce a reliable copy. For our purposes. scene 4
The ancient Greeks knew about rulers.
6. nor even an idea of standardized measurement—nor did the rest of the world. It was even worse when the ruler was a growing child! Think about what it meant to the economy to have a changing unit of length.
Edmund Waller (1605–1687) On a Girdle
. CONSTRUCTIBILITY 5.. They did not have rulers for measurement. was the length of the ruler's (e.
Exercise 6. α − β . William Shakespeare Macbeth. a perpendicular from a point to a line. Fire burn. and.6. a circle. Double. the oracle at Delos told the Athenians that a plague would end if they constructed a new altar to Apollo in the shape of a cube. how to trisect every angle. α/β (if β = 0). Show that α + β . as it comes to us through Theon of Smyrna. both have identities. but with double the volume of the existing one. an angle bisector. you too will show the non-constructibility of the classical Greek objects. they found no way to "double a cube" using straight-edge and compass constructions. and cauldron bubble. a perpendicular bisector. Suppose that α and β √ are constructible numbers. However. they could not figure out how to construct a regular heptagon (a seven-sided figure). and others showed in the 1800s that these constructions are impossible. scene 1
The straight-edge and compass constructions at which the Greeks failed were worked on for over two millennia until Gauss. they could make various geometric figures. For example. a regular hexagon. of course. he declared the meaning of the oracle to be not that Apollo required a new altar. double toil and trouble. the distributive laws hold. According to legend. and every non-zero element has a multiplicative inverse. In Exercise 6. or how to construct a square with the same area as a given circle. such that both operations are commutative and associative. a regular pentagon. Wantzel. Recall that a field is a set that is closed under two binary operations. and α (if α > 0) are constructible. In this chapter. We start by looking at what lengths can be constructed. Lindemann. Their idea was that a number ℓ (a length) is constructible if. CLASSROOM DISCUSSION
61
What the Greek mathematicians did was to start with a given unit of length and to work from there.1. but that the Greeks needed to pay more attention to mathematics. one can construct a line segment of length |ℓ| units in a finite number of steps using only a straight-edge and a compass. Act iv. shows that public relations is an old art. a square.3. αβ . However.1 you proved that the set of constructible numbers is a subfield
. they could construct an equilateral triangle. When Plato was consulted. The story recorded by Eratosthenes. called addition and multiplication. a line through a given point parallel to a given line. starting with a given unit length. By putting such line segments together (again with straightedge and compass). every element has an additive inverse.
. If F is a subset of K . Show that this set is a subfield of the field of real numbers. √ √ Exercise 6.
Matthew Green (1696–1737) The Spleen
Exercise 6. Likewise. b ∈ Q}. a line and a circle. Is this list complete. The constructible numbers are sometimes called the "surds". a circle in F has an equation of the form x2 + y 2 + ax + by + c = 0. b. although in keeping with the cognate word "absurd". a surd is strictly speaking an irrational number. Thus a line in F has an equation ax + by + c = 0.3. √ We can continue to build the surds by finding an extension field of Q[ 2 ] that is also contained in the surds. Since all our straight-edge and compass constructions are done with lines and circles. and two circles can be obtained using only field operations and extraction of square roots. I mind my compass and my way. where a. and c are elements of F . or are there other ways to obtain constructible numbers? For any subfield F of the real numbers we can think of the plane of F as the set of points in the real plane that have both coordinates in F . Precisely how we do this will have to wait for later. Now we have a list of procedures for constructing new surds from old ones.1. and c are elements of F . b. then we say that F is a subfield of K and that K is an extension field of F . all numbers that can be constructed from numbers in F can be obtained from a sequence of intersections of lines and circles in F . Though pleased to see the dolphins play.2. Show that the surds consist precisely of those real numbers that can be obtained from the rational numbers by applying field operations and taking square roots in some order a finite number of times. where a.62
CHAPTER 6. To prove the converse of Exercise 6. you must show that intersections of two lines. Yet what are all such gaieties to me Whose thoughts are full of indices and surds?
Lewis Carroll Phantasmagoria
We can build the surds one step at a time. and both F and K are fields under the same operations. Let Q[ 2 ] = {a + b 2 : a. CONSTRUCTIBILITY
of the real numbers that is closed under taking square roots.
multiplication. For out of the old fieldes.
6. A polynomial of degree n with coefficients in a field F (or in Z) has at most n roots in F (or in Z). as men saithe.5. And out of old bookes. Prove the following theorem and corollaries from the reading. Let R be a ring.6. neither do they spin. Let f.4. and extraction of square roots. but how do you know that it can't? To prove the impossibility. a ∈ R. Cometh al this new science that men lere. Matthew 6:28 We start with some material on polynomial rings as defined in the reading. we need to be able to construct 3 2. f (a) = 0) if and only if the first-degree polynomial x − a is a factor of f . and f ∈ R[x]. Corollary 1 (Remainder Theorem). The number a is a root of the polynomial f (that is. Cometh al this new corne fro yere to yere. which doesn't look like it can be done by a sequence of the operations of addition. division.3. they toil not. Prove the following result.6. where either r = 0 or deg(r) < deg(g ). CLASSROOM DISCUSSION
63
√ In order to double a cube. subtraction. how they grow.2
Polynomials and Field Extensions
Consider the lilies of the field. where F is a field and g = 0 (or F is a ring and g is monic).
. Corollary 2 (Factor Theorem). Then there exist unique polynomials q. g ∈ F [x]. Then there exists a polynomial q ∈ R[x] such that f (x) = (x − a)q (x) + f (a).3. r ∈ F [x] such that f = qg + r. we need to take a side path into the area of algebra called field extensions. line 22
Exercise 6. Theorem (Division Algorithm). Corollary 3. Exercise 6. in good faithe.
Geoffrey Chaucer (1328–1400) The Assembly of Fowles. Exercise 6. Use the first isomorphism theorem and the evaluation homomorphism to show that C ∼ = R[x]/(x2 + 1).
9.7. Exercise 6.3
Constructibility
If you choose to represent the various parts in life by holes upon a table. βn } be a basis for K over E . the doer and the thing done. Prove that if F ⊆ E ⊆ K is a tower of finite field extensions. αm } be a basis for E over F . but we need only the direction stated above and the proof of the other direction uses material we have not studied. If f ∈ F [x] is irreducible. where irr(α. Let α ∈ R. Then α is constructible =⇒ [Q[α] : Q] = 2n for some non-negative integer n.3.64
CHAPTER 6. and let {β1 . . Q)) = 2n . we can prove a theorem that will allow us to test when a number is constructible. Corollary 4. Actually the implication in the above theorem is an equivalence (if and only if). Sydney Smith (1769–1845) Sketches of Moral Philosophy Now that we have the algebra machinery at hand. and α is an element of K that is algebraic over F . we shall generally find that the triangular person has got into the square hole. Can you construct a basis for K over F ?)
6. . . The officer and the office. some square. of different shapes.
. then F [α] ∼ = F [x]/(m). the oblong into the triangular. . and a square person has squeezed himself into the round hole.—and the persons acting these parts by bits of wood of similar shapes. then [K : F ] = [K : E ] [E : F ]. . Exercise 6. α is constructible =⇒ deg(irr(α. Exercise 6. If m is the minimal polynomial of α over F .8. then there is an extension field of F that contains a root of f . Use the preceding theorem to prove the following corollary. Q) is the minimal polynomial of α over Q. CONSTRUCTIBILITY
Theorem. some triangular. (Hint: Let {α1 . Equivalently.—some circular. Prove the above theorem. . . . some oblong. Theorem. Suppose that F is a subfield of K . seldom fit so exactly that we can say they were almost made for each other.
The question then arises of how to find a minimal polynomial for cos θ. Exercise 6. CLASSROOM DISCUSSION
65
Now it is easy to tell if a number is constructible. then cos 3θ = cos 30 = √ 3/2. Since 3/2 is not a rational number. For example. Since cos nθ + i sin nθ = (cos θ + i sin θ)n . However. (Alternatively. x3 − 2 is irreducible over Q. 2. and so we cannot double a cube using only straight-edge and compass. the angle θ is constructible. the binomial expansion implies that cos 3θ = Re((cos θ + 3 2 i sin θ)3 ) = cos3 θ − 3 cos θ sin2 θ = cos θ) = 4 cos3 θ − 3 cos θ. Exercise 6. so cos 10 satisfies the polynomial 3 equation 3/2 = 4x − 3x. For example. Show that there is at least one angle that cannot be trisected using straight-edge and compass. one could directly express cos 6θ in terms of cos θ. 1. Show that a regular pentagon is constructible using straight-edge and compass. In the next exercise you will see what angles are constructible by showing that an angle θ is constructible if and only if cos θ is constructible. squaring both sides shows that cos 10◦ satisfies the polynomial equation 3/4 = 16x6 − 24x4 + 9x2 . √θ − 3 cos θ(1 − cos ◦ ◦ ◦ If θ = 10 √ .11. which is not a power of 2.10. Show that a square with the same area as a circle of unit radius is not constructible with straight-edge and compass. A common way to do this is to choose a positive integer n for which we know that cos nθ is constructible. and to relate cos θ to cos nθ by De Moivre's Theorem. We know √ 3 that 2 is a root of x3 − 2. it follows that cos 10◦ is not constructible. the length sin θ is constructible. we do not yet have a candidate for a minimal polynomial for cos 10◦ over Q. we have in particular that cos nθ = Re((cos θ + i sin θ)n ). Thus 2 is not constructible.3.) An equivalent equation is 0 = 64x6 − 96x4 + 36x2 − 3. and since 64x6 − 96x4 + 36x2 − 3 is irreducible by Eisenstein's criterion. and the other two roots are complex numbers.
. Show that the following are equivalent: the length cos θ is constructible. Since none of the roots is rational. We want to know if 2 is constructible. 3.6. Therefore √ √ 3 3 [Q[ 2] : Q] = 3. consider √ 3 the case of doubling a cube.
66
CHAPTER 6. CONSTRUCTIBILITY
6.4
Problems
In arguing too, the parson own'd his skill, For e'en though vanquish'd he could argue still; While words of learned length and thundering sound Amaz'd the gazing rustics rang'd around; And still they gaz'd, and still the wonder grew That one small head could carry all he knew. Oliver Goldsmith (1728–1774) The Deserted Village
Problem 6.5. Show that a regular heptagon (7-gon) is not constructible using straight-edge and compass. Problem 6.6. Show that a regular 10-gon is constructible using straight-edge and compass. Problem 6.7. Show that a regular 20-gon is constructible using straight-edge and compass. Problem 6.8. Show that a regular 30-gon is constructible using straight-edge and compass. Problem 6.9. Show that an angle of 72◦ is constructible using straight-edge and compass. Problem 6.10. Show that a regular 15-gon is constructible using straightedge and compass. Problem 6.11. Conjecture and prove what you can about the values of n for which a regular n-gon is constructible using straight-edge and compass. (For example, if a regular n-gon is constructible, is a regular 2n-gon, a regular n/2-gon (for n even), a regular 3n-gon?)
" American Mathematical Monthly 64 (1957). Part I. John G. Charles Lamb 69
. Learn the basic principles of the mathematical theory of games.Chapter 7 Game Theory
The game is up.1
Goals
1. 1993.2
Reading
1. scene 2
7. Mathematical Association of America. 2. "Game-theoretic solution of baccarat. 3. Realize that mathematics does not prescribe a best "solution" for all games. 465–469.3
Classroom Discussion
A clear fire. Straffin. 2. William Shakespeare Cymbeline. Game Theory and Strategy. a clean hearth. Be able to apply the principles of game theory to analyze both abstract games and real-life situations.
7. Kemeny and J. and the rigour of the game.
7. Act iii. pages 3–61. Laurie Snell. Philip D.
1
Warm up
The reading introduced the following concepts from the mathematical theory of games. Lord Byron (1788–1824) Age of Bronze. Whose table earth. What does this theorem say about the games you listed in the previous exercise? 2. The minimax theorem of John von Neumann says that every m × n matrix game has a solution. whose dice were human bones. • dominance • saddle point • mixed strategy • value of a game
7. GAME THEORY (1775–1834) Mrs.2. Does this mean that some of these games are uninteresting to play? Exercise 7. Review the definitions of the following concepts. Stanza 3
.2
Examples of games
Whose game was empires and whose stakes were thrones.3. 1. • two-person game • zero-sum game • perfect information Exercise 7.1. List some familiar games.70
CHAPTER 7. To which of them do the above terms apply? Exercise 7.3.3. Battle's Opinions on Whist
7.
Straffin. You may not find the solution satisfactory in all respects. 0)
For instance. In the Monty Hall game. Mathematical Association of America. Straffin. 1993. from Philip D. 1993. Game Theory and Strategy. Since both players want to maximize their payoffs. Mathematical Association of America.4.5.1 A B C D A 1 2 2 2 B C 1 1 1 1 2 1 2 2 D E 2 2 1 2 1 1 1 0
71
Exercise 7. the payoffs are 0 to the row player and 1 to the column player. Exercise 7. two of which conceal goats. the host of a game show offers the contestant the choice of three doors. this strategy would not be optimal for either player. displaying a goat. the host does not open it. 2 Problem 4. 3) B (0. if both players use pure strategy A.
Problem 2. but instead opens one of the other doors. page 11.3. 4) (1.3.6.2 A B C A (0. or of switching and taking whatever is behind the third door. Solve the following 4 × 5 matrix game. 1) (5. it suffices to write the payoffs to the row player. 4) C (2. 2) (1. 0) (1. page 72. or does it matter?
7. Find what you think is a reasonable solution to the above game. since the payoffs to the column player are the negatives of the payoffs to the row player. it is necessary to write the payoffs to both players. Game Theory and Strategy. After the contestant chooses a door.
1
. Should the contestant switch or not. The contestant now is offered the choice either of taking whatever is behind the originally chosen door.3
Generalizations
In a zero-sum game. since non-zero-sum games admit puzzling phenomena such as the prisoner's dilemma. from Philip D. and one of which conceals a new automobile. Here is an example of a 3 × 3 non-zero-sum two-person game. CLASSROOM DISCUSSION Exercise 7. 1) (4. 1) (4. In a non-zero-sum game.7.
What would be a reasonable strategy for playing the three-person zero-sum game shown in Figure 7. Solve the following 4 × 3 matrix game. Game Theory and Strategy. Rose and Colin play a game in which they simultaneously display one side or the other of their cards.1.
Matthew Prior (1664–1721) To the Hon. Richard Brinsley Sheridan (1751–1816) The Rivals. Mathematical Association of America. In either case. Mathematical Association of America.7. 1. from Philip D. If two players could form a coalition to play in concert against the third player. the payoff to the winner is the face value of the winner's card. Rose wins if the colors match.1?3 2. aim At objects in an airy height. which two players would most likely team up?
7. Straffin. page 22. Straffin.2.4
Problems
Our hopes. Game Theory and Strategy.4 A B C D A B 5 2 4 1 3 4 1 6 C 1 3 3 2
Problem 7. scene 3
Exercise 7. 4 Problem 5b. Rose holds a double-faced playing card made by gluing the ♠A back-to-back with the ♥8.72
CHAPTER 7. 1993. like towering falcons. Act v. The little pleasure of the game Is from afar to view the flight. you won't be so cantankerous as to spoil the party by sitting out. page 127. (Here the value of an ace is 1.)
Example taken from Philip D. GAME THEORY As there are three of us come on purpose for the game.
3
. Charles Montague
Problem 7. Colin wins otherwise. Colin has a similar card made by gluing the ♦2 back-to-back with the ♣7. 1993.
Theorems in a formal axiomatic system are statements 75
.
8. 9. Stanislaw Lem. consistency. 1968. Robert Gray. Vilenkin. Ya. even at the most basic level.1 Goals
1.3. Be aware that mathematics is open-ended. no. in Stories about Sets by N. 2.
8. Academic Press. Understand the fundamental theorems of Cantor and G¨ odel about set theory.3
8. 3.1
Classroom Discussion
The axiomatic method
A formal approach to mathematics consists in specifying some undefined terms and some axioms.Chapter 8 Set Theory and Foundations
8. Georg Cantor and transcendental numbers. or the thousand and first journey of Ion the quiet". "The extraordinary hotel. and independence. American Mathematical Monthly 101 (1994). 819–832.2
Reading
1. Understand the notions of axiomatic systems. 2.
an axiom is independent of the other axioms if it is false in some model that satisfies the remaining axioms. Axiom 3 For each trad. A model for a formal axiomatic system is a concrete realization of the undefined terms that satisfies all the axioms. Turin.2. University of Toronto Press. can be derived as theorems from a small set of fundamental principles.3.76
CHAPTER 8. 1.
8. see "The principles of arithmetic. and distributive laws. like chess. trad Axiom 1 Every ag kows at least two trads. Axiom 4 The set of trads is non-empty. He showed how the basic principles of arithmetic.
1
. associative.1
Peano's original work is a booklet Arithmetices principia nova methodo exposita. Exercise 8. at least one ag kows it. In a consistent system. An axiomatic system is consistent if it admits at least one model. SET THEORY AND FOUNDATIONS
about the undefined terms that can be deduced logically from the axioms. Kennedy.1. 2. Axiom 2 There is at least one trad that every ag kows. kow." His Grundlagen der Geometrie presented an axiomatic development of Euclidean geometry. 1889. For an English translation. Undefined terms ag. Construct a model showing that this axiomatic system is consistent. with no intrinsic meaning: "Mathematics is a game played according to certain simple rules with meaningless marks on paper. translated and edited by Hubert C. Peano's five axioms for the natural numbers are the following. Is each axiom independent of the other axioms? Exercise 8. pages 101–134. 1973. Bocca. The axiomatic method is especially associated with the name of David Hilbert. presented by a new method" in Selected Works of Giuseppe Peano.2
Peano's axioms
Giuseppe Peano formalized arithmetic as an axiomatic system. Consider the following formal axiomatic system. Prove that there are at least two trads. who viewed mathematics as a particularly elaborate game. such as the commutative.
• a + 1 = a′ for every natural number a. If 1 ∈ S .4. No sequence of real numbers exhausts an interval. then S is the set of all natural numbers. two sets have the same cardinality if each can be put into one-to-one correspondence with a subset of the other.
8. Two sets have the same cardinality if they can be put into oneto-one-correspondence with each other. Using only Peano's axioms. Peano 5 Axiom of Induction. then a = b if and only if a′ = b′ . CLASSROOM DISCUSSION Peano 1 There is a natural number denoted by 1. Using the induction axiom. the condition a ∈ S implies that a′ ∈ S .3.8. b. a natural number denoted by a′ . Let S be a set of natural numbers. (According to the Schroeder-Bernstein theorem.
. we have a′ = 1.3
Cantor's theorems
Georg Cantor was the first one to understand the notion of cardinality of infinite sets. Peano 3 If a and b are natural numbers. Exercise 8. Peano 4 The natural number 1 is not a successor: for every natural number a. and c. Peano defined addition of natural numbers by the following properties.
77
Peano 2 Every natural number a has a successor.) Cantor showed in particular that there is no one-to-one correspondence between the rational numbers and the real numbers. • a + b′ = (a + b)′ for all natural numbers a and b.3. prove that no natural number is its own successor. and every natural number other than 1 is a successor. Theorem (Cantor). Exercise 8. Prove the associative law of addition: (a + b) + c = a + (b + c) for all natural numbers a.3. and if for every natural number a.
the first two elements of the sequence determine an interval. Both the continuum hypothesis and the generalized continuum hypothesis are consistent with the usual axioms of set theory. Show that the power set of the positive integers has the same cardinality as the set of real numbers. P (S ) cannot be put into one-to-one correspondence with S . by Cantor's nested set procedure.4
The continuum hypothesis
Cantor's continuum hypothesis is the statement that every infinite set of real numbers can be put into one-to-one correspondence with either the integers or the whole set of real numbers.5.6. 4/5. The next two elements of the sequence that are in the first interval determine a second interval. changing each digit to a different one.
. The intersection of the nested intervals contains a number that is not in the sequence.3. .
8. this procedure creates a number that is not in the list. . In other words. 1/4. 1) as 1/2. by Cantor's diagonal procedure. but S can be put into one-to-one correspondence with a subset of P (S ). That is. Cantor's second proof is his famous diagonal argument. 2/5. Then. If done with a little care. Cantor was able to prove neither the continuum hypothesis nor the generalized continuum hypothesis : For every set S . 1/3. . Cantor says. 2/3. Exercise 8. SET THEORY AND FOUNDATIONS
Cantor's first proof is based on the proposition that a sequence of nested closed intervals has a nonvoid intersection. 3/4. Cantor's power set theorem says that the power set P (S ) (the set of all subsets of S ) has larger cardinality than the set S itself. 3/5. . Write the numbers of the sequence as a list of decimals. Theorem (Kurt G¨ odel (1940)). It turns out that there is a good reason for Cantor's failure. 2. 1/5.78
CHAPTER 8. Enumerate the rational numbers in the interval (0. Exercise 8. Read down the diagonal. and so on. there is no set of cardinality between the cardinality of S and the cardinality of the power set of S . there is no set of cardinality strictly between the cardinality of the integers and the cardinality of the real numbers. Construct the first few digits of an irrational number: 1.
8. similarly. If the system can prove its own consistency.9. G¨ odel uses a diagonal argument to show the existence of a number n that is assigned to the statement: "Sentence number n is not provable in the system. In 1931. Theorem (G¨ odel's first incompleteness theorem). Exercise 8. Why? Theorem (G¨ odel's second incompleteness theorem). each proof within the system has an assigned G¨ odel number. if the system is consistent.
. Contradiction. Both the continuum hypothesis and the generalized continuum hypothesis are independent of the usual axioms of set theory. A statement or formula is a string of symbols.") Namely.3.8. ("This sentence is false. then it can prove that it cannot prove proposition n. so the system has proved proposition n after all. What do the preceding two theorems say about the existence of certain models for set theory?
8. But this is just what proposition n states." This statement is true. that it is unprovable. so each statement within a formal system gets a number assigned to it. Indeed. Exercise 8.7. CLASSROOM DISCUSSION
79
Theorem (Paul Cohen (1963)). G¨ odel assigns to each symbol an integer. Kurt G¨ odel showed that any interesting formal axiomatic system must contain undecidable propositions: statements that cannot be either proved or disproved within the system. Exercise 8. and for a good reason. then we know that it cannot prove the above undecidable proposition n (which asserts its own unprovability). but neither it nor its negative can be proved within the formal axiomatic system. A consistent axiomatic system that contains elementary logic and arithmetic cannot prove its own consistency. Does G¨ odel's theorem mean that it is pointless to study mathematics? G¨ odel's proof is based on formalizing the liar paradox. Essentially.5
G¨ odel's incompleteness theorems
Hilbert's program of axiomatizing all of mathematics failed. Any axiomatic system that is consistent and that contains elementary logic and arithmetic must contain undecidable propositions.3.
Nor yet disproven: wherefore thou be wise. Use Peano's axioms to prove the validity of the commutative law a + b = b + a for all natural numbers a and b. Cleave ever to the sunnier side of doubt. How do you know that the proposition cannot be proved and cannot be disproved? Problem 8. Donald M.2. Davis.1. Edmund Landau. Alfred. SET THEORY AND FOUNDATIONS For nothing worthy proving can be proven. Foundations of Analysis: The Arithmetic of Whole. 1982.80
CHAPTER 8. Rational. William Dunham. Formulate a definition of multiplication of natural numbers based on Peano's axioms and use it to prove the distributive law (a + b) × c = (a × c) + (b × c).1. 1990. Birkh¨ auser.
. section 2. Rudy Rucker. Infinity and the Mind.3.5
Additional Literature
1. sections 37–41.1. Chelsea. New York. 1993. Journey Through Genius. Great Moments in Mathematics (After 1650). 1981. Steinhardt. Formulate a proposition that is undecidable in the axiomatic system of Exercise 8. third edition. The Nature and Power of Mathematics. translated from the German by F. Chapters 11– 12. 3. 1966. Problem 8. Princeton University Press. Howard Eves. 4. Mathematical Association of America. 5. Irrational and Complex Numbers. Wiley.
8.4
Problems
Problem 8. 2. Lord Tennyson (1809–1892) The Ancient Sage
8.
1 Goals
1. Does n=0 n! really converge? Infinite series and p-adic analysis.2
Reading
∞ 1 1. people who cooperated on other matters prospered.3
Classroom Discussion
Human life was once like a zero-sum game. more for one group meant less for another.Chapter 9 Limits
9. Where pastures. American Mathematical Monthly 103 (1996). Burger and Thomas Struppeck. K. Be able to apply your knowledge to new situations. According to the Humpty Dumpty 81
. 565–577. but to cooperate and build. and so our ancestors learned not just to grab. net benefits summed to zero. farmland.
9. and hunting grounds were concerned. number 7. Still. Renew your acquaintance with the notion of a limit. Eric Drexler. Humankind lived near its ecological limit and tribe fought tribe for living space.
9. Because one's gain roughly equaled the other's loss. Engines of Creation
The notion of a limit is a fundamental concept in the realm of continuous mathematics (calculus and analysis). 2. Edward B.
Indeed.3. . Sir Arthur Conan Doyle Sussex Vampire Victor Hugo Les Mis´ erables William Shakespeare Romeo and Juliet.1
Intuitive limits
Oscar Wilde Picture of Dorian Gray
To define is to limit. ii. 2/3." However. "it means just what I choose it to mean—neither more nor less. 1 For example.
The sequence of rational numbers 1/2. 10−k of 1 when n > 10k . l. LIMITS
principle. Charles Lamb Last Essays of Elia • There is a limit to a mother's patience. the nth term of the sequence equals n/(n + 1). the word "limit" most often conveys the idea of a boundary or a restriction." In ordinary discourse. II. Here at the quiet limit of the world. say. Here are some examples of this usage." Through the Looking-Glass
1
. evidently approaches the limit 1. the expression lima→0 a x−1/2 dx might be expressed in words as "the limit of the integral as the lower limit tends to zero. • A pun is not bound by the laws which limit nicer wit. Lord Tennyson Tithonus.1 the correspondence between concepts and words is not a one-to-one correspondence. the word "limit" has more than one mathematical meaning. or equivalently 1 − (n + 1)−1 . in rather a scornful tone." Humpty Dumpty said. . so we can be sure that the terms are within. This second notion is not disjoint from the first one: it may happen that a limiting value is also an extreme point. we shall be dealing with the notion of "limit" as a value to which one approaches arbitrarily closely. Alfred. 7
9. 4/5.82
CHAPTER 9. 3/4. • For stony limits cannot hold love out. . • Human thought has no limit.
"When I use a word. 67
This is the sense in which mathematicians use the word "limit" in the phrase "limit of integration.
and the sequence of partial sums is monotonically increasing.171717 . . How would you answer the question: "Does it ever get there?" Understanding the definition of the limit concept does not necessarily mean being able to compute numerical values of limits. What would its value be. . Indeed.9. so we can invoke the fundamental property of the real numbers that a bounded increasing sequence converges (in fact. . expressed as an ordinary rational number? 3. We say that this repeating decimal equals 17/99 because the value of the limit is 17/99.3. even without computation. 1." L. then I'll be through with them. Exercise 9. Marilla? There must be a limit to the mistakes one person can make. Suppose that 0. implicitly defines a limit: namely. That's a very comforting thought.171717 .
2. Montgomery Anne of Green Gables Exercise 9. . .2. converges to its least upper bound).3. For example. In what sense do the numbers n−1 cos(πn/2) get "closer and closer" to 0 as the integer n increases? The repeating decimal 0. 10 100 1000 10000 It is evident. the verification that √ nn e−n 2πn =1 lim n→∞ n!
.171717 . each partial sum is certainly less than 1. the sum of the infinite series 7 1 7 1 + + + + ··· .1. . Exercise 9. This is good intuition. don't you see. CLASSROOM DISCUSSION
83
It is often said that the sequence {n/(n + 1)}∞ n=1 has limit 1 because the terms get "closer and closer" to 1. that the limit exists. M. Aren't the terms of the sequence {n/(n + 1)}∞ n=1 also getting "closer and closer" to π ? "Oh. but suspiciously imprecise. and when I get to the end of them. Verify the value of the repeating decimal 0.. is reinterpreted as an expansion in base 8.
Immanuel Kant Critique of Pure Reason The most familiar way to represent real numbers is via decimal expansions. Show that lim
n→∞
k=1
n2
π n = .4. starting from an equilateral triangle with sides of length 1. as it ties together properties of the natural numbers (represented by the factorial function) with the special numbers e and π . although it may exist in a state of infinite rarefaction. The formula is indeed a beautiful one. 2 +k 4
The von Koch fractal snowflake curve is defined by an iterative process. we cannot. At each stage of the construction. which (as observed above) are limits of sums.3.
n
Exercise 9. it is advantageous instead to represent real numbers as limits of quotients. For many purposes. Edgar Allan Poe Hans Phaall
9. the very conception of which implies that it consists of knowledge altogether non-empirical and a priori. 1. ascend as high as we may. The first four stages of the construction are shown in Figure 9. What is the area enclosed by the limiting curve? 3.
. an outward-pointing equilateral triangle is erected on the middle piece. and that middle segment is deleted. every straight line segment is subdivided into three equal parts.1. arrive at a limit beyond which no atmosphere is to be found. LIMITS
is sufficiently subtle that the equality bears the name Stirling's formula.2
Continued fractions
I will then limit my assertion to pure mathematics. literally speaking. What is the perimeter of the limiting curve? It is therefore evident that.5. In what sense does the snowflake construction converge to a limiting curve? 2.84
CHAPTER 9. Exercise 9. I argued. It must exist.
1. Continued fractions play a minor role in one of the most romantic stories in the history of mathematics. − = −2π 2 2 e 1+ e−4π 1+ e−6π 1+ 1 + ···
. similarly for the odd-order truncations. Assuming that the limit exists. 3. H. Hardy in England. determine what its value must be.88
CHAPTER 9. 4. One of these formulas was a closed-form expression for a certain continued fraction: √ √ 1 5+ 5 5 + 1 2π/5 e . a poor Indian clerk who had flunked out of college.10.4. Srinivasa Ramanujan. similarly for the odd-order truncations. Use the fundamental property of the real numbers that a bounded monotonic sequence converges. and find its value. Exercise 9. Show that the limit exists in the above example.416¯ 6. determine what its value must be.5. Show that the even-order truncations form a monotonic sequence. 2 1 1+ = 1. wrote down some mathematical formulas that he had discovered and sent them to the famous G. In 1913. LIMITS
Some truncated versions of this fraction are 1 1 + = 1. More generally. assuming that the even-order truncations have a limit. 1+ 1 2+ 1 2+ 2 It would be reasonable to assign to the unending continued fraction the value of the limit of the truncated fractions. if the limit exists. 2. as follows. 1 2+ 2 1 = 1.
I had never seen anything in the least like them before. we need to be able to say when two quantities are close to each other. 2. Formally. but bein' only eyes. no one would have had the imagination to invent them.3. 137–155.3
The p-adic numbers
'Yes. . y ) = d(y.3. y ) ≥ 0 for all x and y (with equality if and only if x = y ). and he died three years later. Exercise 9. . 1. a2 . A single look at them is enough to show that they could only be written down by a mathematician of the highest class. p'raps I might be able to see through a flight o' stairs and a deal door. an ] (written as a fraction in lowest terms with positive denominator). Hardy immediately brought Ramanujan to England to work with him. . we need a distance function or metric. wondering what to make of this communication from an unknown Indian. you see my wision's limited. if they were not true. d(x. I have a pair of eyes. Of this continued fraction and two related ones. If x has the continued fraction expansion [a1 . Ramanujan's health failed in 1917. Cambridge University Press. . ch.]. with mixed success.11. Symmetry: d(x. 'and that's just it. let sn (x) and tn (x) denote the numerator and denominator of the nth approximant [a1 . CLASSROOM DISCUSSION
89
Hardy. . tried to prove Ramanujan's formulas. 1936.
2
. Recall the following defining properties of a metric d. American Mathematical Monthly 44 (1937). They must be true because. 1940. x) for all x and y .' Charles Dickens Pickwick Papers. Tragically. .
9. If they wos a pair o' patent double million magnifyin' gas microscopes of hextra power. published as The Indian mathematician Ramanujan.9. and reprinted in Hardy's book Ramanujan. Guess recursive formulas for sn (x) and tn (x).' replied Sam. . and prove your formulas by induction. Hardy later said2 that they defeated me completely.
Lecture delivered at the Harvard Tercentenary Conference of Arts and Sciences on August 31. 34 In order to talk about limits.
1. y ) = |x − y |. z ) + d(z. you will see only rational numbers. y ) for all x. then we may have a different set of convergent sequences. What are the convergent sequences of real numbers in the discrete metric? It is worthwhile to keep in mind that the familiar real numbers are actually a quite abstract notion. Verify that the discrete metric on the real numbers does satisfy the three properties of a metric. and z . Exercise 9.
So far we have been dealing only with the usual metric on the real numbers: d(x.4
√ You might be able to display an irrational number geometrically : 2 is the length of the diagonal of a unit square. 2. the elements of the completion are equivalence classes of convergent sequences.3 We normally exhibit irrational real numbers as limits of convergent sequences of rational numbers. A function with these properties is often called a norm in the context of vector spaces and a valuation in the context of fields. What valuation on the real numbers induces the discrete metric? 3. These properties are the following.
A standard example of a nonstandard metric is the discrete metric defined by d(x. nonrepeating decimal in numerical form without resorting to a limiting operation.90
CHAPTER 9. It is useful to isolate the properties of the absolute value function that enable us to define a metric in this way. There is no explicit way to display a non-terminating. This process of generating new numbers by taking limits of convergent sequences is known as completion. 2. 3. |x · y | = |x| · |y | for all x and y . y ) = |x − y | are consequences of the three properties of an absolute value. Triangle inequality: d(x. 4 More precisely. On a calculator or a computer. two sequences being equivalent if their difference tends to zero. |x| ≥ 0 (with equality if and only if x = 0). and their limits will define a new number system.
3
.13. If we change the metric on the rational numbers. y ) = 1 if x = y . Show that the three properties of a metric d given by d(x. y . y ) ≤ d(x. 1.12. Triangle inequality: |x + y | ≤ |x| + |y | for all x and y . Exercise 9. LIMITS 3.
|5|2 = 1. |25/96|2 = 32. 1975. pages 38–41. For example. the difference between the nth partial sum and the (n + k )th partial sum has dyadic absolute value no more than 1/2n . |4|2 = 1/4. Verify that | · |2 is a valuation on the rational numbers. Convergence with respect to the dyadic valuation |·|2 is dramatically different from convergence with respect to the ordinary absolute value. say 2k . for which rational numbers x and y is |x + y |2 = max(|x|2 . The series 1 + 2 + 4 + 8 + · · · + 2n + · · · converges dyadically to an integer: which integer?
This is the Greek philosopher of "Eureka" fame. When does equality hold in the strong triangle inequality? That is. | · |2 satisfies the strong triangle inequality : |x + y |2 ≤ max(|x|2 . then |n|2 ≤ 1. Extract from the product the power of 2. see "The Death of Archimedes" in his Apocryphal Stories. consider the infinite series 1 + 2 + 4 + 8 + 16 + · · · + 2n + · · · .17. Archimedes was killed by a Roman soldier during the sack of Syracuse in ˇ 212 b. Exercise 9. |6|2 = 1/2. which tends to zero as n increases without bound.3. non-standard valuation on the rational numbers.16. If n is an integer.15.
5
. For example. Penguin Books. This series converges dyadically! Indeed. Exercise 9.9. Completing the rational numbers when they are equipped with the discrete metric is uninteresting. A non-zero rational number r can be factored as a product of primes (some powers being negative). Why? We can create an interesting new number system by using a non-trivial. Exercise 9. they fail the axiom of Archimedes5 that if a positive quantity is added to itself a sufficient number of times. |y |2 ). CLASSROOM DISCUSSION
91
Exercise 9. we can define a valuation | · |2 (which we might call the dyadic valuation) on the rational numbers in the following way.18. and define |r|2 to be the reciprocal 1/2k . its value becomes bigger than any specified value. For a modern retelling of this legend by Karel Capek (inventor of the word "robot"). For example. Moreover. As related by Plutarch in his biography of Marcellus. For example.c.14. |y |2 )? The rational numbers with the dyadic valuation appear strange at first sight. Exercise 9.
a series of the form ∞ j =−∞ aj 2 . For example. Evidently. for the beginning is a limit. |17/99|3 = 9 and |100/33|5 = 1/25.20. the series 1+ 1 1 1 + + ··· + n + ··· 2 4 2
CHAPTER 9. but in general they converge to elements of the dyadic completion of the rational numbers. but not every limit is a beginning.
1. Thus a p-adic number can be represented as a series a− 1 a− k + ··· + + a0 + a1 p + a 2 p 2 + · · · k p p where each coefficient aj is an integer between 0 and (p − 1) inclusive. LIMITS
j diverges dyadically since |1/2n |2 = 2n . where each aj is either 0 or 1. Find the dyadic expansion of 2/3. The p-adic numbers are the completion of the rational numbers with respect to | · |p .92 On the other hand. Such series may converge to rational numbers. Use long division to find this expansion.19. A non-Archimedean valuation | · |p can be defined similarly for every prime number p. converges dyadically if and only if there are only a finite number of non-zero coefficients aj with negative indices. and yet more. Indeed. Aristotle Metaphysics
.
2. Find the dyadic expansion of the current year. Exercise 9. 'limit' has as many senses as 'beginning'.
Exercise 9. therefore. The quotient 1 · 30 + 1 · 31 + 1 · 32 + 1 · 33 + 1 · 34 + 1 · 35 + · · · 1 · 30 + 2 · 31 + 1 · 32 + 2 · 33 + 1 · 34 + 2 · 35 + · · · of 3-adic expansions can itself be expressed as a 3-adic expansion.
sich zu beschr¨ anken und zu isolieren. For which values of x larger than 1 does the sequence xx . the last and greatest art is to limit and isolate oneself. every isosceles right triangle is replaced by two isosceles right triangles whose sides have half the length. xx . This problem was suggested by a note of Allen J. x x xx . What can you say about the limiting behavior of the sequence of numbers sin n as n runs through the natural numbers? Does it matter if the argument is measured in degrees or in radians? Problem 9. √ Consequently.3. The first stage shows an isosceles right triangle whose hypotenuse is a √ horizontal line segment of length 1. PROBLEMS
93
9. College Mathematics Journal 28 (1997).2.4
Problems
For the rest of it. . . number 1. so the two slanted sides together have length 2.4.9. Since the seesaw √ curves approach the horizontal line of length 1 as their limit. . Schwenk: "Introduction to limits. (The other two √ sides each have length 1/ 2.4 (The limits of computers7 ). converge to a (finite) limit?
x
Problem 9.2. Consider the iterative construction shown in Figure 9. 51.2:
√
2=1
Problem 9. the total length of the slanted sides at each stage is always 2.6 Goethe Conversations with Eckermann
Problem 9. What went wrong?
Figure 9.) At each subsequent stage.1.
Im u ¨brigen ist es zuletzt die gr¨ oßte Kunst.
7
6
. we deduce that 2 = 1. or why can't we just trust the table?".
1: A table of values for (sin x2/5 )/x2/5
1.003. 1940.01 ±0.009.99934 0. x (sin x
2/5
)/x
2/5
±0.3. They purportedly prove that π = 47. . make a similar table of values for the function f defined for x = 0 by f (x) = cos 1 · tan−1 x 1 x . Did they? If not.006. then circumscribe a circle around the square and a regular pentagon around the new circle. 0.) Problem 9.94
CHAPTER 9. LIMITS
Beginning calculus students often think that their instructors' discussions of limits are pedantic. Bentley's theorems. . What really happens in the limit?
8
Mathematics and the Imagination. Make sense out of the numbers that the computer generated.03.99998
Table 9.00001 0. .99584 0. 4.9. when x = 0. What is the true value of limx→0 f (x)? Problem 9. .001 ±0. ." a "proof" done by a group of entering freshman in a special enrichment program the summer before they started. .6. and so on.
What does this table suggest for the value of limx→0 f (x)? 2. Using a calculator or computer. . . 0.09. 0.99989 0. What do these tables suggest for the value of limx→0 f (x)? 3. Circumscribe an equilateral triangle around a unit circle. Make analogous tables of values of f (x) when x = 0. and when x = 0.97379 0. then circumscribe a circle around the triangle and a square around the new circle. page 312.
. .1). . See Figure 9. Simon and Schuster.6. .06. where did their argument go wrong? (Find ALL errors. 0.1 ±0. .0001 ±0. for it seems perfectly obvious from numerical evidence what the value of a limit like limx→0 (sin x2/5 )/x2/5 must be (see Table 9. They are wrong. 0. In the reading handouts is "Chapter 47. Edward Kasner and James Newman state8 that the construction converges to a limit circle whose radius is about 12. 0.5.3.
. Show that the rearranged series converges to zero: 1− 1 1 1 1 1 1 1 1 1 1 − − − + − − − − + − .
1. show that 1− 1 1 1 1 1 1 1 + − + − + − + . Rearrange the preceding conditionally convergent series so that each positive term is followed by the next four negative terms. . .577. By considering Riemann sums. 2.3: What happens in the limit?
Problem 9. The numerical value of γ is about 0. This limit.9. By using the preceding part twice.7. 3.4. 2 3 4 5 6 7 8
This series is often called the alternating harmonic series. . PROBLEMS
95
Figure 9. 2 4 6 8 3 10 12 14 16 5
. = 0. = log 2. show that the limit
n n→∞
lim
k=1
1 − k
n 1
1 dt t
exists. is known as Euler's constant. usually denoted by the Greek letter γ . but nobody knows if γ is a rational number or an irrational number.
5
Additional Literature
1.10. (This amounts to showing that the limit of the even-order truncations equals the limit of the odd-order truncations. for all x and y . Claude Brezinski. whatever the values of the positive integers aj . Consequently. the strong triangle inequality might equally well be called the isosceles triangle inequality. A. then every triangle is isosceles with respect to | · |. where the bar denotes a repeating block). . Show that if |·| is a valuation that satisfies the strong triangle inequality. 1964. Show that if x is an eventually periodic continued fraction (that is. . C. Continued Fractions. Kurt Mahler. p-adic Analysis. Prove that every unending continued fraction [a1 . Does every rational number have a p-adic expansion whose coefficients are eventually periodic?
9. ] converges. 1984.13. Ultrametric Calculus. Schikhof. 6. Continued Fractions. 2.
. . then x is a quadratic surd: an irrational number that is the root of a quadratic equation with integral coefficients.12. 1991. p-adic Numbers. but −1 is a 5-adic square. D. (The converse is also true. LIMITS √ 3. Problem 9. Problem 9. Neal Koblitz. Olds. Springer. In other words. |y |. an . 4. 3.8. Find the continued fraction expansion of
Problem 9. Random House. second edition. . .96
CHAPTER 9. 5. bk ]. x = [a1 . a2 . . Cambridge University Press. and |x − y | are equal. W. Springer. but harder to prove. . Introduction to p-adic Numbers and Their Functions. . History of Continued Fractions and Pad´ e Approximants. Ya. Khinchin.9. .) Problem 9. H. 1973. at least two of the numbers |x|. b1 . University of Chicago Press. and Zeta-Functions.
Problem 9. a2 . . 1984. Cambridge University Press. . 1963.) Problem 9. Show that −1 is not a 3-adic square.11.
" Learn from the Masters. 253. Mathematical Association of America. Pomona group (provided by one of the original members of the group). John Fauvel. Carlson.Chapter 10 Functions
10. Special Functions of Applied Mathematics." Mathematics Magazine 68 (1995). 1977. eds. C. 1995. Wayne Barrett..1 Goals
1. 2. B. pages 1–6. 2. Academic Press. Learn some of the history and applications of special functions.
10. "It had to be e". 4. Solidify your understanding of functions.. 15. 97
. Zeev Barel. "bentley's theorems". 5.2
Reading
1. 3. Frank Swetz et al. number 1. especially transcendental functions: the different ways in which they arose and their various definitions. Mathematics Magazine 68 (1995). "A mnemonic for e. pages 39–48. Chapter 47. number 4. "Revisiting the history of logarithms.
" Mathematics Magazine. Victor J.3. Frank Swetz et al. FUNCTIONS 6. it does so in a clumsy enough way by pasting several linear functions together. xxxiv Gottfried Wilhelm Leibniz1 introduced the term "function.98
CHAPTER 10.. 1995. Our federal income tax law defines the tax y to be paid in terms of the income x..
10. 8. 68 (1995). 7. The uncertainty persists today among undergraduate students who wonder if a piecewisedefined function is really one function or two. Katz." Mathematics Magazine 68 (1995). William Wordsworth The River Duddon. Jacques Redway Hammond. number 4." but historically. the Function never dies." Learn from the Masters. pages 29–37 and 98–101. 294–295. Concise Spherical Trigonometry. Mathematical Association of America. Leibniz lived 1646–1716. Make up a strange function that illustrates a subtlety of your definition. Victor J. David Shelupsky. "Napier's logarithms adapted for today's classroom. pages 49–55. The Form remains. and shall for ever glide. 9. Houghton Mifflin Co.2. each valid in another interval or bracket of income. Exercise 10. Exercise 10. 163–174. (1943). number 3. "Limitless integrals and a new definition of the logarithm.3
10. An archeologist who. five thousand years from
1
Co-inventor with Newton of the calculus. Katz. "Ideas of calculus in Islam and India. there was considerable uncertainty about its meaning. eds.1
Classroom Discussion
The function concept
Still glides the Stream.1.
.. What is a function? Is it a formula? a rule? a set? all of the above? none of the above? Formulate a definition of "function" that satisfies you.
Hermann Weyl.3. pn such that the function pn (x)f (x)n + pn−1 (x)f (x)n−1 + · · · + p1 (x)f (x) + p0 (x) is identically zero. p1 . How can one define a transcendental function. Vol. The algebraic numbers are the numbers that satisfy polynomial equations with integral coefficients. Exercise 10. Edmund Burke Reflections on the Revolution in France. the function f defined by f (x) = (x2 − 1)/(x2 + 1) is a rational function.4. show that the exponential function ex is transcendental. . CLASSROOM DISCUSSION now. The function tion does it satisfy? x2 − 1 is algebraic. certainly before Galileo and Vieta. Exercise 10. the rational functions are defined to be ratios of polynomials. a function f is algebraic if there are polynomials p0 . III The simplest functions that come to mind are polynomials. and the exponential function. the logarithm function. 1940
99
10. will probably date them a couple of centuries earlier. Just as the rational numbers are defined to be ratios of integers. For example.2
Transcendental functions
In their nomination to office.10. they will not appoint to the exercise of authority as to a pitiful job. . By considering growth rates as x → ∞. .3. but as to a holy function. . Some examples of transcendental functions are the trigonometric functions. shall unearth some of our income tax returns together with relics of engineering works and mathematical books.3. That is. the algebraic functions are the functions that satisfy polynomial equations with polynomial coefficients. Similarly. given that it does not satisfy any algebraic equation? There are several common ways to define such functions:
. What polynomial equax2 + 1
Functions that are not algebraic are called transcendental.
• as solutions of differential equations. FUNCTIONS
For example. In calculus class. Exercise 10.5 (Trigonometric functions). • via power series. 1.1). Was there a good reason for this.1: Geometric definition of the trigonometric functions
solution of the functional equation f (x + y ) = f (x)f (y ) satisfying f (1) = 2. What power series expansions are there for trigonometric functions? 3. What differential equations do the trigonometric functions satisfy? 4. • as solutions of functional equations. the trigonometric functions may be defined as lengths (see Figure 10. The exponential function 2x may be defined as the unique continuous
1 x
sin x
tan x
cos x
Figure 10. how many need to be defined before all six are determined through functional relationships? 2. Of the six trigonometric functions.
CHAPTER 10.100 • geometrically. or could you just as well have viewed the tangent and cosecant (for example) as the "basic" functions?
. you viewed the sine and cosine functions as the "basic" trigonometric functions.
iii. first learned in a lady's eyes. Compare with the area of a sector of a unit circle with central angle t. Princeton University Press. 1985. 327–332 The identity cos2 t + sin2 t = 1 shows that the trigonometric functions are connected with the unit circle x2 + y 2 = 1. 1. Exercise 10. Lives not alone immured in the brain. What power series expansions are there for the exponential and logarithm functions? 2. Boyer. What functional equations characterize the exponential and logarithm functions? 4. and the line joining the origin to the point (cosh t. And gives to every power a double power.7 (Hyperbolic functions). IV.
2
Carl B. the x-axis. CLASSROOM DISCUSSION
101
5. What differential equations do the exponential and logarithm functions satisfy? 3. page 504. The hyperbola x2 − y 2 = 1 is connected with the hyperbolic functions cosh t = 1 (et + e−t ) and sinh t = 2 1 t (e − e−t ).10.6 (ex and log x). Above their functions and their offices. Exercise 10.3. 2. and tanh t as lengths of certain line segments related to the graph of the hyperbola x2 − y 2 = 1. Draw a diagram analogous to Figure 10. 1.1. sinh t. Subsequently. Find the area bounded by the hyperbola. Vincenzo Riccati studied the hyperbolic functions in the middle 2 of the eighteenth century. Courses as swift as thought in every power. with the motion of all elements. But. William Shakespeare Love's Labour's Lost. and identify cosh t. A History of Mathematics. sinh t). these functions were popularized and given their modern names by Johann Heinrich Lambert (a modest man who is supposed to have replied "All" to Frederick the Great's inquiry of which science he knew best2 ).
. Show that the geometric definition of the trigonometric functions agrees with the power series and differential equation definitions. What geometric characterizations are there for these functions? But love.
the two coordinates are latitude. Longitude ranges from 0◦ to 180◦ .076 feet (compared to the statute mile of 5. it has been important to know where on Earth you are. and this is still the most common convention in physics books. either East or West of the zero meridian or prime meridian that passes through Greenwich. how to get somewhere else. FUNCTIONS
10. and how to draw illustrative maps of regions of the Earth's surface.852 meters or 6.3 A standard unit for measuring distance on the surface of the Earth is the nautical mile.3.102
CHAPTER 10. that is. the polar angle measured in the plane of the equator. there is some ambiguity about the precise value of the nautical mile. mathematics textbooks denoted the azimuthal angle by φ and the co-latitude by θ. but this has been replaced by the standard international nautical mile of 1852 meters. and a location on this two-dimensional surface can be specified by two coordinates. Unfortunately.
3
. nautical mile was 1853. The U. attributed to David Brinkley Since antiquity. which is the azimuthal angle. Mathematicians ordinarily use the angle complementary to the latitude. and circles where the longitude is constant are meridians. which is the angle of elevation above the plane of the equator. and longitude. but most college mathematics texts have now switched to using φ for the co-latitude and θ for the azimuthal angle. the surface of the Earth is a sphere.25 meters or 6080 feet. the co-latitude.html. which is the length of one minute of arc (1/60th of a degree) along a great circle. with an additional designation of North (for angles of elevation above the equator) and South (for angles below the equator).4
The notation is now a hopeless muddle that will be resolved only when the topic of spherical coordinates goes out of fashion in the undergraduate curriculum. England. which is measured down from the North Pole instead of up from the equator.nist. Circles where the latitude is constant are parallels of latitude. 4 See
Mercator's map and rhumb lines
The one function TV news performs very well is that when there is no news we give it to you with the same emphasis as if there were. At one time. Since the equatorial radius and the polar radius of the Earth differ by about 1 part in 300.S. there is no standard convention for the designation of the spherical angles. most commonly by two angles. To a first approximation. Normally latitude ranges from 0◦ to 90◦ .gov/cuu/Units/outside. In geography.280 feet). This works out to be about 1.
the standard meter was defined by a platinum-iridium bar in Paris. How many Martian meters are there in a Martian nautical mile? In steering a ship at sea.html.8 It is important to know how to choose the course to navigate between two locations of prescribed latitude and longitude.5 The National Institute of Standards and Technology provides definitive information about the International System of Units (abbreviated SI from the French Syst` eme International d'Unites ).nist. and the "drome" is a Greek root referring to running. Due to its rotation. 8 The "loxo" is a Greek root meaning "oblique" or "slanting".3.nist. the meter is defined to be 1/299. 792. Suppose that a Martian nautical mile is defined to be the length of one minute of arc along a great circle on the surface of Mars. It is easy to determine the loxodromic distance (in nautical miles) between two points on the globe if you know the difference in latitudes and the angle that the loxodromic path between them makes with the meridians. so for many years. or shapes. areas. Currently.6 Exercise 10. in other words. There are various techniques in use for projecting the round Earth
See Exercise 10. html. 458 times the distance that light travels in one second in a vacuum. and a Martian meter is defined to be 10−7 times the distance from the equator of Mars to a pole. The Earth is not a satisfactory standard for very precise measurements. See the online references at Drawing a planar map of the spherical surface of the Earth is problematical: there is no way to do it without distorting either distances. and how to determine the loxodromic distance between two points. a course that makes a fixed angle with each meridian of longitude. it is most convenient to follow a fixed compass bearing. Such a course is a rhumb line 7 or loxodrome.8. 7 The word "rhumb" comes from the same Greek root that gives us "rhombus".
6 5
. and the first standard meter was slightly short because of a miscalculation.gov/cuu/Units/bibliography. Find a formula. as in "hippodrome". the Earth would not be a perfect sphere even if its surface were devoid of geographical irregularities such as mountains and valleys. CLASSROOM DISCUSSION
103
The meter itself was originally intended to be 10−7 times the distance from the Equator to the North Pole along a meridian through Paris.10.
1.
9
. the meridians are spread apart to be parallel lines. conic equal-area projection. bipolar oblique conic conformal projection. where gd is the Gudermannian function of Problem 10. The Mercator map is constructed to be conformal. Exercise 10. He lived 1512–1594 and should not be confused with Nicolaus Mercator.)
Kremer's Latinized name was Mercator. Briesemeister elliptical equal-area projection. (This quantity is also gd−1 (x). oblique cylindrical projection. with names such as azimuthal equal-area projection. Mercator projection. Conformality demands that the length magnification be the same in the direction of a meridian as in the direction of a parallel.9 In the Mercator map.6 below. named for the Flemish surveyor Gerhard Kremer. but not evenly spaced: the Mercator map badly distorts distances in the Arctic and in the Antarctic. A glance at a modern atlas reveals a multitude of methods. conic projection. who was born in Denmark and lived 1620–1687. Bonne projection. the distance distortion is the same in all directions. What is needed for a map to be conformal is that near every point. The most famous method for drawing a flat map of the Earth is the Mercator projection. azimuthal equidistant polar projection. and sinusoidal projection. 4 2
where the unit of length is the radius of the globe on which the map is based. On the globe.104
CHAPTER 10. even though it distorts distances. cylindrical equal-area projection. oblique conic conformal projection. meridians converge at the poles. FUNCTIONS
onto a flat map. but on Mercator's map. cylindrical projection. The parallels of latitude appear as horizontal straight lines.10 (The Mercator projection). polar projection. Lambert conformal conic projection. how should you choose the length magnification factor along a parallel at latitude x? 2. meaning that locally it preserves shapes (angles). Lambert azimuthal equal-area projection. Integrate and use trigonometric identities to show that a point at positive latitude x radians has distance from the equator on Mercator's map equal to ln tan π x + . To achieve this effect. polyconic projection. This second Mercator is the one who found an infinite series expansion for the logarithm function. the meridians of longitude appear as evenly spaced vertical straight lines.
We want to show that f (x) = f (a) + f ′ (a)(x − a) + f ′′ (a) f (3) (c) (x − a)2 + (x − a)3 2 3!
for some point c between a and x.4. William Shakespeare Macbeth. Form and function should be one. f ′ (x) = f ′ (a) + f ′′ (a)(x − a) + f (3) (c) 2 Now repeat the antiderivative process to get f (x) = f (a) + f ′ (a)(x − a) + f ′′ (a) (x − a)2 (x − a)3 + f (3) (c) . Start with the mean-value theorem applied to the function f ′′ : namely. Portugal (38◦ 42′ 0′′ N. 80◦ 7′ 49′′ W) and Lisbon. 9◦ 5′ 0′′ W). f ′′ (x) = f ′′ (a) + f (3) (c)(x − a) for some c. Shakes so my single state of man that function Is smother'd in surmise. Find the loxodromic distance between Miami Beach.4
Problems
Form follows function—that has been misunderstood.
. My thought. Florida (25◦ 47′ 25′′ N. the meridians are parallel straight lines.11. Frank Lloyd Wright Problem 10. and nothing is But what is not. so loxodromic curves are also straight lines. PROBLEMS
105
On Mercator's map. I. Exercise 10. 139–142
10.1. joined in a spiritual union. iii. The following is purportedly a proof of Taylor's Theorem of order two with remainder. Mercator's map makes it straightforward to determine the proper course heading for a rhumb line between two specified points. and construct appropriate counterexamples to show that the errors you identified are indeed errors. 2 3!
Find all the errors in this alleged proof.10. whose murder yet is but fantastical. Take the antiderivative of this formula with an appropriate integration constant to get (x − a)2 .
where i = −1. wants to compute the product 48. Find the derivative d gd /dx.
2. How can he evaluate this product (to nine significant figures) without much work? Problem 10. Calculate the shortest distance on the globe (great circle route) between Miami Beach. Robinson Crusoe.4. Show that tanh(x/2) = tan(gd /2). Show that the logarithm function is transcendental. Show that gd(x) = 2 tan−1 (ex ) − π/2. Problem 10.7. Express each of the six hyperbolic functions of x in terms of the six trigonometric functions of gd. Historical Challenge: Why was Gudermann interested in the Gudermannian function?
Gudermann lived 1798–1852 and is mainly remembered as a teacher of the great Karl Weierstrass. Show that the Maclaurin series expansions of tan x and sec x have coefficients that are all non-negative rational numbers. Florida and Lisbon. The Gudermannian function gd(x) is defined implicitly via sinh x = tan gd. This quantity appeared in Exer4 2
. 3. FUNCTIONS
Problem 10. 962 × 258. 480.
10
π gd(x) + . 1. and differential equation definitions for the logarithm function are equivalent. 1. Problem 10.3. 819.8. Problem 10. Portugal. Show that the geometric. Show that x = ln tan cise 10. 5. He has salvaged a table of values of the cosine function (Table 10. Show that the trigonometric functions are transcendental.6 (The Gudermannian).1). infinite series.106
CHAPTER 10. 2. shipwrecked on a desert island. 045. 4.
2. Problem 10.5. Christoph Gudermann10 discovered that it is possible to relate the trigonometric functions and the hyperbolic functions without employing complex numbers. 1.10. with −π/2 < gd < π/2. Show that the infinite series and differential equation definitions for the exponential function are equivalent. The trigonometric functions are related √ to the hyperbolic functions via cos(ix) = cosh(x). Problem 10.2.
1
Goals
1.
11. Polya How To Solve It
11. 109
. 3.
You are familiar with two ways to describe a curve in the two-dimensional plane: you can specify the curve either geometrically. and cubic equations and their geometric significance. quadratic.Chapter 11 Plane geometry
Geometry is the art of correct reasoning on incorrect figures. 2. Appreciate the interplay between analytic and algebraic geometry. a circle is the locus of points at fixed distance from a specified center point (geometric definition). or the set of points with coordinates (x.2. or by a formula. G. y ) satisfying an equation of the form (x − a)2 + (y − b)2 = r2 (analytic definition). For example.2
11. Renew and deepen your acquaintance with linear. Learn about non-Euclidean geometries.1
Classroom Discussion
Algebraic geometry
Euclid (to Ptolemy I)
There is no royal road to geometry.
1 for whom Cartesian coordinates are named. However. Taking as the parameter t the negative of the slope of the line joining (0. PLANE GEOMETRY
The idea of connecting the two descriptions is due to Ren´ e Descartes.110
CHAPTER 11. 1) to a point on the circle x2 + y 2 = 1. There are infinitely many essentially distinct such triples. 13). Cassius. Caesar said to me 'Darest thou. it may seem a bit odd to parametrize an algebraic curve with transcendental functions. and it is possible to describe all of them. derive the parametrization x= 2t 1 + t2 and y= 1 − t2 .2. and that is because the poser of the world always works in circles. There is a way to parametrize the circle with rational functions. and each point has two coordinates. why are there three parameters A. now Leap in with me into this angry flood. which is given by a polynomial equation of first degree: Ax + By + C = 0. Black Elk The most familiar way to parametrize the unit circle x2 + y 2 = 1 is by using trigonometric functions: x = cos t and y = sin t. And swim to yonder point?' William Shakespeare Julius Caesar.
1
Lived 1596–1650. Since a line is uniquely determined by two points in the plane. with quotients of polynomials of degrees 1 and 2. I ii Exercise 11.
. 4. 12. In the case that the formulas are given by polynomial equations. The simplest plane curve is a straight line. B . 5) and (5.1. The interplay between the geometry of curves and their analytic descriptions is the subject of analytic geometry. and C in the general equation of a line? Everything an Indian does is in a circle. and everything tries to be round. 1 + t2
The rational parametrization of the circle has interesting applications. For example. indeed. the subject is algebraic geometry. you are familiar with Pythagorean triples such as (3. which correspond to the equalities 32 + 42 = 52 and 52 + 122 = 132 . Exercise 11.
caress it. x = 0). il deviendra vicieux! The meaning of vicious circle is the same in French as in English. number 3.4. The recent proof of Fermat's last theorem2 confirmed that when n is an integer larger than 2. and c such that an + bn = cn . ibid. and reconstitute your answer as a function of x. CLASSROOM DISCUSSION
111
Exercise 11. x2 + y 2 + 1 = 0).3.11..3 Eug` ene Ionesco The Bald Soprano
dx into the integral of a rational 1 + cos x function of t by using the rational parametrization of the circle. c) of positive integers such that a2 + b2 = c2 . it is given as a magical formula without motivation.
Andrew Wiles. there are no positive integers a. 443–551. If this substitution is presented at all. Ring-theoretic properties of certain Hecke algebras. b. In degenerate cases. or the empty set (for instance. A topic that is falling out of fashion in the second-semester calculus course is a substitution that converts the integral of a rational function of cos x and sin x into the integral of a rational function of a new variable t. b. a single point (for instance. so called because the ellipse. hyperbola. a conic can reduce to a line (for instance. Annals of Mathematics (2) 141 (1995). caressez-le. a pair of parallel lines (for instance. Can you reconcile your result with the answer tan(x/2) that Maple gives? Exercise 11. x2 + y 2 = 0). Use the rational parametrization of the circle to find all triples (a. 3 Prenez un cercle. Such an equation describes a conic curve. Evidently. Convert the integral The most general polynomial equation of degree two in variables x and y has the form ax2 + bxy + cy 2 + dx + ey + f = 0. Modular elliptic curves and Fermat's last theorem. the substitution is nothing more than the rational parametrization of the circle. a pair of intersecting lines (for instance.
2
. one cannot expect in general to find a rational parametrization of a plane curve given by a polynomial equation of degree larger than 2.2. Evaluate the integral. x2 − 1 = 0). 553–572. and parabola can be obtained by slicing a cone with a plane at different angles. In fact. and it will turn vicious. xy = 0). Take a circle. Andrew Wiles and Richard Taylor.
1). −1). and f does the above equation describe an ellipse? a hyperbola? a parabola? Two points determine a line. Find two different ellipses passing through the four points (1. then there is at least one conic (possibly a degenerate one) passing through all five points.6. 1). Find two distinct (degenerate) conics passing through the five points (0. Show that no conic contains the six points (1.112
CHAPTER 11. 1). (−1. (−1. 1). do not determine a general conic curve. (2.8. Three points determine a circle (Problem 11.
. 1). and (1. Exercise 11. Such curves begin to have sufficient complexity that it becomes tedious to sketch their graphs by hand. however. 1. b. e. c. S. The billiard sharp whom any one catches. 0). (0. 2. six points will not lie on a conic unless the points are in special positions. 0). And there he plays extravagant matches In fitless finger-stalls On a cloth untrue With a twisted cue And elliptical billiard balls. On the other hand. A cubic curve is specified by a polynomial in x and y of degree 3. (0. 0). −1). 0). Four points.4). Exercise 11.1). (−1. Under what conditions on a. Gilbert The Mikado
Exercise 11. 0). but you can easily display pictures of such curves using a graphing calculator or computer (Problem 11. (3. PLANE GEOMETRY
Exercise 11. (1.7.
W. (1. −1). Show that if five points are specified in the plane. −1). but in degenerate cases the conic may not be uniquely determined. Five points in general position determine a conic. d. His doom's extremely hard— He's made to dwell— In a dungeon cell On a spot that's always barred.5. 0). and (1. (−1.
This is illustrated for the specific cubic 4y 2 = (x + 4)(x2 + 1) in Figure 11. William Shakespeare Macbeth. For the cubic 4y 2 = (x + 4)(x2 + 1) shown in Figure 11. If P and Q are points on the cubic. CLASSROOM DISCUSSION
113
P
Q
P+Q
Figure 11. The reflection of this point with respect to the x-axis is defined to be the sum P + Q. rebellious arm 'gainst arm. There is no such point on the cubic. How many points would you expect to need to specify in order to determine a cubic curve? Nondegenerate cubic curves have the remarkable property that their points carry a natural group structure.1. The victory fell on us.1: The cubic curve 4y 2 = (x + 4)(x2 + 1)
Exercise 11. use the group law to find the sum of the two points (−4. I ii
.1.10.9. A group is supposed to have an identity element E with the property that P + E = P for every P .2. It intersects the cubic at a third point. 0) and (0. The addition law is determined in the following way. 1).11. Curbing his lavish spirit: and. lapp'd in proof. Confronted him with self-comparisons. Till that Bellona's bridegroom. but we can supplement the cubic with an idealized point "at infinity" that will serve as the identity element if we agree that all vertical lines pass through the point at infinity. draw the line through them. to conclude. Point against point. Exercise 11.
2
Non-Euclidean geometry
The eye is the first circle. The (not so easy) verification of associativity is left for Problem 11.6. 5.2. Augustine described the nature of God as a circle whose centre was everywhere. that there is always another dawn risen on mid-noon. The next exercise is a typical example of that kind of reasoning. Ralph Waldo Emerson Essays. that every action admits of being outdone. in considering the circular or compensatory character of every human action. Verify that the addition law described above does provide the cubic curve with the structure of a commutative group. that there is no end in nature. PLANE GEOMETRY
Exercise 11.
. the horizon which it forms is the second. but every end is a beginning. Another analogy we shall now trace.11.
11. What is the additive inverse of a point P ? 6. We are all our lifetime reading the copious sense of this first of forms. A group law is supposed to be associative. The rule needs to be modified when the line joining P and Q is tangent to the curve at P . Our life is an apprenticeship to the truth. Circles You probably saw in high school some basic notions from Euclidean geometry such as the principle of similar triangles and the construction of a perpendicular bisector of a line segment. It is the highest emblem in the cipher of the world. and under every deep a lower deep opens. and its circumference nowhere. x. How should P + P be defined? 2. How should P + Q be defined in this case? 3. The rule specifying the addition law needs to be modified for the case P = Q.114
CHAPTER 11. Check that the addition is commutative. St. One moral we have already deduced. 4. and throughout nature this primary figure is repeated without end. that around every circle another can be drawn. 1. Check that the point at infinity does act as an identity element.
2. A circle can be drawn with any center and any radius.2 the two intersecting lines that are
. By considering a circle inscribed in a triangle. By making a copy of a triangle four times as big as the original. As Lines so Loves Oblique may well Themselves in every Angle greet: But ours so truly Parallel. One such model is hyperbolic geometry in the unit disk. Euclid's Elements builds up the theory of ordinary planar geometry from a few basic assumptions and "common notions. "lines" are arcs of circles that are orthogonal to the boundary unit circle at both intersection points. Parallel postulate: through a given point not on a given line can be drawn exactly one line parallel to the given line.2." Euclid's axioms are the following. Diameters of the unit circle also count as "lines. In fact. All right angles are equal. 1. In other words. CLASSROOM DISCUSSION
115
Exercise 11. 5. show that the perpendicular bisectors of the three sides of a triangle meet at a common point. it is possible to construct a geometric model that satisfies the first four axioms but not the fifth. Though infinite can never meet. show that the lines bisecting the three angles of a triangle meet at a common point. A straight line segment can be drawn from any point to any other point. A straight line segment can be extended continuously to a straight line. Figure 11. 4. the fifth axiom is independent of the others. and some thought that they succeeded.2 shows a picture of some lines in the "Poincar´ e unit disk.
Andrew Marvell The Definition of Love
Euclid's fifth axiom—the parallel postulate—was a source of controversy for centuries.12." In the hyperbolic disk. show that the three altitudes of a triangle meet at a common point. 3. 2. By considering a circle circumscribed around a triangle. 3. however.11. Many people tried to prove it from the other axioms. 1." Notice in Figure 11.
For precept must be upon precept. Compute the sum of the angles of the hyperbolic triangle with vertices at (0. See Figure 11.3.2: The Poincar´ e unit disk
disjoint from (that is. precept upon precept.
. and there a little. the boundary is at infinite distance from any point of the disk. 0).14. here a little. Thus Euclid's fifth axiom does not hold in the hyperbolic disk. The distance between points in the hyperbolic disk is obtained by integratds ing along the "line" joining the points. Isaiah.116
CHAPTER 11. line upon line. 1/2). line upon line. The boundary circle is not included as part of the hyperbolic disk. "parallel to") the other two lines. 28:10 Exercise 11. What! will the line stretch out to the crack of doom? William Shakespeare Macbeth. Distances become very 1 − x2 − y 2 large near the boundary of the disk. IV i Exercise 11. so Euclid's second axiom is satisfied. (0. Indeed. PLANE GEOMETRY
Figure 11. and (1/2.13. Verify Euclid's first axiom for the hyperbolic disk. 0). Angles are computed in the usual way: the angle of intersection of two curves is the angle between their tangent lines.
He has many friends. Euclid's fifth postulate fails not because there are too many parallel lines. then one mile due east.16. turns due north. walks one mile and is back at camp. Old Foss is the name of his cat: His body is perfectly spherical. you would not wish to quit the sphere in which you have been brought up. takes a picture of a bear. laymen and clerical.11. What color was the bear?
. Find a spherical triangle with three right angles. CLASSROOM DISCUSSION
117
Figure 11.3: A hyperbolic triangle
"If you were sensible of your own good. while in spherical geometry. It is more convenient.10). to think of this geometry on the sphere itself." Jane Austen Pride and Prejudice (Lady Catherine de Bourgh. A photographer for National Geographic sets out to capture a bear on film. to Elizabeth Bennet) One can also put a "spherical geometry" on the plane by using stereographic projection (Problem 11.15. but because there are none! In hyperbolic geometry.2. Exercise 11. The "lines" are great circles on the sphere (circles whose radius is the same as the radius of the sphere). the angles of a triangle sum to more than 180◦ . In spherical geometry. the angles of a triangle sum to less than 180◦ . She walks one mile due south from base camp. however. He weareth a runcible hat.
Edward Lear Nonsense Songs
Exercise 11.
2. Produce pictures (by computer. 598–600. edited by Gerald L. Observe that the three graphs look very different from each other. I am here. Problem 11. Show that if two points P and Q on the cubic shown in Figure 11.6. Problem 11. Prove that the points must all lie on a line. Alexanderson and Dale H. According to legend. Bulletin of the American Mathematical Society 51 (1945). 56–57. Mathematical Association of America.1 have coordinates that are rational numbers. a different story appeared in the Mathematical Intelligencer 14 (1992).1 is associative. Show that if three points in the plane are not all on the same line. Show that if five points are specified in the plane.3.118
CHAPTER 11. and the review of that paper was ghost-written for Kaplansky by the editor of Mathematical Reviews.) However. pages 33–34. Problem 11. PLANE GEOMETRY The wheel is come full circle. inscription on Plato's door Problem 11. this problem was posed as part of the eighteenth William Lowell Putnam mathematical competition. William Shakespeare King Lear. and if no four of the points are on the same line. then there is a unique conic (possibly degenerate) passing through the five points. the distance separating them is an integer. 1995.3
Problems
Let no one enter who does not know geometry. y 2 = x3 + x2 .4. V iii
11. number 1. if you like) of the graphs of the three cubics defined by the equations y 2 = x3 + x2 + 2x + 1. Prove that the addition law for points on the cubic curve shown in Figure 11. and y 2 = x3 + x2 − 2x − 1.
In February 1958. then there is one and only one circle passing through them.1. Suppose given an infinite sequence of points in the plane with the property that for every pair of points. (See Lion Hunting and Other Mathematical Pursuits.5. then the coordinates of the sum point P + Q are again rational numbers.4 Hint: what sort of curve is the locus of points whose distances from two specified locations have a fixed difference? Problem 11. Mugler.
4
. Anning and Paul Erd˝ os. a follow-up paper giving a simpler solution was ghost-written for Erd˝ os by the reviewer Irving Kaplansky. Problem 11. Integral distances. The result first appeared in a paper by Norman H.
Imagine a sphere resting on the plane with its south pole at the origin5 and a light source at the north pole. Introduction to Hyperbolic Geometry. (QA 685 C7. Marcel Dekker. 1995.10. Prove the remarkable fact that these "straight lines" in the plane are actually ordinary circles (unless the great circle passes through the poles. how might you define area in the hyperbolic disk? Problem 11. University of Toronto Press. in which case its projection is an ordinary straight line).
11. ADDITIONAL LITERATURE Problem 11. 1942.11.
5
.11. A point on the sphere casts a shadow on the plane. 2 Problem 11.4. M.
119
Problem 11.2 W44 1985)
An alternate version of stereographic projection places the center of the sphere at the origin. Arlan Ramsay and Robert D. Springer-Verlag. Weeks. Show that the (hyperbolic) area of the hyperbolic triangle shown in Figure 11. Knowing how to compute length in the hyperbolic disk. Jeffrey R. Show that the area of a spherical triangle on a sphere of radius 1 is ∆ − π . (QA 685 R18 1995) 3. Coxeter. H. The "straight lines" in the plane are defined to be the shadows of great circles.7. S. Non-Euclidean Geometry.8.8) 2.9. The planar model of spherical geometry is obtained from the spherical model by stereographic projection. (QA 612. The Shape of Space. Verify Euclid's third axiom for the hyperbolic disk.3 equals 1 arctan(1/4). where ∆ is the sum of the angles (in radians) of the triangle.4
Additional Literature
1. 1985. Richtmyer. Problem 11.
.
" The College Math Journal. Journey through Genius: The Great Theorems of Mathematics. 94–102. 2.2
Reading
1. A History of Algebra. (1997). horrible shadow! Unreal mockery. Scene iv
12. William Dunham. 28. Act III. and 177–186. Mayer. 3. Chapter 6: Cardano and the Solution of the Cubic. 3. van der Waerden. properties. Uwe F. pages 52–62. pages 133–154. 1990.
12. Prove the Fundamental Theorem of Algebra and appreciate its beauty.Chapter 12 Beyond the real numbers
Hence. L. and applications of the complex numbers. Learn some of the history. 1985. Learn some of the history. page 58. 2. number 1. Springer-Verlag. 121
. Wiley. "A Proof that Polynomials have Roots.1
Goals
1. properties. B. and applications of the quaternions. hence! William Shakespeare Macbeth.
Dover. 1959. Such expressions are added and multiplied by following the usual rules of "high school algebra". A source book in Mathematics." How would you respond? What are the complex numbers? There are several ways to think about them. "When I enter the number −1 on my calculator and push the square-root key. I get an error message. Francis. Duc de La Rochefoucauld (1613–1680) Reflections. where a and b are ordinary real numbers. "Gauss". associative. it is a species of commerce out of which self-love always expects to gain something.3
12. Exercise 12. The most familiar description is that the complex numbers consist of all expressions a + bi.122
CHAPTER 12. in fact.3.2. and distributive laws. we can say that the complex numbers are a field. what is the additive identity? the multiplicative identity?
. In more abstract language. an extension of the field of real numbers. or Sentences and Moral Maxims Maxim 83 Exercise 12. In the field of complex numbers. David Eugene Smith. with the additional rule that all occurrences of i2 should be replaced by −1. Show that if a and b are not both zero.
12. Consequently. A high school student says. and an exchange of good offices.1 (Warm up). Friendship is only a reciprocal conciliation of interests.1
Classroom Discussion
The complex numbers
Exercise 12. BEYOND THE REAL NUMBERS
4.3. and the imaginary unit i has the property that i2 = −1. it is easy to see that the operations of addition and multiplication of complex numbers satisfy the usual commutative. Therefore the equation x2 + 1 = 0 has no solution. pages 292–306. then the reciprocal 1 of a complex number is again a complex number: it can be rewritten a + bi in the form A + Bi for suitable real numbers A and B .
d) = (ac − bd. where a and b are real numbers. Explain why there is not a conflict with your preceding answer.4. What familiar geometric object is the set of complex numbers z with the property that |z − i| = 4? Exercise 12. ones? 1. the argument of a complex number is determined only up to integral multiples of 2π . called its argument and often written arg(a + bi). multiplication of complex numbers is a commutative operation. i/(−2 − 2i). CLASSROOM DISCUSSION
123
An alternative definition of the complex numbers—first made explicit by the nineteenth-century Irish mathematician William Rowan Hamilton—is the set of all ordered pairs (a. and an associated angle arctan b/a. ad + bc).6. Exercise 12. b) means that the complex numbers can be viewed geometrically as points in the ordinary Euclidean √ plane.c. a complex number a + bi has an associated length a2 + b2 .12. b + d) and the multiplication law (a.
. Prove the triangle inequality. However. Find the modulus √ and the argument of the following complex numbers: 1.8. In what sense is this definition the same as the previous
2. b) of real numbers subject to the addition law (a. and interpret it geometrically by representing complex numbers as vectors in the plane. In what sense is this definition the same as the usual one? The identification of a complex number a+bi with a vector (a.5. −2/(1 + 3 i). d) = (a + c.) The truth is always the strongest argument.3. called its modulus and often written |a + bi|. The complex numbers can also be represented as the set of all 2 × 2 mab trices −a b a .7. The triangle inequality says that |z + w| ≤ |z | + |w| for all complex numbers z and w. Exercise 12. the addition and multiplication operations being the usual ones for matrices.
Exercise 12. Exercise 12. b) · (c. Consequently. (Just like the angle in polar coordinates. b)+ (c. i. 1 + i. Matrix multiplication is a basic example of a noncommutative operation. Sophocles 496–406 b.
BEYOND THE REAL NUMBERS
One motivation for introducing the complex numbers is to force the "unsolvable" quadratic equation x2 + 1 = 0 to have a solution. we know that two linearly independent solutions of the differential equation g ′′ (x) = −g (x) are cos(x) and sin(x). Verify that this is an equivalence relation. This idea can be used to give the complex numbers an algebraic definition. define an equivalence relation on the set of polynomials with real coefficients by declaring polynomials p and q to be equivalent if and only if there exists a polynomial r (which could be a constant.
. we would expect eix to be a function whose derivative is ieix and whose second derivative is −eix . Which equivalence class corresponds to the complex number i? We know what i2 means (namely. (In the notation of abstract algebra. or even 0) such that p(x) − q (x) = (x2 + 1)r(x) for all x. Namely. deduce Euler's formula eix = cos(x) + i sin(x). we expect there to be (complex) constants c1 and c2 such that eix = c1 cos(x) + c2 sin(x). Now define the complex numbers to be the set of all equivalence classes. Confirm that this definition makes sense and is compatible with the other definitions of the complex numbers. Exercise 12. express 2i in the form a + bi. i × i. or −1). Exercise 12. Assuming that the complex exponential function suggested by the preceding heuristic argument does exist. The operations of addition and multiplication are inherited from the corresponding operations on polynomials.10.) Exercise 12. On the other hand. Observing that 2 = elog 2 . One way to characterize the usual real exponential function is that ekx is the unique solution of the differential equation f ′ (x) = kf (x) satisfying the initial condition f (0) = 1.12.9. but what might 2i mean? We are free to give this symbol whatever meaning seems reasonably consistent with the notion of exponentiation of real numbers. where a and b are real numbers.11. Exercise 12. If there is going to be a reasonable generalization of the exponential function to complex numbers. this definition says that the complex numbers are the quotient ring R[x]/(x2 + 1).124
CHAPTER 12. Consequently.
In view of the periodicity of the complex exponential function.14. elog x = x and log ex = x for every real number x. and so dream all night without a stir. is very far from being one-to-one: every point of its image has infinitely many pre-images. Tall oaks. eiπ = −1. Use Euler's formula to show that every nonzero complex number can be written in the form reiθ . so that the imaginary part of z lies between −π and π . Write (1 + i)99 in the form a + bi. √ 2. where a and b are real numbers. Book I
Exercise 12. Find them.
The real logarithm function is often defined as the inverse of the real exponential function: namely. Different choices of domain lead to different branches of the logarithm function. Exercise 12. Determine the principal value of ii . where r equals the modulus and θ equals the argument. John Keats 1795–1821 Hyperion.12. The principal branch of the logarithm corresponds to taking this strip to be centered on the real axis. the imaginary unit. and the ratio of the circumference of a circle to its diameter. To define a complex logarithm function as an inverse of the complex exponential function. The complex number 1 − 3 i has two square roots. Dream. the most natural restricted domain is a horizontal strip of height 2π in the complex plane.15. Exercise 12. however. The complex exponential function.13. Show that the complex exponential function z → ez is periodic with period 2πi. 1. What are the other possible values of ii (corresponding to different branches of the logarithm)?
. branch-charmed by the earnest stars. Exercise 12. Those green-robed senators of mighty woods.16. we need to restrict the domain of the exponential function to a set on which the function is one-to-one. This definition makes sense because the real exponential function is strictly increasing and hence one-to-one. CLASSROOM DISCUSSION
125
Specializing Euler's formula to x = π gives a formula regarded by many as one of the most beautiful in mathematics: Here we see in one formula the base of the natural logarithms.3.
Act I. It turns out that there are formulas for the solutions of cubic and quartic equations.
1
√ . A first step is to eliminate the quadratic term in a cubic equation.
William Shakespeare Macbeth. Jacques Hadamard 1865–1963 (attributed) It is a remarkable phenomenon that complex numbers arise naturally in problems about real numbers.19. Exercise 12. how does 97 + 9408 simplify? If you are stuck.2
The solution of cubic and quartic equations
The shortest path between two truths in the real domain passes through the complex domain. Exercise 12.3 + 2 slauqe tI
.126
CHAPTER 12. It is not obvious how to simplify complicated expressions √ 4 involving roots. Scene iii
Exercise 12.1)
The insane root That takes the reason prisoner. For example. but the formulas may involve complex numbers even when the solutions turn out to be real numbers. Show how to choose d so that the change of variable x = y − d transforms the general cubic equation y 3 + Ay 2 + By + C = 0 to the "reduced" cubic equation x3 − 3ax − 2b = 0.18. BEYOND THE REAL NUMBERS
12. Verify that formula (12. This phenomenon was first observed in the Renaissance. look at the answer in the footnote for a clue.1) gives a correct real-valued solution of the reduced cubic equation x3 − 3x − 2 = 0. when solving cubic and quartic equations was a hot research topic. The formula due to Tartaglia and Cardano is that a solution of the reduced cubic equation x3 − 3ax − 2b = 0 is given by x=
3
b+
√
b 2 − a3 +
3
b−
√
b 2 − a3 .
(12.3.17.1 The key to solving a cubic equation is to use some trickery to reduce the problem to solving an associated quadratic equation.
Exercise 12. Just as one can solve a cubic equation by reducing it to an associated quadratic equation. This manipulation yields the new equation
4
(x2 + z )2 = (2z − a)x2 − bx + z 2 − c. Exercise 12.
(12. A first step is to eliminate the cubic term in a quartic equation. Deduce formula (12.1) initially produces answers involving complex numbers. Scene iii Exercise 12. Show that the condition for the right-hand side of equation (12.12.1) to find the three real solutions of the cubic equation x3 − 6x − 4 = 0. simplification leads to real answers.2)
The right-hand side will also be a perfect square if z is chosen appropriately. Fran¸ cois Vi` ete had the following idea for solving a reduced quartic equation 2 x + ax + bx + c = 0: introduce a second variable z and add 2x2 z + z 2 to both sides of the equation to create a perfect square on the left-hand side.20. Observe that although (12. Act II. Solve the simultaneous equations by eliminating v and solving a quadratic equation for u3 . Exercise 12. Use (12. which will certainly be true if u and v satisfy the simultaneous equations u3 + v 3 = 2b uv = a.3.
. I have not kept my square.2) to be a perfect square is a certain cubic equation in z . Read not my blemishes in the world's report. CLASSROOM DISCUSSION
127
One way to derive the formula is to introduce two auxiliary variables u and v such that u + v = x. Show how to choose d so that the change of variable x = y − d transforms the general quartic equation y 4 + Ay 3 + By 2 + Cy + D = 0 to the reduced quartic equation x4 + ax2 + bx + c = 0. but that to come Shall all be done by the rule. William Shakespeare Antony and Cleopatra.23.21.1). one can solve a quartic equation by reducing it to an associated cubic equation. Substituting into the cubic equation and simplifying gives u3 + v 3 − 2b + (u + v )(3uv − 3a) = 0.22.
the quintic equation x5 + 15x + 12 = 0 has a unique real root. Solve x4 − 2x2 − 16x + 1 = 0. S.25. There actually is a formula for solving those quintics that can be solved by a formula. Exercise 12.
2
. Roosevelt (1882–1945) Of course.4 For example. 4 Although this statement sounds tautological.24. from which the example is taken. a political revolutionary who was arrested for threatening the life of king Louis Philippe. to say that the general quintic equation cannot be solved by radicals is not to say that every quintic equation is unsolvable. Why?
A Norwegian mathematician.3 A radical is a man with both feet firmly planted in the air.3. The procedure for solving a quartic equation can also be codified into a formula.1) for solving a cubic equation.1). you derived formula (12. a victim of tuberculosis.
12.2). He introduced the word "group" in its modern algebraic sense. Dummit. A characterization of when a polynomial equation is solvable by radicals ´ was found by Evariste Galois. Then by taking square roots in (12. Exercise 12. but he failed to express himself in a way that his contemporaries could understand. Hint: the associated cubic equation has an integer solution that is easier to find without using formula (12. Solving solvable quintics. we get a quadratic equation for x. It is a surprising theorem of Niels Abel2 that there is no general formula for writing down the solution of every fifth degree polynomial equation in terms of radicals. but the formula is too complicated to be of practical use. which we can solve too. He died tragically young.128
CHAPTER 12. it has nontrivial content! See D. Franklin D.3
The fundamental theorem of algebra
In the previous section. 387–401. Galois made revolutionary contributions to algebra. Mathematics of Computation 57 (1991). Galois died in a duel—the circumstances of which are still under dispute by historians—before reaching his twenty-first birthday. we can determine a suitable value for z . 3 In his short life (1811–1832). BEYOND THE REAL NUMBERS
Since we already know how to solve a cubic equation. Abel lived 1802–1829.
. and a polynomial q such that p(z ) = p(a) + b(z − a)k + (z − a)k+1 q (z ) for all z . To confirm that this candidate works. of course. Thus. we need to analyze the local behavior of the polynomial p near a. the problem of explicitly finding roots of polynomial equations is a difficult one. then lim|z|→∞ |p(z )| = ∞. Evidently. real-valued function on a closed. Exercise 12. Here |z | → ∞ means that z escapes from every disk centered at the origin. bounded subset of the plane attains a minimum on the set. There exist a positive integer k . the complex numbers form an algebraically closed field. a nonzero complex number b. the polynomial is a constant).3. It is therefore useful to have an existence theorem. every polynomial equation with real or complex coefficients does have a solution in the complex numbers (unless. The goal of this section is to work through a short one based on ideas from advanced calculus. In other words.26. there is a point a in the plane such that |p(a)| ≤ |p(z )| for every complex number z . CLASSROOM DISCUSSION
129
The real root of this quintic equation is given by the following expression involving square roots and fifth roots: − 1 5
5
√ 1875 + 525 10 +
5
√ 1875 − 525 10
5
+
√ −5625 + 1800 10 −
5
√ 5625 + 1800 10
(where the real fifth root is taken in each term). but so do the equations x28 + 37x14 + 92x6 + 13 = 0 and x4 + (2 + 3i)x2 + (9 − 2i) = 0. then we should look close to home. If p is a nonconstant polynomial. There are many proofs of the fundamental theorem of algebra.27. Evidently our candidate for a solution to the equation p(z ) = 0 should be the point z = a where |p(z )| attains a minimum. not only does the equation x2 + 1 = 0 have a solution in the complex numbers. According to the fundamental theorem of algebra. The following exercise says that if we are looking for a point in the complex plane where a polynomial is equal to zero.12. Deduce that since a continuous. Exercise 12.
transforming x + iy into x + iy = x − iy ) corresponds to reflection with respect to the x-axis. it follows that multiplication by eiϕ corresponds to rotation by the angle ϕ in the positive (counterclockwise) direction.
Complex numbers provide a convenient notation to describe transformations of the Euclidean plane. seeking a contradiction. From the representation of a complex number in the form reiθ . Exercise 12.30. Using the notation of complex numbers. We can transform the plane into itself by sliding the paper around on the desk. then the transformation z → z + w is a translation of the plane a units to the right and b units vertically. BEYOND THE REAL NUMBERS
Exercise 12. Nor knowest thou what argument Thy life to thy neighbor's creed has lent. we have a contradiction. Suppose we view the complex plane as a sheet of newspaper spread out on a desk.130
CHAPTER 12. All are needed by each one. Since |p(z )| was supposed to have a global minimum at z = a.29.3.28. This shows that p(a) must be zero after all. Nothing is fair or good alone. For example. write a formula describing reflection with respect to a line making an angle ϕ with the x-axis. Let θ denote the difference between the argument of p(a) and the argument of b. Show that if ǫ is a sufficiently small positive real number. that p(a) = 0.
12. Evidently the set of all sliding motions forms a group. An element of this
.4
Reflections and rotations
Sir Walter Scott (1771–1832) Chronicles of the Canongate. then |p(a + ǫei(θ−π)/k )| < |p(a)|. taking the complex conjugate (that is. and so the fundamental theorem of algebra is proved. Why can the polynomial p be represented in this way? Suppose.
Ralph Waldo Emerson (1803–1882) Each and All
Exercise 12. Chapter iv
But with the morning cool reflection came. If w = a + bi is a fixed complex number.
and q ′ . A reflection across the perpendicular bisector of the line segment joining p and T p maps p to T p and q to some new point q ′ . and so every rotation can be realized as the composition of two reflections.
. then it reverses the sense of angles.31. T p. moreover. Show that if a rigid motion of the plane is the composition of two reflections.12. and every isometry can be obtained as the composition of three reflections. It is illuminating to see which two reflections generate a rotation by angle θ. while if it is the composition of three reflections.3. By rewriting eiθ z as e−iθ/2 z ¯eiθ/2 . pick any two points p and q in the plane. the two possible locations for the image of r are reflections of each other across the line through T p and T q . If we admit the possibility of picking the sheet of paper up and flipping it over. A rotation by angle θ about the origin is described by z → eiθ z . Now consider any third point r in the domain plane. the perpendicular bisector of the line segment joining T q and q ′ is also the angle bisector of the angle determined by T q . Because both T and reflection preserve distances. since there are only two triangles congruent to the triangle pqr and having T p and T q as vertices. To see this.30 that the rotation is the composition of reflections in two lines making an angle of θ/2 with each other. we find from Exercise 12. the line segment joining T p to T q has the same length as the line segment joining T p to q ′ . A distance-preserving transformation that takes p to T p and q to T q can take r to one of only two locations. CLASSROOM DISCUSSION
131
group can be viewed as the composition of some finite sequence of translations and rotations. Therefore T is the composition of either two or three reflections. Exercise 12. Consequently. line 225 A rotation is a rigid motion. It is remarkable that every sliding motion can be obtained as the composition of just two reflections. A reflection across this perpendicular bisector therefore takes q ′ to T q while keeping T p fixed. Denote their images under an isometric transformation T by T p and T q . then it preserves the sense of angles. then we obtain a larger group of rigid motions: namely. Remembrance and reflection how allied! What thin partitions sense from thought divide! Alexander Pope (1688–1744) Essay on Man. Epistle i. the group of isometries (distance-preserving transformations) of the plane.
132
CHAPTER 12.33.34. Show that a rotation by angle θ about an axis in threedimensional space can be realized as the composition of two reflections in planes whose line of intersection is the rotation axis and whose normal lines make an angle of θ/2 with each other. where a. However. b. Act iv. Three-dimensional vectors equipped with the usual addition and with the cross product fail to form a field. • the usual associative and distributive laws. Which properties of a field are lacking?
. because no such operation exists. Scene 1 Exercise 12. • i2 = −1. Exercise 12.5
Quaternions
The complex numbers arise from the real numbers by adjoining a quantity i with the property that i2 = −1. A famous story from the history of mathematics relates that William Rowan Hamilton spent years looking for a way to multiply and divide triples of real numbers. cross'd with adversity.3. It is natural to consider the possibility of extending the complex numbers by adjoining another quantity j to form objects of the form a + bi + cj . and c are all 0. William Shakespeare The Two Gentlemen of Verona. if all of the following properties are to hold: • closure under multiplication. there is no reasonable way to make such an extension. • a + bi + cj = 0 if and only if a. One of his attempts was what we now call the cross product. A man I am. BEYOND THE REAL NUMBERS
Exercise 12. b. and c are real numbers.
12. Recall that the cross product of two vectors in three-dimensional space is a vector perpendicular to both and with length equal to the product of the lengths of the two vectors times the sine of the angle between them.32. Show that there is no consistent way to define multiplication on objects of the form a + bi + cj . He failed.
then Q Exercise 12. Hamilton would say in lugubrious tones. Q ¯ so QQ is the length of Q. Hamilton had discovered the quaternions : the set of objects a + bi + cj + dk . c. b. which he termed a "vector". Show that if Q1 and Q2 are quaternions. Hamilton had a flash of insight while strolling with his wife along the Royal Canal: he suddenly realized that the problem of multiplying and dividing vectors could be solved in four -dimensional space. a game in the Hamilton household was for the children to ask at mealtimes. Show that if a. b.36. "No. c. One of the revolutionary aspects of Hamilton's work was the noncommutativity of the multiplication law. although nowadays it would bring a fine for defacing public property. and D. c. quaternions have a conjugation operation. You are probably familiar with the idea of representing vectors in threedimensional space as linear combinations of orthogonal basis vectors i. then ¯ is defined to be a − bi − cj − dk . but with the additional rules that i2 = j 2 = k 2 = ijk = −1. and d are real numbers. have you found a way to divide triples?" In response. He viewed a quaternion a + bi + cj + dk as consisting of a scalar part a and a "pure quaternion" part bi + cj + dk . It is easy to see that QQ ¯ = a2 + b2 + c2 + d2 . where a. the operations of conjugation and multiplication interact with a twist. B .12. j . and d are not all zero.35. 1843. If Q = a + bi + cj + dk . ¯ 1Q ¯ 2 = Q2 Q1 . and d are real numbers. C . and ki = j = −ik . where a. October 16. CLASSROOM DISCUSSION
133
Each cursed his fate that thus their project crossed. and k . According to the next exercise.
. Thus. multiplied according to the usual rules of high school algebra. I can only multiply them.3. b. In analogy with the complex numbers. "Papa. He was so taken with his discovery that he carved it into the stones of Brougham Bridge. Exercise 12. Show that the rules i2 = j 2 = k 2 = ijk = −1 imply the properties ij = k = −ji. How hard their lot who neither won nor lost! Richard Graves (1715–1804) The Festoon Reportedly. jk = i = −kj . quaternions can be used as a tool for the analysis of three-dimensional vectors.37. Exercise 12." The story continues that on the morning of Monday. then the reciprocal 1 of a quaternion is again a quaternion: it can be written in a + bi + cj + dk the form A + Bi + Cj + Dk for suitable real numbers A. This idea comes directly from Hamilton's work. an action that historians seem to consider romantic.
then V W = −V · W + V × W. V W V = −(V · W )V + (V × W )V = −(V · W )V − (V × W ) · V + (V × W ) × V = −(V · W )V + (V × W ) × V. while the multiplication operations on the right-hand side are the scalar product ("dot product") and vector product ("cross product") of the associated vectors. Show that if V and W are pure quaternions. By (12. then V W V = W .3)
The multiplication on the left-hand side is quaternionic multiplication.39.
. Since every vector can be decomposed into the sum of a vector orthogonal to V and a vector parallel to V . BEYOND THE REAL NUMBERS
Exercise 12. if Q denotes the quaternion cos(θ/2) + V sin(θ/2). In summary.38. Show that if the vector W is orthogonal to the unit vector V . According to Exercise 12. (12. We can now investigate how the transformed vector V W V compares to W .134
CHAPTER 12. For example.32. An interesting application of quaternions is to describe motions of threedimensional space. we can write V2 V1 = − cos(θ/2) − V sin(θ/2). then ¯ represents rotation of a vector W by angle θ the transformation W → QW Q about the direction of the vector V . According to the preceding exercise. On the other hand. If vectors V1 and V2 are unit vectors orthogonal to the planes. then the operation corresponding to the composition of reflections is W → V2 V1 W V1 V2 . where V is a unit vector. the quaternionic product V W V of vectors (pure quaternions) is again a pure quaternion. Exercise 12. and consider the transformation that sends a variable vector W into V W V .
(12. In particular. it follows that the transformation W → V W V is reflection in the plane perpendicular to the unit vector V .3). (12. suppose V is a fixed unit vector.4)
the last step following because the vector V × W is orthogonal to the vector V . where V is a unit vector in the direction of the rotation axis. the composition of reflections in two different planes making an angle θ/2 is a rotation of angle θ about the line of intersection of the planes.4) implies that V V V = −V .
12.4. LITERATURE
135
Quaternionic multiplication gives an easy way to compute the composition of rotations. Namely, we multiply the quaternions that represent the two rotations, and the product is a quaternion representing the composite rotation. Exercise 12.40. Rotate space by π/2 radians about the z -axis, then by π radians about the y -axis, and then by π/2 radians about the x-axis. The composite motion is a rotation by what angle about what axis?
Research one of the following paradoxes, write a paper about it, and make a presentation in class about it. You are expected to do more than simply research the topic in the library and on the Internet and report your findings. You should include some original contribution of your own. For example, you might present illustrative examples and your opinion (with explanatory reasons) about a solution of the paradox. A Of course your paper should be computer printed, preferably by L TEX, the de facto standard for mathematical typesetting. Please consult your instructor if you have any questions. Banach-Tarski paradox A ball the size of a pea can be cut into finitely many pieces that can be reassembled into a ball the size of the sun. Braess's paradox Adding capacity to a network can make it less efficient. Carl Hempel's paradox of the ravens Every black raven that Edgar Allan Poe observes gives him more confidence in the truth of the statement: "All ravens are black." The contrapositive statement "Everything that is not black is not a raven" is equivalent. Therefore a purple cow is also a confirming instance of the proposition that all ravens are black. Jacoby's paradox in backgammon Sometimes it can become advantageous for a player to double when the opponent's position improves. 137
If one box contains Y dollars and the other contains 2Y dollars. they may arrive at an outcome that is inferior for all of them. it cannot be on any of the preceding days either. half the time you will gain Y by switching. By induction. Argument 2: Suppose your box contains X dollars. Box B is empty if and only if a Being with superior predictive powers has guessed that you will take both boxes.138
CHAPTER 13. and half the time you will lose Y by switching. Compare "the tragedy of the commons. If heads comes up for the first time at toss n. so you should always switch. Color blindness is not an allowable resolution of the paradox!
. or you may take both boxes. What should you do? The exchange paradox You are offered a choice of one of two identical boxes. you win 2n dollars. You open your choice and observe how much money it contains. then you deduce that it will be on Friday. If you switch." Surprise examination paradox There will be a surprise examination one day next week. Your expected gain by switching is zero. and you are given the information that one of the boxes contains twice as much money as the other one. half the time you will end up with 2X (a gain of X ). Hence the examination cannot be on Friday.000. Now you are offered the chance to switch and take the other box. box B has $1.000 or $0. If the examination has not been given by Thursday. Nelson Goodman's grue-bleen paradox The adjective "grue" applies to all objects examined prior to the resolution of the Riemann hypothesis just in case they are green and to all other objects just in case they are blue. Petersburg paradox A fair coin is flipped repeatedly until heads comes up. it makes no difference whether or not you switch. PROJECTS
Newcomb's problem Box A has $1. Your expected gain by switching is therefore (X − X/2)/2 = X/4. and half the time you will end up with X/2 (a loss of X/2). Should you switch? Argument 1: By symmetry. Confirming instances of the statement "All emeralds are green" are also confirming instances of the statement "All emeralds are grue". so it will not be a surprise.000 in it. You may take either box B alone. What is a fair price to play the game? Prisoner's dilemma In a competitive situation where all parties act rationally.
To determine if all customers are drinking legally.
A 0 4 0 4 4 4
B 3 3 3 3 3 3
C 2 2 2 2 6 6
D 5 1 1 1 5 5
. . the second one is significantly easier to solve. which cards must you turn over?
' $ ' $ ' % & $ '
$ %
25
&
16
% &
Beer
Coke
% &
The Efron dice Consider the following four "unfolded" dice with sides labelled as indicated.1. In a bar. then there is an even number on the other side". . PARADOXES . James Joyce Finnegans Wake page 23
139
The content effect in the Wason selection task Although the following two problems are logically equivalent.13. there is a card in front of each customer with the customer's age written on one side and the name of the customer's drink written on the other side. which cards must you turn over?
' ' $ ' $ ' $
$ %
5
&
8
% &
A
% &
F
% &
2. Each card has an integer written on one side and a letter written on the other side. Why? 1. . To determine the truth or falsity of the statement "If there is a vowel on one side. like a rudd yellan gruebleen orangeman in his violet indigonation . .
. then half the time still remaining must elapse. To test Reality we must see it on the tight-rope. then half . One of them says that this class will never end: before the end of class. PROJECTS If two of these dice are rolled in competition with each other. Perhaps it was. When the Verities become acrobats. the way of paradoxes is the way of truth. . C beats D two times out of three. .
Zeno's paradoxes These are the grandparents of all paradoxes." Oscar Wilde The Picture of Dorian Gray
.140
CHAPTER 13. then A beats B two times out of three. Well. and there are infinitely many halves. and D beats A two times out of three. B beats C two times out of three.
"Was that a paradox?" asked Mr. "I did not think so. we can judge them. Erskine. half the remaining time must elapse.
Your lesson should include homework (and solutions). • Bessel functions • elliptic functions/integrals • Gamma function • Hermite polynomials/functions • hypergeometric functions • Laguerre polynomials/functions • Legendre polynomials/functions • Riemann zeta function • Weierstrass ℘ function • Lambert's W function defined by W (z )eW (z) = z • other (please get the instructor's approval) Some questions you might want to consider are: • How is the function defined? • When was it first defined? • Who first defined it? • What question or event motivated the inventor to think about this function? • What other uses does this function have? • What are some examples that illustrate the importance or application of this function?
.13. SPECIAL FUNCTIONS
141
13.2
Special Functions
Research one of the following special functions. write a paper about it. and prepare a lesson about it that you will present in class.2.
You are also responsible for preparing a homework set for the other students to work to solidify their knowledge of your special function. You should be creative and not simply do a literature search.
.142
CHAPTER 13. PROJECTS
These questions are to get you started. After you write your paper and the instructor reads and returns it. you will teach the rest of the class about your special function using group exercise techniques.
41–51.mit. Chapter 13 is Gardner's original Scientific American column.gz contains an extensive bibliography that was omitted from the published version. New York. is Chapters 13–14 of Martin Gardner's book Knotted Doughnuts and Other Mathematical Entertainments (W. pages 11–23. Blachman and D. with references. Newcomb's problem One source. In the bibliography at the end of the book. New York. the instructor may need to provide hints. American Mathematical Monthly 105 (1998).ps. H. the electronic version available at Chow. "The surprise examination or unexpected hanging paradox". A bibliography follows Chapter 14. Here are some entry points to the literature.edu/~tchow/ unexpected. 3. Marc Kilgour. Another source is T.Appendix A Sources for projects
In case the students are unable to find satisfactory sources for the projects. there are many additional references to the literature on this paradox. Chapter 1. 1986). is titled "The paradox of the unexpected hanging". 1969). Freeman. Surprise examination paradox A source for references about the surprise examination paradox is Chapter 1 of Martin Gardner's book The Unexpected Hanging (Simon and Schuster. The exchange paradox A source for references for the exchange paradox is "Elusive optimality in the box paradox" by Nelson M. 143
. 171–181. no. and Chapter 14 is Robert Nozick's follow-up guest column. Mathematics Magazine 74 (2001). | 677.169 | 1 |
CCSS High School: Geometry (Modeling with Geometry)
[b]This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful remediation worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom.
[color=#1551b5]None of the work contained in this GeoGebra book is my own. This book contains contributions from the following individuals/organizations:
Raul Manuel Falcon Ganfornina ([url]
Dr. Ted Coe ([url]
Walch Education ([url]
EDC in Maine ([url]
Terry Lee Lindenmuth ([url]
Without their amazing talent and creativity, this resource would not have been possible. [/b][/color]
[color=#000000]Teachers can use these resources as a powerful means to naturally [br][br][/color][b][color=#0000ff]1) Foster Discovery Learning[br][/color][color=#0000ff]2) Provide Meaningful Remediation[br]3) Differentiate Instruction, &[br]4) Assess students' understanding.[/color][color=#000000] [/color][/b][br][br][color=#000000]Since any curriculum is [/color][b][i][color=#980000]always[/color][/i][/b][color=#000000] a fluid document, these books, too, will continue to remain works in progress.[/color][br][br][][/color][br][b][color=#0000ff]Students:[/color][/b][color=#000000] [br]It is my hope that these resources help you discover & help reinforce mathematics concepts in a way that makes sense to you. [/color]
[b][color=#980000]These GeoGebra books display the amazing work from several esteemed members of the GeoGebra community. I am truly humbled and amazed by their talents. These comprehensive resources would not have been possible without their contributions. [br][br]I would like to express a [u]HUGE THANK YOU[/u] to[br][br][/color][/b][url= C.M. OR[/url][br][url= Phelps[/url][br][url= Silverman[/url][br][url= Ted Coe[/url][br][url= Cruz[/url][br][url= Lee Lindenmuth[br][/url][url= Manuel Falcon Ganfornina[/url] [br][url= Education[/url][br][url= in Maine[/url][br][/url][color=#1e84cc
[][br][/color][b][color=#0000ff]Students:[/color][/b][color=#000000] [br]It is my hope that these resources help you discover & help reinforce mathematics concepts in a way that makes sense to you.[br][br][/color][b]I would like to express a HUGE THANK YOU to[/b] [url= C.M.Or[/color][/url] [b]and[/b] [color=#0000ff][url= Phelps[/url], [/color][br][b]whose work also appears in this project. [/b][/url][color=#1e84cc][br | 677.169 | 1 |
Math
Introduction
The purpose of this page is to summarize this past year. In 10th grade we have learned Quadratics, Probability, and Similarity. So one thing here that different than other schools is that it's not just memorization skills we have to do lots of critical thinking and working with others to find solutions, ''There won't be any need of the quadratic formula outside of this class, but explaining your thinking and collaboration are things you'll need to do every day.''
Work
So for my quadratic portfolio I had to write about ways that grew in the ways of a mathematician which is: Questioning, GrowthMindset, Hypothesizing/ Experimenting, Explaining/Justifying, Finding Patterns, and Collaborating. I learned how change sets of numbers into another to work on it from there.
For Probability I learned how to calculate the chances of something happening, and the chance of things that had happened.
We also did a project on probability. We made a casino made of casino games that we made using our knowledge of probability. I was one of the people in charge of the design of the casino.
In Similarity I learned a lot about triangles and how find out the height of things using similarity between triangles.
Math was fun this year, although it did have it's up's and downs. Some stretches of mine were asking questions in the beginning of the year, but I did get more comfortable asking questions towards the end of the year. Yet other than that, math was pretty easy for me over all . I don't think I would change anything either, I had a fun time gaining the experience that I did. To push my-self further in math next year I'm thinking about joining Math Honors. Some good moments in math class were helping my classmates understand when they had trouble. I fell that I've grown more comfortable with asking questions and being wrong. | 677.169 | 1 |
ALEX Resources
Title: Logarithms: Undo the Exponential
Description:
In
Standard(s): [MA2015] PRE (9-12) 24: (+) Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. [F-BF5]
Subject: Mathematics (9 - 12) Title: Logarithms: Undo the Exponential Description: In | 677.169 | 1 |
Kaseberg/Cripe/Wildman's respected INTERMEDIATE ALGEBRA is known for an informal, interactive style that makes algebra more accessible while maintaining a high level of mathematical accuracy. This new edition introduces two new co-authors, Greg Cripe and Peter Wildman. The three authors have created a new textbook that introduces new pedagogy to teach you how to be better prepared to succeed in math and then life by strengthening your ability to solve critical-thinking problems. This text's popularity is attributable to the author's use of guided discovery, explorations, and problem solving, all of which help you learn new concepts and strengthen your skill retention.
"synopsis" may belong to another edition of this title.
About the Author:
Alice Kaseberg combines 30 years of secondary school and community college teaching experience with an avid mathematical curiosity, constantly seeking to answer the commonly asked question, "What is mathematics good for?" Her BA in Business Administration, MA in Mathematics, and MS in Engineering Science reflect her keen interest in academics and mathematics applications. She began her career as a teacher in Australia. Then, as a junior high teacher, she class-tested materials from the Mathematics Resource Project, Lane County Math Project, School Mathematics Project (of Great Britain), and Bob Wirtz's Drill and Practice at the Problem Solving Level. Her mentors were early advocates of the discovery approach, laboratory activities, and membership/involvement in professional organizations. Kaseberg has always reached out to mathematics students in a variety of situations. She was a high school teacher for the hearing impaired, where she taught contrasting classes such as Advanced Placement Calculus and remedial mathematics. Always eager to take on new projects, she organized weekly problem-solving activities for the entire department. As a community college teacher (15 years), she has taught a range of courses from Beginning Algebra through Calculus, as well as Strength of Materials and Statics in Engineering. While on sabbatical from teaching, she worked 15 months for the EQUALS Project, Berkeley, CA, writing Odds-on-You and coauthoring the EQUALS Handbook. In addition to her teaching at the school and community college level, she has developed and presented numerous workshops on mathematics applications, content strategies, problem solving, art and math, and teaching strategies. Group work and intense participation by attendees characterize her workshops and her classroom. Her goal in writing textbooks is to help students succeed in mathematics, learn how to learn, and become actively aware of the mathematics in their daily lives. | 677.169 | 1 |
Precalculus with Calculus Previews
Browse related Subjects ...
Read More includes rotation of conics in the rectangular coordinate system, sequences and series, mathematical induction, the Binomial Theorem, systems of equations, partial fractions, systems of inequalities, a brief introduction to counting and probability, and a full discussion of complex numbers. This student-friendly, full-color text offers numerous exercise sets and examples to aid in students' learning and understanding, and graphs and figures throughout serve to better illuminate key concepts. The exercise sets include engaging problems that focus on algebra, graphing, and function theory, the subtext of so many calculus problems. The authors are careful to use the terminology of calculus in an informal and comprehensible way to facilitate the student's successful transition into future calculus courses. New to the Sixth Edition: - Increased number of student aids throughout the text including, examples, figures, marginal annotations, in-example guideline annotations, and Notes from the Classroom sections - Many sections have been rewritten to increase clarity and improve student understanding - Increased number of Calculator/Computer Problems and For Discussion Problems throughout the text - An expanded Final Examination consists of fill-in-the-blank questions, true/false questions, and review exercises - And much more! Key Features: Translating Words into Functions section illustrates how to translate a verbal description into a symbolic representation of a function and demonstrates these translations with actual calculus problems. Chapter Review Exercises include problems that focus on algebra, graphing and function theory, the sub-text of so many calculus problems. Review questions include conceptual fill-in-the-blank and true-false questions, as well as numerous thought-provoking exercises. The Calculus Preview found at the end of each chapter offers students a glimpse of a single calculus concept along with the algebraic, logarithmic, and trigonometric manipulations that are necessary for the successful completion of typical problems related to that concept. Available with WebAssign.
Read Less
Very Good. 1284077268 Great condition with light wear! Supplemental materials such as CDs or access codes may NOT be included regardless of title. Expedited shipping available (2-4 day delivery)! Contact us with any questions! | 677.169 | 1 |
Geometry: Fundamental Concepts and Applications pdf
Geometry: Fundamental Concepts and Applications by Alan Bass
ISBN-10: 0321473310
Paperback: 176 pages
ISBN-10:ISBN-13:
Publisher: Pearson; 1 edition (April 29, 2007)
Language: English
This Geometry workbook makes the fundamental concepts of geometry accessible and interesting for college students and incorporates a variety of basic algebra skills in order to show the connection between Geometry and Algebra. Topics include: A Brief History of Geometry 1. Basic Geometry Concepts 2. More about Angles 3. Triangles 4. More about Triangles: Similarity and Congruence 5. Quadrilaterals 6. Polygons 7. Area and Perimeter 8. Circles 9. Volume and Surface Area 10. Basic Trigonometry
Recommended books
Enjoy Jerk Chicken, Curry Chicken, Fried Snapper, Jamaican Wings, and Many of the Best and Easiest Jamaican and West Indian Recipes. Get your copy of the best and most unique Jamaican recipes from C... | 677.169 | 1 |
Assessment Book Summary
Assessment Book Description
Andrew Er's Maths Worksheets is carefully designed with each exercise arranged in increasing level of difficulty.
Diagnostic tests and examination papers are included to help the pupil prepare for examinations.
The mathematical problems have been tried and tested on many pupils who have shown tremendous improvement after they have practised on them.
Advertisements
Assessment Book Review
It has different difficulty levels and some enjoyed doing the exercises.
Levels: P1-P6 Author: Andrew Er Publisher: EPH Where to buy: Popular Bookstore or at Amazon.com. | 677.169 | 1 |
I'm just thinking if someone can give me a few pointers here so that I can understand the concepts behind free printable third grade math word problems. I find solving problems really difficult. I work part time and thus have no time left to take extra classes. Can you guys suggest any online resource that can help me with this subject?
Hi, Thanks for the instantaneous answer . But could you let me know the details of reliable sites from where I can make the purchase? Can I get the Algebrator cd from a local book mart available near my residence ?
algebraic signs, quadratic equations and side-side-side similarity were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have used it frequently through many algebra classes – College Algebra, Algebra 1 and Basic Math. Simply typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I truly recommend the program. | 677.169 | 1 |
In a lesson I would use this diagram as an introduction to the topic, and also as a follow up to it. It clearly lays out the steps that students need to perform to solve a 2-step equation. I would give the students a copy of this diagram so they could have it to reference when doing these types of problems.2. This helps enhance students learning of the topic because it provides clear step by step instructions on how to solve a 2-step equation.
Zambelletti, dana, two step equations
1.
Two Step Equations Intended for grades 7-8 By Dana Zambelletti
2.
Technology Integration• Rationale – Algebra is a topic that students will be dealing with throughout high school, and it is important that they understand it. Solving equations can be tricky, especially when it comes to multi-step equations, so it is helpful to have a step by step approach to guide students through the process. – By using technology, we can reach a broader range of students. Students learn in all different ways so having multiple outlets available can help tremendously.
3.
Website Reliability• The websites used in this presentation have been thoroughly examined for usefulness and reliability.• The websites are also hyperlinked throughout the presentation for reference.
4.
Khan Academy Video • This video demonstrates how to do a two step equation, and gives students a more visual way to look at it.
5.
Blog• This blog would be a useful resource. It provides different tips and lesson ideas to use in the classroom.
6.
Internet Podcast• This podcast features different topics from algebra, one of them dealing with solving equations. This would be a great resource overall for students because it covers everything they will be learning in an algebra class
7.
Internet Resources• Khan Academy – Khan Academy is a non-profit organization that provides free tutorials. It is a great resource for math, and this link takes you directly to a page where you can practice solving 2-step equations.• FREE – Federal Resources for Educational Excellence provides resources for both student and teacher on different academic topics.
8.
Internet Resources• Wolfram MathWorld – This website is a great resource for any math topic. It features different innovative and interactive ways to explore math problems.
12.
QuizStar• This would be a different way to administer a quiz. This would also be a helpful tool for students who maybe prefer learning online.
13.
Subject-Specific Resource• This website provides different lesson plan ideas based on subject and grade level, as well as provides different activities that are related to the topic.
14.
Additional Resources• Math Tools- this website provides different resources for teaching algebra. It gives different lesson ideas, activities, and technology resources.• Cool Math-this website provides extra practice with equations. It allows you to click on a type of equation you want to practice and then generates different examples for you to solve. This would be great for students who want extra practice, or who might be struggling.
15.
SlideShare• Using Slideshare, I am able to upload my presentation or notes, and students would be able to access them outside of the classroom. This is helpful for students who may not have been in class, or students who may have missed something while taking notes. | 677.169 | 1 |
October 2, 2015
Just in time for your looming first math exams, here are some
expert tips from math professors and graduate students on how to approach your
work and improve your understanding of math:
Making
Connections, Moving Past Procedures. Let's face it: a monkey can plug
numbers into a calculator; that's not your goal. You want to understand what
you're doing.
The
biggest difficulty is that students are often much more focused on a "plug
and chug" learning that demonstrates they can put numbers into a formula
and calculate an answer. But, hitting the right buttons on the calculator does
NOT mean that you fully understand the concept and when/how it is used. That
type of learning can be useful for quizzes, but does not prepare you for the
conceptual understanding and applying concepts that is required on exams.
First-year
Rutgers students have the advantage of taking courses with highly-esteemed
mathematicians knowledgeable about the applications of content as well as the
logic behind mathematical algorithms and procedures. A conceptual understanding of
mathematics allows one to make sense of the WHY's and HOW's of mathematical
procedures for a more interconnected knowledge base of the discipline.
Effective
studying does require some advanced planning. Many mathematics instructors will include a list of
suggested and required homework problems to complete in their syllabi. Take a
look at the amount and complexity of homework problems that are assigned for
the course. Use this list to gauge your competency with the material PRIOR
to lecture and recitations so you will be ready to pose your most pressing
struggles with the material.
PRE-READING the section before attending the lecture helps
you to follow along and will either give you more confidence or will begin to
signal areas of difficulty.Next,
complete as many problems from a particular section as you can before preparing
for the next section.Keep note of the
questions that you can't answer on your own and try to look them up online or
in the solutions manual.
EXAM
REVIEW is essential to patching any holes in your understanding before moving
forward. Review the exam and all problems that you got wrong; meet with your TA
to review any errors that you don't fully understand. The only thing worse than
losing points for a mistake on the first exam is losing points on subsequent
exams for the same mistake!
Getting Help. Once
you've made a real effort, go to office
hours! If you're still not getting it, you will be able to ask questions about
specific problems or ideas, and get more effective help from the TA. Even if
you feel confident in your approach, office hours are a terrific opportunity to
hear other students' questions and deepen your understanding of mathematical content beyond what can be
addressed in an 80-minute class period. Meet with the TA, professor, and/or
tutor at the campus learning center prior to an examination (and throughout the
semester).Remember that tutoring is not
just to learn concepts that you don't understand; it's also for really
mastering material and seeing connections between concepts. It's about gaining
conceptual – versus rote – knowledge.
For exam preparations, find out the structure of the exam in
advance.Is it proof-based? Calculator,
no calculator, or half and half? These are valid questions. "What's on the
exam?" is not. Practice problems under actual exam conditions. Also, know when and where your exams
are! Some exams are given in the usual classroom and at the usual class time.
And others, generally called common-hour exams, are not in the usual time and
place, so you need to pay particular attention to those details. After all, you
can only be successful at a test that you actually take.
Additionally, the mathematics department posts samples of
past examinations. Search the "CourseMaterials" section of the Math website to find review sheets and
sample exams. YouTube or Khan Academy can be used for additional support,
particularly in peer study groups. These sites may provide a new way of
"seeing" a problem or concept with which you're struggling in
understanding.
Keep in mind that math is a truly cumulative subject. With
strong understanding of how basic concepts are interconnected, you can build
higher levels and form a conceptual understanding of mathematics, but with a weak
foundation, you'll struggle in your work as you move higher.
Thanks to John Kerrigan, Alice Seneres,
and Luis Leyva for the words of wisdom! | 677.169 | 1 |
A-Levels Maths: Books that students will need in A-Level Mathematics
Textbooks are a student's best friends. There is generally a need for textbooks while studying A-Levels Mathematics. The whole class would be studying from the same maths books. Let's have a quick glance on some undeniable importance of books in a student's life.
Your topics are organized, so you don't have to search where you left and what you have read. The topics are also in most cases arranged in increasing level of difficulty.
All the topics and subtopics of a particular lesson are present with some practice questions.
The information provided is just in the right way. Textbooks are always compiled keeping in mind the level of the students who are going to study them.
Not just for students, textbooks are an excellent aid for teachers as well.
There is a chance that in some case that book doesn't satisfy your needs. The way out is to seek help from new books and maybe new teachers. If you are studying privately then the following books, will definitely help you in preparing you for your exam. The maths books would cover all the topics that are in your syllabus. I studied these books in my A- Levels too.
Preparing for Pure Mathematics exam – Maths books
Pure Mathematics 1
You would be studying Pure Mathematics 1 in your AS Level.All the Units in the syllabus are provided in the book:
Quadratics
Functions
Coordinate Geometry
Circular measure
Trigonometry
Vectors
Series
Differentiation
Integration[divide icon="circle"]
Pure Mathematics 2 & 3
Units in Pure Mathematics 2
Algebra
Logarithmic and exponential functions
Trigonometry
Differentiation
Integration
Numerical solution of equations
Units in Pure Mathematics 3
Algebra
Logarithmic and exponential functions
Trigonometry
Differentiation
Integration
Numerical solution of equations
Vectors
Differential equations
Complex Numbers[divide icon="circle"]
Paper 4 : Mechanics 1
Units in Mechanics 1
Forces and Equilibrium
Kinematics of motion in a straight line
Newton's laws of motion
Energy, work and power[divide icon="circle"]
Paper 5 : Mechanics 2
Units in Mechanics 2
Motion of a projectile
Equilibrium of a rigid body
Uniform motion in a circle
Hooke's law
Linear motion under a variable force[divide icon="circle"]
Paper 6: Probability and statistics(Unit S1)
Representation of data
permutations and combinations
probability
Discrete Random Variables
The normal distribution[divide icon="circle"]
Paper 7: Probability and statistics(Unit S2)
The Poisson distribution
Linear combinations of random variables
Continuous random variables
Sampling and estimation
Hypothesis tests[divide icon="circle"]
The best and the easiest combination for maths books to go with is:
AS Level is: P1 & M1
A Level is : P3 & S1
[divide icon="circle"]
You can buy these maths books online from Google books, Amazon and other online sellers too. | 677.169 | 1 |
(Original post by Dingo749)
I think you've clicked on maths and further maths
Hope this helps
Ohhhhhhhhh, that makes a lot of sense!! Hopefully I can still get an A* with a lot of work then! | 677.169 | 1 |
3d math
This book shows how to build complete 2D and 3D games with all essential components
from scratch; shapes, image effects, animation, 3D model creation and use,
graphics math, collision detection, 3D audio, split-screen, and networked games. All
code examples are presented in an easy-to-follow, step-by-step format. This book
targets development for the PC and Xbox 360 and introduces development for the
Zune.
Includes a chapter on developing games in Flash for the iPhone!
Gary Rosenzweig's ActionScript 3.0 Game Programming University, Second Edition is the best hands-on tutorial for learning ActionScript 3.0, the programming language behind Flash Professional CS5. You will master all the basics of ActionScript programming by building 16 robust games. One step at a time, you'll learn techniques (and get tested code) that can be adapted to virtually any project, from games to training and advertising....
Drawing a cartoon house can be quite difficult if you don't break down the steps and work with basic geometric shapes. However, if you start simple, you can have a lot of fun building a house that you'd like to live in. Put on your hard hat, and let's get started.
Step 1 - Building the Frame Okay, you took math, and you can draw a fairly straight line. We're going to go back to geometry and make some shapes. If you'd like, you can reference the 1 point perspective tutorial available HERE and learn how to draw in 3D....
This tutorial shows how to use Maple both as a calculator with instant access to hundreds of high-level math routines and as a programming language for more demanding tasks. It covers topics such as the basic data types and statements in the Maple language. It explains the differences between numeric computation and symbolic computation and illustrates how both are used in Maple. Extensive "how-to" examples are used throughout the tutorial to show how common types of calculations can be expressed easily in Maple. | 677.169 | 1 |
You are here
Effective Approaches to Teaching for Developing Algebraic Thinking
Category
Mathematics
Targeted Audience
Gr 6- 10
Course Code
M-MATH-DAT
Title
Effective Approaches to Teaching for Developing Algebraic Thinking
Description
This module addresses the challenge of teaching Algebra to middle school students. Based on three key research-based recommendations, participants will be exposed to misconceptions of what best practice is, potential roadblocks in the implementation of these approaches and the effect of teacher-guidance on students' understanding. This module will also involve an exchange of resources, active collaboration and opportunities to learn from others' experience.
Start Date
Wednesday, November 8, 2017
Dates
Nov 8, 22, 29, Dec 6, | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
156 KB|16 pages
Share
Product Description
This multiple choice assessment contains 32 problems. This document contains a pre and post test. Concepts found on this test include:
Using substitution to evaluate expressions.
Isolating a variable to solve an equation.
Setting up an equation or an expression from a word problem.
Graphing Inequalities.
Skills required for this test include:
Adding and subtracting decimals
Multiplying fractions.
Finding the perimeter of a triangle.
Finding the area of a rectangle.
Multiplying and dividing decimals.
Evaluate numbers with exponents.
Using the distributive property.
Combining like terms. | 677.169 | 1 |
Free Math Answers! (get any answer to any math problem)! - YouTube
This has started the practice of providing online tutoring through various web sites which specialize in providing these kinds of services to students. These online tutoring web sites supplement the lessons a student gets in a college or university classroom. These sites are especially helpful when it comes to homework assignments. They provide homework solutions to the students for the projects they get in their classes -be it economics homework help or physics homework help. The backbone of these sites is the qualified and experienced tutors who work with the online tutoring companies to provide quality guidance to students. They are equipped to provide assistance on almost every subject and any topic. Students can get answers to math problems, for example -algebra homework help or calculus homework help, accounting homework help, economics homework help or college essay help. Students can also find free* tutoring online or college online courses on the Internet.
How to get answers to Math Problems
WebMath - Solve Your Math Problem
Back in May, Phil Daro, who was on the writing team of Common Core standards, told parents at a community education night meeting in San Mateo that the old curriculum promoted students just trying to get answers to math problems. Common Core changes students' ways of thinking from 'if I get the answer right, I'm done' to explaining how they got to the answer, he said, according to a meeting video. This method of studying doesn't allow students in the United States to actually learn math, he said.
Ask any math question and get an answer from our subject ..
Of all subjects, students learn when growing up, math is the most difficult. It is not something that obviously can be connected to everyday life, and students often get bored when talking about it and learning it. In addition to learning about math for an hour or more a day, students have to take home math homework. This is often a waste of time, especially if a student can also do well on a test without studying. To save time to work on other, more important things, it is best to get answers to math problems elsewhere. There are many tactics you can use to do this. | 677.169 | 1 |
book covers the advanced mathematical techniques useful for physics and engineering students, presented in a form accessible to physics students, avoiding precise mathematical jargon and laborious proofs. | 677.169 | 1 |
Mathematics
Mathematics Department
The Mathematics program at Pius XI provides a high level of rigor through best practices. In this department, students learn the analytical and quantitative skills necessary for them to succeed in a 21st century economy.
Most students will begin in one of three classes: Accelerated Algebra, Algebra or Pre-Algebra. Incoming students can be placed into Accelerated Geometry upon successful completion of the Pius XI Algebra Skills Test.
"Preparation for the ACT is integrated throughout the math curriculum to provide students the greatest opportunity to perform at a high level on this test."
Catherine Drasch, Mathematics Faculty
Most colleges and universities require three or four years of college preparatory mathematics. Students are strongly encouraged to go beyond the minimum graduation requirement of three years.
The Math Program Enables Students to Develop the Ability to:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Recognize regularity in repeated reasoning.
Program Highlights:
Median test scores for Pius XI students on the AP Calculus exam were 4 out of a possible 5.
74% of the students taking the Advanced Placement exam received a score of 3 or higher.
The math department was awarded a Phi Delta Kappa – Innovation and Impact Grant to integrate the use of MATLAB (a programming language for numerical computing) into junior level courses.
Pius XI was one of only three schools chosen nationally for this grant.
Technology is incorporated into all Geometry courses through the use of Geometer's Sketchpad.
The Pius XI Math Club – the Mathletes – participate in a variety of competitions including Trig Star and Wisconsin Math League contests.
Prospective students are encouraged to participate in the 8th Grade Math Scholarship Competition.
Students are reminded that for those classes requiring a calculator, they need to purchase a TI-84+ (preferred) or a TI-84 Silver Edition Graphing Calculator. | 677.169 | 1 |
Claim the "An introductory account of certain modern ideas and methods in plane analytical | 677.169 | 1 |
A Graduate Course in Probability
A Graduate Course in Probability
Suitable for a graduate course in analytic probability theory, this text requires no previous knowledge of probability and only a limited background in real analysis. In addition to providing instruction for graduate students in mathematics and mathematical statistics, the book features detailed proofs that offer direct access to the basic theorems of probability theory for mathematicians of all interests. The treatment strikes a balance between measure-theoretic aspects of probability and distribution aspects, presenting some of the basic theorems of analytic probability theory in a cohesive manner. Statements are rendered as simply as possible in order to make them easy to remember and to demonstrate the essential idea behind each proof. Topics include probability spaces and distributions, stochastic independence, basic limiting operations, strong limit theorems for independent random variables, the central limit theorem, conditional expectation and Martingale theory, and an introduction to stochastic processes, particularly Brownian motion. Each section concludes with problems that reinforce the preceding material. | 677.169 | 1 |
6111 Linear Algebra
Course Description
1.Matrices: Definition and Notation, Matrix Algebra, Terminology and Notation for Systems of Linear Equations, Elementary Row Operations and Row-Echelon Matrices, Gaussian Elimination, Gauss-Jordan Elimination, Elementary Matrices and the LU Factorization.
2.Determinants: The Definition of a Determinant, Properties of Determinants, Cofactor Expansions, Cramer's Rule.
3.Vectors: fundamental Concepts of Vectors, Vector Product of two Vectors, Angle between of two vectors, Cross Product of two Vectors, Work Done by Force using Dot Product of two Vectors.
Learning Objective
At the end of the semester, students should know basic concepts of matrices and they enable to solve systems by using different methods such as Gaussian Elimination, Gauss-Jordan Elimination, and Inverse of a Square Matrix. We expect that the students should able to use some properties of determinant and students will use it to solve linear systems. They are also expected to work out some operations of vectors such as dot product of vectors and cross product of two Vectors. | 677.169 | 1 |
About this product
Description
Description
Helping students master secondary school mathematics just got a whole lot easier! Bestselling authors Cheryl Rose Tobey and Carolyn B. Arline provide 25 detailed and grade-level specific assessment probes that promote deep learning and expert maths instruction. Learn to ask the right questions to uncover where and how students commonly get confused. You'll learn how to: * Quickly diagse students' common misconceptions and procedural mistakes * Help students pinpoint areas of struggle * Plan targeted instruction that builds on students' current understandings while addressing difficulties with algebra, functions, logarithms, geometry, trigometric ratios, statistics and probability, and more * Elicit the skills and processes related to the Standards for Mathematical Practices You'll find sample student responses, extensive Teacher Notes, and research-based tips and resources, as well as the QUEST Cycle for effective, hands-on implementation, to help instil new mathematical ideas. This is a great teaching resource with easy-to-implement tools and ideas to build solid mathematics proficiency.
Author Biography
Cheryl Rose Tobey is a senior mathematics associate at Education Development Center (EDC) in Massachusetts. She is the project director for Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students (FACETS) and a mathematics specialist for Differentiated Professional Development: Building Mathematics Knowledge for Teaching Struggling Students (DPD); both projects are funded by the National Science Foundation (NSF). She also serves as a director of development for an Institute for Educational Science (IES) project, Eliciting Mathematics Misconceptions (EM2). Her work is primarily in the areas of formative assessment and professional development. Prior to joining EDC, Tobey was the senior program director for mathematics at the Maine Mathematics and Science Alliance (MMSA), where she served as the co-principal investigator of the mathematics section of the NSF-funded Curriculum Topic Study, and principal investigator and project director of two Title IIa state Mathematics and Science Partnership projects. Prior to working on these projects, Tobey was the co-principal investigator and project director for MMSA's NSF-funded Local Systemic Change Initiative, Broadening Educational Access to Mathematics in Maine (BEAMM), and she was a fellow in Cohort 4 of the National Academy for Science and Mathematics Education Leadership. She is the coauthor of six published Corwin books, including seven books in the Uncovering Student Thinking series (2007, 2009, 2011, 2013, 2014), two Mathematics Curriculum Topic Study resources (2006, 2012), and Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction and Learning (2011). Before joining MMSA in 2001 to begin working with teachers, Tobey was a high school and middle school mathematics educator for ten years. She received her BS in secondary mathematics education from the University of Maine at Farmington and her MEd from City University in Seattle. She currently lives in Maine with her husband and blended family of five children. Carolyn B. Arline is a secondary mathematics educator, currently teaching high school students in Maine. Carolyn also works as a teacher leader in the areas of mathematics professional development, learning communities, assessment, systematic school reform, standards-based teaching, learning and grading, student-centered classrooms, and technology. She has previously worked as a mathematics specialist at the Maine Mathematics and Science Alliance (MMSA) and continues her work with them as a consultant. Carolyn is a fellow of the second cohort group of the Governor's Academy for Science and Mathematics Educators and serves as a mentor teacher with the current cohort. She participated as a mathematics mentor in the NSF-funded Northern New England Co-Mentoring Network (NNECN) and continues her role as a mentor teacher. She serves as a board member of the Association of Teachers of Mathematics in Maine (ATOMIM) and on local curriculum committees. Carolyn received her B.S. in secondary mathematics education from the University of Maine. | 677.169 | 1 |
Chaos and Fractals : An Elementary Introduction Paperback
Share
Description
For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia Sets and the Mandelbrot Set, power laws, and cellular automata. | 677.169 | 1 |
(Original post by intellectual1)
Is there an offical list of calculators which are allowed? | 677.169 | 1 |
Summary: Maths in Practice gives every pupil
the confidence to become a successful learner. The course lays the foundations
for good teaching and learning of mathematics through the key concepts outlined
in the secondary curriculum review. These concepts include competence in
mathematical procedures, creativity, and understanding and using mathematics.
The full range and content of the curriculum are covered and reinforced
through practice of essential mathematical skills and processes. Each chapter
covers a particular attainment target (Number, Algebra, Geometry and Measures
or Statistics) and they are presented in a possible teaching order. This
Practice Book Book covers work at levels 4 to 5 and supports Pupils Book
2.
There is a CD included which contains Personal Tutor examples. A Network
Edition Dynamic Learning CD is also available, including interactive activities
and animations as well as Personal Tutor examples. Dynamic Learning enables
users to navigate interactive pages or menus and launch a wealth of resources.
It makes personalised learning a reality, arranging resources in a way that
suits you, whether you are a student or a teacher, bringing learning to life.
The tools and resources provided also enable teachers to build their
own lessons, populate a VLE with content and use a digital whiteboard to
full effect. Visit our website to find out more about this unique electronic
resource.
A brand new suite of resources written for the new Key Stage 3 Programme
of Study for Mathematics
Provides full support for specialist, non-specialist and newly-qualified
teachers through fully-integrated print and digital resources
Offers flexible routes through the curriculum and provides teachers with
everything they need to create their own schemes of work
Experienced author team comprising of subject and teaching experts
Functional skills clearly highlighted
Full-colour teacher guides give comprehensive guidance and support
Practice books offer ready-made homework or can give extra practice for
use in class
Dynamic Learning CD includes lesson plans with interactive activities and animations
as well as Personal Tutor examples to support the work in the Pupil's Books | 677.169 | 1 |
Avails our Combinatorics Help and See the Positive Results
Students get various combinatorics homework and combinatorics assignment at high school, college and university levels. Combinatorics is that field of math, which covers distinct objects. Combinatorics homework and combinatorics assignment can include other kinds of mathematic topics too, namely algebra and geometry and statistical subjects namely physics and computer science.
Troubles faced by many of the students:
students are unable to understand the basic concept of combinatorics homework;
many students do not even know the basic tools of combinatorics;
accurate online information on combinatorics is never available to students.
Students can avail our services at any time. We complete your combinatorics homework with top quality guaranteed. You can approach us and simply say "do my combinatorics homework". Our expert writers believe in clarifying all the basic concepts to students. They help solving all the problems related to combinatorics. We cover various topics on probability theory, graph theory, design theory and functions.
Our special features:
our writers have immense experience and problem solving ability;
you just need to walk up to us and avail our services;
our assignments are completely original and free of plagiarism.
We provide you with best quality and non-plagiarized assignments. We also offer our services to students at all levels. You just need to ask for and we will provide you every possible combinatorics solution and combinatorics help. Our writers make full use of their experience to make your combinatorics homework and combinatorics assignment the best.
Easy to reach:
you can select the writer of your choice for completing your assignment;
we will complete your combinatorics homework with utmost dedication;
make the payment, the chosen writer will start working on your assignment at once.
Students can avail completely non-plagiarized, unique and professional services. We have highly competent writers having ability to complete anything and meeting any deadline. Buy our combinatorics solutions and combinatorics assignment and benefit in multiple ways. We have set some standards and every writer has to maintain them, if he or she desires to be part of our writing company. Our writers can provide you the lowest possible rate and after an order is placed, you can stay in touch with the writer to stay informed about your project's status. You need not be shy in approaching us, as we can assist in completing any given task in the set time without compromising the quality.
Numbers accompany us from birth until the last days and we can't imagine the life without them. Among all the numbers,… Read more…
APPROVED BY CLIENTS | 677.169 | 1 |
books.google.com - For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated... Is Mathematics?
What Is Mathematics?: An Elementary Approach to Ideas and Methods
LibraryThing Review
User Review - nealjking - LibraryThing
This is a fantastic book for introducing sophisticated mathematical thinking with "elementary" material. It would be a great present for a gifted child with a taste for mathematical thinking.Read full review | 677.169 | 1 |
Mathematics and National Curriculum - Essay Example
Extract of sample Mathematics and National Curriculum
This difference in the opinions enables the abstract mathematics intellect to perform mathematical operations for the sake of mathematics itself, and to use mathematics as a tool to actually resolve the real problems. According to Kister, mathematics has grown into a tremendous structure constituting more than sixty classes of mathematical activities (Kister, 1992). The ideologies of mathematics possess a distinctly extensive verve. For instance, the Babylonian explanation for quadratic equations holds the same significance as it had past 4,000 years (The Georgia Framework, 1996). In the vein of other sciences, mathematics imitates the decrees of the material vicinity around us and serves as an authoritative instructional implement for comprehending nature. Nevertheless, mathematics is yet again classified by its autonomy from the material world. The intangible behavior of mathematics gave rise in relic to the essential difference in opinions of mathematics as a substance of discourse and also as an element for implementation. Mathematical notions are long-lasting and keep on expanding with time. ...Show more
Summary
Mathematics is a tremendously scientific sphere of influence which is featured by the fact that the things which comprise of it are idealized rational theories. These things can, in no way, be recognized straightly through the logics…I will discuss about inequality within the education system during the past and how it has affected the students and pupils. I will then explore the inequality in the National Curriculum and discuss the changes that have taken place to make the differences there are in the system today.
The aim of education has undergone changes from developing the reasoning power of the individual to introducing social reform in the country. Hence curriculum theory has developed from the classics based curriculum of the nineteenth century to the present day emphasis on multiculturalism in the curriculum | 677.169 | 1 |
ALEX Resources
Title: Polynomial Subtraction
Description:
Students Polynomial Subtraction Description: Students
Title: Conquering Polynomials
Description:
In Conquering Polynomials Description: In
Title: Polynomials Divided by Monomials
Description:
Investigation of division of polynomials by mononials.
This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI projectTitle: What's The Real Cost of That Car?
Description:
This is a Commerce and Information Technology lesson plan. A project requiring research, critical thinking and complex decision-making about factoring all the costs of purchasing a large ticket item... a car.
Standard(s): [MA2015] DM1 (9-12) 9: Determine a minimum project time using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms. (Alabama)
Subject: Business, Management, and Administration (9 - 12), or Mathematics (9 - 12) Title: What's The Real Cost of That Car? Description: This is a Commerce and Information Technology lesson plan. A project requiring research, critical thinking and complex decision-making about factoring all the costs of purchasing a large ticket item... a car.
Title: A Geometric Investigation of (a + b)2
Description:
This Title: A Geometric Investigation of (a + b)2 Description: This Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 | 677.169 | 1 |
Prentice hall geometry homework help
Free answers to ALL your math homework. Correct. Pre-algebra Algebra Integrated math Geometry Algebra 2 Trigonometry Precalculus Calculus.The subjects covered in our short videos correspond to the chapters in your book and can help you work through a tough homework.Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry,. geometry, and calculus problem.
Prentice Hall Geometry: Online Textbook Help Course
Instructors are independent contractors who tailor their services to each client, using their own style.Digits Homework Helper Volume 1 Grade 7, 2013, 0133276317, 9780133276312, Prentice Hall,.
Homework Help / Helpful Videos - myips.org
geometry 12 4 - Geometry homework help
Your best solution is to contact MajesticPapers writing service.
Prentice hall geometry homework help | 100% Original
Our class just started discussing a new topic in algebra regarding prentice hall algebra 2 homework help and I did pretty good for most assignments we got but the.This is another assignment that students ask us to help them with.
prentice hall mathematics algebra i homework help
This Report emphasizes on the feedback. the prentice hall geometry homework help. | 677.169 | 1 |
Calculator Set Up Make sure that your calculator has the Plots Off, Y= functions cleared, the MODE and FORMAT are set at "stage left", and the lists are cleared. Turn the Diagnostics on. Press 2nd [...]
Conditional Statements My goal for this unit is to help students see the goal of becoming more logical, methodical, and developing their pre-frontal cortex. After the chapter 1 test, I gave them a copy of [...]
Reviewing Parent Functions In Pre-Calculus class we dove right in and reviewed five of our parent functions. Screen shots of Reviewing Parent Functions Geogebra Activity are shown below. I will likely rework it next year [...] | 677.169 | 1 |
valuable resource of n-syllabus material for mathematics and science teachers at secondary school level, teenagers and parents. It contains written versions of Royal Institution masterclasses on a wide selection of topics in pure and applied mathematics, and very little kwledge is assumed. Topics include chaos theory, meteorology, storage limitations of computers, population growth and decay, and the mechanics of disaurs. This book shows that mathematics can be fun! | 677.169 | 1 |
Maths Past Exam Question Papers With Solutions
Related Book PDF Book Maths Past Exam Question Papers With Solutions: - Home - Its Another Fine Day Adventures In Odyssey - Itil For Beginners The Complete Beginners.IIT JEE and AIEEE Previous Years Papers with Solutions. year solved papers with solution and AIEEE past year. examination contains questions from maths,.Level Examination Past Papers with Answer Guides: Maths India Edition.
CBSE Class 12th Mathematics | Solved Question Papers
PDF 63,80MB Maths Past Exam Question Papers With Solutions
Help Center Detailed answers to any questions you might have.Secondary 2 Maths, Solution: Maths. to these past year papers and solutions.On this page you will find all available past Edexcel Linear Mathematics A GCSE Papers, Mark Schemes. exam papers on.The following IGCSE Additional Maths fully worked solutions.PDF Book Library Maths Past Exam Question Papers With Solutions Summary: Size 49,59MB Maths Past Exam Question Papers With Solutions Ebook Scouting for Maths Past.IB Higher Level Video Past Paper Worked Solutions. June 30,.Documents Similar To Vectors Questions with Answers Skip carousel. IGCSE Maths Past Examination Papers Classified by Topic.A Level Zimsec Past Exam Questions.pdf. Zimsec O Level Maths Past Exam Papers Students have had these since September 2013 to use throughout the year.
Related Book Epub Books Maths Past Exam Question Papers With Solutions: - Home - Simon Rising An After The Crash Superhero Novel Volume 1 - Simon Schuster Handbook.Get Fully Solved Specimen Questions for All Papers of CIE IGCSE Mathematics.A web page with links to a complete set of hand-written solutions to all MEI Core 4 papers from 2005 to 2013.
Practice Exam Questions – Level 2, Adult Numeracy.
Each free Set of GCSE Maths Past Papers contains between 50 and 110 Higher and Foundation questions.Find CBSE Class XII Board Exam Mathematics Solved Question Papers for the. they can do it easily by checking out previous year question papers and their solutions.
Math Exams With Solutions
Exam papers are located in the Exam Paper tab while solutions are located.UC Merced old calculus exams with solutions, Math 21. Dr. Math has a list of questions and answers.Answers to practice percentage questions for Level 2 numeracy. | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.