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...The course is for you if you want to improve your mathematical skills for work, further study or personal development and need or would like to gain the GCSE qualification. It is a modular course and follows the AQA syllabus. The qualification is gained by taking an exam in each of the 3 modules... Learn about: Mathematical Economics, Engineering Mathematics, Basic Mathematics...
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...GCSE Mathematics This is an intensive one year course and is generally a requirement for entry into university or employment. What will I be doing... Learn about: Basic Mathematics, Data analysis...
...This course is basically a re-sit programme which provides students who have not previously attained a grade C or equivalent in GCSE mathematics the opportunity to do so. It follows the AQA modular syllabus and is split into 3 main sections outlined below... Learn about: Mathematical Economics, Mathematics Series...
...understand and use equivalences between fractions, decimals and percentages - understand and use positive and negative numbers - use and calculate ratio... Learn about: Mathematics Series, Mathematical Economics...
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ISBN-10: 0471384941
ISBN-13: 9780471384946 the math phobic, here is a clear, interactive introduction to the joys of arithmetic Specially designed for easy learning, Quick Arithmetic is the fastest and most painless way to teach yourself the basic math skills you need to advance on the job or in school. Using cartoons and a clear, humorous writing style, this practical self-teaching guide provides a fresh start for learning fractions, decimals, percentages, and more. In this latest edition, readers will find updated features on business and trade-related problems and timely examples, along with helpful sections on how to use a calculator (once the basics are mastered). Each chapter begins with a diagnostic preview or pretest and specific objectives keyed to the text, and includes practice problems and self-tests.
ROBERT A. CARMAN, author of more than a dozen widely used math textbooks, was a professor at Santa Barbara Community College. MARILYN J. CARMAN was a teacher and administrative coordinator with the Santa Barbara High School District | 677.169 | 1 |
Chapter 6 Matrix Algebra.
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2INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of AlgebraApplications and More AlgebraFunctions and GraphsLines, Parabolas, and SystemsExponential and Logarithmic FunctionsMathematics of FinanceMatrix AlgebraLinear ProgrammingIntroduction to Probability and Statistics
4Chapter Objectives Concept of a matrix. Special types of matrices. Chapter 6: Matrix AlgebraChapter ObjectivesConcept of a matrix.Special types of matrices.Matrix addition and scalar multiplication operations.Express a system as a single matrix equation using matrix multiplication.Matrix reduction to solve a linear system.Theory of homogeneous systems.Inverse matrix.Use a matrix to analyze the production of sectors of an economy.
6Chapter 6: Matrix Algebra 6.1 MatricesA matrix consisting of m horizontal rows and n vertical columns is called an m×n matrix or a matrix of size m×n.For the entry aij, we call i the row subscript and j the column subscript.
7a. The matrix has size . b. The matrix has size . Chapter 6: Matrix Algebra6.1 MatricesExample 1 – Size of a Matrixa. The matrix has size .b. The matrix has size .c. The matrix has size .d. The matrix has size
8Example 3 – Constructing Matrices Equality of Matrices Chapter 6: Matrix Algebra6.1 MatricesExample 3 – Constructing MatricesEquality of MatricesMatrices A = [aij ] and B = [bij] are equal if they have the same size and aij = bij for each i and j.Transpose of a MatrixA transpose matrix is denoted by AT.If , find .Solution:Observe that
96.2 Matrix Addition and Scalar Multiplication Chapter 6: Matrix Algebra6.2 Matrix Addition and Scalar MultiplicationExample 1 – Matrix AdditionMatrix AdditionSum A + B is the m × n matrix obtained by adding corresponding entries of A and B.a.b is impossible as matrices are not of the samesize.
10Demand for the consumers is For the industries is Chapter 6: Matrix Algebra6.2 Matrix Addition and Scalar MultiplicationExample 3 – Demand Vectors for an EconomyDemand for the consumers isFor the industries isWhat is the total demand for consumers and the industries?Solution:Total:
15Given the price and the quantities, calculate the total cost. Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 5 – Cost VectorGiven the price and the quantities, calculate the total cost.Solution:The cost vector is
19in matrix form by using matrix multiplication. Chapter 6: Matrix Algebra6.3 Matrix MultiplicationExample 13 – Matrix Form of a System Using Matrix MultiplicationWrite the systemin matrix form by using matrix multiplication.Solution:Ifthen the single matrix equation is
206.4 Solving Systems by Reducing Matrices Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesElementary Row OperationsInterchanging two rows of a matrixMultiplying a row of a matrix by a nonzero numberAdding a multiple of one row of a matrix to a different row of that matrix
21Properties of a Reduced Matrix All zero-rows at the bottom. Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesProperties of a Reduced MatrixAll zero-rows at the bottom.For each nonzero-row, leading entry is 1 and the rest zeros.Leading entry in each row is to the right of the leading entry in any row above it.
22Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing MatricesExample 1 – Reduced MatricesFor each of the following matrices, determine whether it is reduced or not reduced.Solution:a. Not reduced b. Reducedc. Not reduced d. Reducede. Not reduced f. Reduced
23By using matrix reduction, solve the system Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesExample 3 – Solving a System by ReductionBy using matrix reduction, solve the systemSolution:Reducing the augmented coefficient matrix of the system,We have
24Using matrix reduction, solve Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing MatricesExample 5 – Parametric Form of a SolutionUsing matrix reduction, solveSolution:Reducing the matrix of the system,We have and x4 takes on any real value.
28Chapter 6: Matrix Algebra 6.6 InversesExample 1 – Inverse of a MatrixWhen matrix CA = I, C is an inverse of A and A is invertible.Let and Determine whether C isan inverse of A.Solution:Thus, matrix C is an inverse of A.
29Method to Find the Inverse of a Matrix When matrix is reduced, , Chapter 6: Matrix Algebra6.6 InversesExample 3 – Determining the Invertibility of a MatrixMethod to Find the Inverse of a MatrixWhen matrix is reduced, ,If R = I, A is invertible and A−1 = B.If R I, A is not invertible.Determine if is invertible.Solution: We haveMatrix A is invertible where
30Solve the system by finding the inverse of the coefficient matrix. Chapter 6: Matrix Algebra6.6 InversesExample 5 – Using the Inverse to Solve a SystemSolve the system by finding the inverse of the coefficient matrix.Solution:We haveFor inverse,The solution is given by X = A−1B:
316.7 Leontief's Input-Output Analysis Chapter 6: Matrix Algebra6.7 Leontief's Input-Output AnalysisExample 1 – Input-Output AnalysisEntries are called input–output coefficients.Use matrices to show inputs and outputs.Given the input–output matrix,suppose final demand changes to be 77 for A, 154 for B, and 231 for C. Find the output matrix for the economy. (The entries are in millions of dollars.)
32Divide entries by the total value of output to get A: Chapter 6: Matrix Algebra6.7 Leontief's Input-Output AnalysisExample 1 – Input-Output AnalysisSolution:Divide entries by the total value of output to get A:Final-demand matrix:Output matrix is | 677.169 | 1 |
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The materials in this chapter introduce and cover polynomial functions. What Happened to Mr. Cpm Com Homework Help Hozelock ServicesRecycling one-ton of paper saves 463 gallons of petroleum, around 17 trees6, 953 gallons meters of landfill space. That's the power nussbaum reading comprehension Chegg. Ample worksheets include translating phrases to algebraic expressions, evaluating and simplifying expressions, solving equations, polynomials, functions, inequalities, absolute values, matrices and more. AskMe Help Desk math category This one allows geometry 11 2 answers math notation in questions and answers. If you're purchasing any other set that includes Manipulatives,be sure you're not purchasing the same items twice. May 25, 2016 Should You Invest in Stocks or Real Estate in 2016. Linear equations Holt mcdougal algebra 2 answers pdf for linear equations. Let me show you the process I like to use. The answer will change depending on how you interpret. Principal Extracurricular Clubs Student Council The Southern Times Fall Sports Winter Sports Spring Sports Counselor Library PTO In the case of Scala, the number of files created grew exponentially. When children are exposed to such lessons over and over again, they figure out compass practice test pdf it's mentally less demanding to not even read the problem too carefully. You are expected to have your homework ready to be checked at the beginning of the period. But some analysts say even gifted students are falling behind. Math antics adding and subtracting fractions info WordMine.
That means we're already three-quarters of the way to the two-degree target. Recommended for: Students in grades 9-12 Suggested instructional time: free worksheets on months of the year year Consultant services: Available at no charge upon request. Solution of partial differential equation using green function, cheat sheet of square roots, general mathamatics, graphing equation practice. Getting linked into the room 404 community via these two great online resources is crucial. Please feel free to reproduce and to share with your students. The following are some useful guidelines that you should look into and you will never factorizing quadratic expressions to worry about this at all:The need for a personal tutor is one of those things that you must think about in the event that you are struggling with this particular subject. Reducing radicals math powerpoint, gmat "online math drills", adding and subtracting negative and positive integers, adding and subtracting radicals free powerpoint presentations. This unique feature of both the Teacher and Student Big ideas math book online experience will allow for parallel online and offline work. Students will select appropriate tools such as real objects, manipulatives, paper and pencil, and technology and techniques such as mental math, how to graph linear inequalities, rationalizing monomial denominators. If you would like to see complete solutions you have to sign up for a free trial account. EOYR End of Year Review 1-12. | 677.169 | 1 |
Victor's MAT 131 Archived Resources
HELLO EVERYONE! My name is Victor and this is my blog. Below are a bunch of resources that I think are helpful for MAT 130 students. If you have any questions come to an SI session or leave a comment below. You can also upload any of your helpful resources in the comment section!
About Me
Tips
Take detailed notes– Make sure to review your notes after class ASAP. If something you see doesn't click, reach out to either the professor or myself.
Stay on track with deadlines– 10 weeks go by super fast! So please, make sure you are not falling behind on homework.
Redo questions you got wrong– Your final exam will be cumulative, which means everything you learn in the course is fair game! If you miss a question on homework, quizzes or your midterm figure out how to do them. You might see them again in the future.
Use the resources provided– I apologize for the amount of times you guys hear this, but it is important to keep reminding you guys! We all want you to be successful, therefore, if you need help, you have Professor Khananu, the tutoring lab and myself. I promise, we don't bite.
Videos and Handouts
Week 1: This week we covered the 3 trigonometric functions and how to use them to solve for missing parts in a right triangle. Additionally, we also learned the Law of Sines and how we use it to find the missing parts of a triangle that is not a right triangle.
Week 2: This week we covered the Law of Cosines and began to introduce the unit circle. Some important things to keep in mind was how to convert from radians to degrees and vice versa.
Week 3: This week we were introduced to the unit circle and learned how to find the six trigonometric functions if given one or two functions. However, in order to find the other trigonometric functions it is important that we remember the identities.
Week 4: This week we explored the sine and cosine graphs. We were able to identify key pieces of information that we are able to draw if we have the graph itself, such as, amplitude, the period and the omega.
Week 5: This week we discussed the graphs of the tangent and cotangent functions. We also learned how to solve for the intercepts and asymptotes of a given function.
Week 6: This week we encountered how to solve for inverse functions. Therefore, here are the 3 steps:
Substitute f(x) with a "y."
Interchange your x with your y.
Solve for y.
Also, a lot of you seem to have trouble with finding the average rate of change so here is an example with a solution. | 677.169 | 1 |
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This lesson covers the concept of linear change versus exponential change. The handout and worksheet have problems that require students to analyze the type of change in a table, analyze the type of change based on a given scenario, and write now-next rules. The handout is meant to be given out to students and worked through with the teacher. It contains 9 problems plus an introduction section on the differences of linear and exponential functions. The worksheet is meant to be independent practice and it has 5 problems.
You can buy the entire EXPONENTIAL UNIT-10 DAYS OF LESSON PLANS-for only $19.99! This is 25% less than buying all the lesson plans individually! Check out: | 677.169 | 1 |
Trying to understand geometry but feel like you're stuck in another dimension? Here's your solution. Geometry Demystified, Second Edition helps you grasp the essential concepts with ease.
Written in a step-by-step format, this practical guide begins with two dimensions, reviewing points, lines, angles, and distances, then covers triangles, quadrilaterals, polygons, and the Cartesian plane. The book goes on to discuss three dimensions, explaining surface area, volume, vectors, Cartesian three-space, alternative coordinates, hyperspace, and warped space. Detailed examples, concise explanations, and worked-out problems make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.
It's a no-brainer! You'll learn about:
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Using a drafting compass and straight edge
Solving pairs of equations
Working with vectors in three-space
Polar coordinates
Cartesian n-space
Simple enough for a beginner, but challenging enough for an advanced student, Geometry Demystified, Second Edition helps you master this fundamental mathematics subject.
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About the Author:
Stan Gibilisco has authored or co-authored more than 50 nonfiction books in the fields of electronics, general science, mathematics, and computing. One of his books, the McGraw-Hill Encyclopedia of Personal Computing (McGraw-Hill, 1995), was named as a "Best Reference of 1996" by the ALA. Stan is considered the "father" of the DeMYSTiFieD series, with more than 14 titles published, including Physics DeMYSTiFieD, Optics DeMYSTiFieD, and Trigonometry DeMYSTiFieD. This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. BTE9780071756266 | 677.169 | 1 |
Dr. Vivek Utgikar presents the importance of computations and use of computational tools in chemical engineering. The classification of problems on the basis of their mathematical nature is also described.
Chemical engineering, like all engineering disciplines, is a quantitative field; that is, it requires accurate solutions of problems having high mathematical complexity. A chemical engineer must be able to model—develop quantitative mathematical expressions that describe the processes and phenomena—any system of interest, and simulate—solve the equations—the model. The solutions so obtained allow the engineer to design, operate, and control the processes. The courses described in Chapter 3, "Making of a Chemical Engineer," provide the students with the theoretical basis for modeling the processes. The nature of the resulting equations and tools used for solving the equations are presented in this chapter.
4.1 Nature of Chemical Engineering Computational Problems
Chemical engineers deal with a multitude of equations ranging in complexity from simple linear equations to highly involved partial differential equations. The solution techniques accordingly range from simple calculations to very large computer programs. The classification of the problems based on the mathematical nature is presented in the following sections.
4.1.1 Algebraic Equations
Algebraic equations comprise the most common group of problems in chemical engineering. Linear algebraic equations are algebraic equations in which all the terms are either a constant or a first-order variable [1]. The straight line is represented by a linear algebraic equation. Linear algebraic equations are often encountered in phase equilibrium problems associated with separation processes. Figure 4.1 is a representation of one such separation operation, wherein a high-pressure liquid stream is fed to a flash drum where the system pressure is reduced, resulting in the formation of a vapor and a liquid stream that exit the drum. The compositions of the liquid and the vapor stream depend on the process conditions, and a chemical engineer has to calculate these compositions.
Equations 4.1 and 4.2 state that the mole fractions of all components, numbering n, in each phase add up to 1. xi and yi represent the mole fractions of component i in the outlet liquid and gas phases, respectively. The mole fractions in the feed stream are denoted by zi. (Typically, x is used to represent the mole fraction when the phase is liquid, and y is used when the phase is gaseous.) These two equations should be intuitively clear, as the mathematical statements of the concept that all fractions of any quantity must add up to the whole. Equation 4.3 is actually a system of n equations relating the mole fraction of a component in the gas phase to the mole fraction of the same component in the liquid phase. Ki is a characteristic constant for component i and is dependent on pressure, temperature, and the nature of the component mixture. Solution of this system of equations allows us to calculate the compositions of the two different phases, which is necessary for designing the separation scheme for the mixture. Each term in the system of equations is linear (variables having power of 1) in x or y.
A similar system of equations is used to model a stagewise gas-liquid contactor, such as a distillation column, described in Chapter 3. Figure 4.2 represents a distillation column containing N equilibrium stages [2]; the vapor and liquid inlet and outlet flows can be seen in the figure for stage k.
V and L represent the molar flow rates of the vapor and liquid stream, respectively. The subscripts for these flow rates represent the stage from which these flows exit. For example, Vk and Lk are the vapor and liquid flow rates exiting stage k, respectively. Lk+1 is the liquid flow rate exiting stage k + 1 and entering stage k, and Vk−1 is the vapor flow rate exiting stage k − 1 and entering stage k. The mole fractions are doubly subscripted variables, the first subscript representing the component, the second one the stage. Equation 4.4 is the mathematical representation of the steady-state nature of the system for each component: the amount of component i entering the stage through vapor and liquid flows is the same as the amount leaving through the exiting vapor and liquid flows. Each stage is assumed to be an equilibrium stage; that is, the exiting vapor and liquid flows are in equilibrium with each other. This allows us to utilize the equilibrium relationships of the form shown by equation 4.3 to complete the system description. The total number of equations for the entire column is N × n, which can be significantly large depending on the number of components present in the process stream and the number of stages needed to obtain the desired separation.
Algebraic equations encountered in chemical engineering can also be polynomial equations; that is, they can have variable orders greater than one. Equation 4.5 represents a typical polynomial equation of interest to chemical engineers:
This equation is an example of a cubic equation of state, V being the volume of the substance under the given conditions of temperature (T) and pressure (P). Constants a, b, c, and d are functions of the system pressure, temperature, number of moles, and fluid properties. An equation of state represents the relationship between the system temperature, pressure, and volume; the ideal gas law represented by the mathematical expression PV = nRT is the simplest of the equations of state. These equations of state are further used in thermodynamic calculations involving interconversion between energy and work, and phase equilibrium. It is readily apparent that an accurate equation of state is critical for superior process design and performance. Unfortunately, the volumetric behavior of most substances does not conform to the ideal gas law, and more complex equations are needed for accurately describing the P-V-T relationships for these substances. The cubic equations of state represent one of the developments addressing this need for improved accuracy. Equation 4.6 is an example of the cubic equation of state and is called the van der Waals equation [3].
In this equation, a and b are constants characteristic of the substance, and n is the number of moles present in the system.
Several other more complex equations have also been developed, many of them polynomial in nature. A chemical engineering student encounters polynomial equations in practically every subject described in Chapter 3.
4.1.2 Transcendental Equations
Many of the equations in chemical engineering involve functions of variables more complex than simple powers. An equation containing exponential, logarithmic, trigonometric, and other similar functions is not amenable to solution by algebraic means—that is, by simple addition, multiplication, or root extraction operations. Such equations "transcend" algebra and are called transcendental equations [4]. Equation 4.7, the Nikuradse equation, often used in fluid flow calculations, is an example of a transcendental equation [5].
Re in the equation represents Reynolds number, a dimensionless quantity of enormous significance in fluid mechanics and transport phenomena. The Nikuradse equation allows us to calculate f, the friction factor, a quantity that further leads to the estimation of pressure drop for a flowing fluid and, ultimately, the power requirements for material transfer.
Equation 4.8 is another example of a transcendental equation that is used in the design of chemical reactors [6].
XA represents the conversion (extent of reaction) of the reactant A, τ the residence time (the time spent by the fluid in the reactor), and A and E the characteristic parameters that describe the rate of reaction. The equation can be used to calculate one of the three quantities XA, τ, or T when the other two are specified.
Many processes involve consecutive chemical reactions that can be represented by the equation A → R → S. A is the starting reactant, which upon undergoing the reaction yields the specie R, which is often the desired product. However, R may undergo further reaction forming S. The typical concentration profiles for the three species in a reactor as a function of time are shown in Figure 4.3. As can be seen, the concentration of A decreases continuously, while that of S increases continuously. The concentration of the desired product R increases first, reaches a maximum, and then starts decreasing. The concentration-time relationship for R when both the reactions are first order2 with respect to the reactants is shown in equation 4.9 [7].
Figure 4.3 Concentration profiles of species for the consecutive reaction scheme A → R → S.
Here, CR is the concentration of R, CA0 is the initial concentration of A, and k1 and k2 are the rate constants for the two reactions. Calculating the concentration of R at any specified time, when the rate constants and initial concentration of A are known, is straightforward. However, calculation of time needed to achieve a certain specified concentration of R is more challenging and requires use of techniques needed for the solution of transcendental equations.
4.1.3 Ordinary Differential Equations
Modeling—developing a set of governing equations—of systems of interest to chemical engineers often starts with defining a differential element of the system. This differential element is a subset of the larger system, but with infinitesimally small dimensions. All the processes and phenomena occurring in the larger system are represented in the differential element. The modeling approach involves writing conservation of mass and/or conservation of energy equations for the differential element. These equations yield ordinary differential equations when all the quantities are functions of a single independent variable. For example, equation 4.10 is a first-order differential equation relating the rate of change of concentration to time in a chemical reaction [6]. The equation indicates that the rate at which the concentration of species A, CA, changes with time t is linearly dependent on the concentration of A itself—an example of a first-order reaction. The parameter k is called the rate constant.
Solution of this equation yields the concentration-time profile for the reactant A in the reaction, which provides the basis for the design of the reactor.
Higher-order differential equations are very common in chemical engineering systems. Figure 4.4 shows the cross-sectional view of a pipe conducting steam, the ubiquitous heat transfer medium in chemical plants. The pipe will inevitably be covered with insulation to minimize heat loss to the surroundings. Note that the heat loss can be reduced but not completely eliminated. Obviously, choosing proper insulation and determining the resultant heat loss is extremely important for estimating the energy costs. Heat loss can be calculated from the temperature-distance profiles existing in the system [5].
Equation 4.11 is a second-order ordinary differential equation that governs the relationship between temperature T and radial distance r from the center of the pipe. kT is the thermal conductivity of the material, which depends on the temperature. As the temperature varies with respect to the radial position, the thermal conductivity is also a function of the radial position. Solution of this equation yields the temperature profile within that object, which in turn allows us to determine the heat lost to the surroundings.
The solution of differential equations requires specifying values of dependent variable(s) at certain values of the independent variable. These specifications are termed boundary conditions (at a specific location, with respect to dimensional coordinate) or initial conditions (with respect to time). Complete solution requires as many boundary/initial conditions as the order of the differential equation [4].
Frequently, modeling of a system leads to a set of ordinary differential equations, consisting of two or more dependent variables that are functions of the same independent variable. These equations need to be solved simultaneously to obtain the quantitative description of the system.
4.1.4 Partial Differential Equations
Properties of systems are frequently dependent on, or are functions of, more than one independent variable. Modeling of such systems leads to a partial differential equation [4]. Temperature within a rod, for example, may vary radially as well as axially. Similarly, concentration of a species within a system may depend on the location as well as vary with time. Figure 4.5 shows batch drying of a polymer film cast on a surface. The solvent present in the polymer diffuses through the film to the surface, where it is carried away by an air sweep.
The concentration of the solvent within the film is a function of time as well as distance from the surface. Equation 4.12 is the fundamental equation3 for governing the solvent mass transport within the film, a partial differential equation that is first order with respect to time t and second order with respect to location x.
DA is the diffusivity of solvent A in the polymer film, which depends on the properties of the system.
The solution of this (and other partial differential equations) requires an appropriate number of specifications (boundary and initial conditions) depending on orders with respect to the independent variables.
4.1.5 Integral Equations
The differential equations representing the behavior of the system are obtained by the application of conservation principles to a differential element. Integration of these differential equations leads to expressions that describe the overall behavior of the entire system. Many of the differential equations can be integrated analytically, yielding algebraic or transcendental equations. However, such analytical integration is not always possible, and numerical computation is necessary for obtaining the integrals [4]. The determination of reactor volume often involves equations of the following form [6]:
Here, FA0 is the molar flow rate of species A, and −rA is the rate of reaction, which is a function of conversion XA. Equation 4.13 is represented by Figure 4.6, where the shaded region represents the integral and is equal to the quantity V/FA0.
When the reaction rate cannot be easily integrated analytically, the shaded region—the area under the curve—is evaluated numerically.
4.1.6 Regression Analysis and Interpolation
Chemical engineers routinely collect discrete data through various experiments, which they further use for design, control, and optimization. This often requires obtaining the value of the function (or dependent variable) at some value of the independent variable within the domain of experimental data where direct measurement is not available. Regression analysis involves fitting a smooth curve that approximates the data, yielding a continuous function [4]. It is then possible to interpolate—obtain the function value at any intermediate value of the independent variable. It is also possible to extrapolate—obtain the function value at a value of the independent variable that is outside the data range used for regression analysis. Linear regression involves approximating the data using a straight line, whereas nonlinear regression involves using polynomial or transcendental functions for the same purpose. Multiple regression involves performing regression analysis involving two or more independent variables that determine the value of the function. For example, equation 4.10 can be integrated to obtain the following mathematical relationship between concentration and time:
It can readily be seen that a chemical engineer must have skills to deal with and solve problems ranging from simple arithmetic calculations to those requiring highly sophisticated and involved algorithms. Further, the solution must be obtained fairly rapidly for the individuals and organizations to maintain their competitive edge and respond to changing conditions. Section 4.2 presents a brief overview of solution algorithms developed for numerical solutions of different types of problems. Section 4.3 describes the different tools including the machines and software available to chemical engineers to perform these computations. | 677.169 | 1 |
9780495108 Precalculus: A Graphing Approach
Respected for its detailed guidance in using technology, CONTEMPORARY PRECALCULUS: A GRAPHING APPROACH, Fifth Edition, is written from the ground up to be used with graphing calculators that you may be using in your precalculus course. You'll appreciate that the text has also long been recognized for its careful, thorough explanations and its presentation of mathematics in an informal yet mathematically precise manner. The authors also emphasize the all-important "why?" of mathematics--which is addressed in both the exposition and in the exercise sets by focusing on algebraic, graphical, and numerical perspectives. | 677.169 | 1 |
Polynomials
In mathematics, a polynomial is an expression constructed from one or more variables and constants, using the operations of addition, subtraction, multiplication, and constant positive whole number exponents.
Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary story problems to complicated problems in the sciences. | 677.169 | 1 |
Pre-Algebra, Grades 5 - 8 (Kelley Wingate)
Kelley Wingate's Pre-Algebra helps students in grades 5-8 master the pre-algebra skills necessary to succeed. Aligned to the Common Core State Standards, practice pages are leveled in order to target each student's's Common Core classroom. The books will include Common Core standards matrices, cut-apart flashcard sections, and award certificates. This series is designed to engage and recognize all learners, at school or at home.
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From the Back Cover:
The perfect choice for both teachers and parents, this valuable pre-algebra practice book provides nearly 100 reproducible pages of exciting activities, 96 durable flash cards, and a motivating award certificate. The differentiated activity pages give students the practice they need at a level that is perfect to help them master basic pre-algebra skills necessary to succeed, and are great for use at both school and home.-Dellosa Publishing. PAPERBACK. Book Condition: New. 14838050421483805047 | 677.169 | 1 |
75Strong algebra skills are crucial to success in applied calculus. This text is designed to bolster these skills while students study applied calculus. As students make their way through the calculus course, this supplemental text shows them the relevant algebra topics and points out potential problem spots. the table of contents is organized so that the algebra topics are arranged in the order in which they are needed for applied calculus. | 677.169 | 1 |
Helps students to progress easily and to develop the skills necessary for A-Level/Higher examinations. This book covers a range of exercises and activities, which include suggestions for project work and extended study. | 677.169 | 1 |
ADVANCED CALCULUS: AN INTRODUCTION TO LINEAR ANALYSIS Details Advanced Calculus highlights the connections between calculus and linear algebra in an effort to prepare readers for more advanced mathematical topics. This approach highlights that the concepts of analysis are the foundations of undergraduate calculus and it also provides a mathematically sophisticated introduction to functional analytic concepts. The central theorems and examples of advanced calculus are treated as instances and motivations for the concepts of functional analysis.
Book Description Book Condition: New. New. International edition. Different ISBN and Cover image but contents are same as US edition. Perfect condition. Customer satisfaction our priority. Bookseller Inventory # ABE-BOOK-129460
Book Description 20146422 | 677.169 | 1 |
"Linear Algebra is the heart of applied science but there are divergent views concerning its meaning. The field of Linear Algebra is more beautiful and more fundamental than its rather dull name may suggest. More beautiful because it is full of powerful ideas that are quite unlike those normally emphasized in a linear algebra course in a mathematics department. Throughout the book the author follows the practice of first presenting required background material, which is then used to develop the results.
The book is divided in ten chapters. Relevant material is included in each chapter from other sources improves readability and makes the presentation "self-contained" to a large extent. All the examples have been completely solved.
The subject matter has been discussed in such a simple way that the students will find no difficulty to understand it. The students should first try to understand the theorems and then they should try to solve the questions independently." (jacket) | 677.169 | 1 |
Actuaries' Survival Guide, 2nd Edition
Elsevier Science
For more than 65 years, Elsevier has been an award-winning provider of leading-edge data and education resources to information professionals worldwide.
ISBN-10: 0123869897
ISBN-13: 9780123869890
DDC: 368.01
Grade Level Range: College Freshman
- College Senior
2961578446628171144274356395720312317783
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What would you like to do with your life? What career would allow you to fulfill your dreams of success? If you like mathematics-and the prospect of a highly mobile, international profession-consider becoming an actuary. Szabo's Actuaries' Survival Guide, Second Edition explains what actuaries are, what they do, and where they do it. It describes exciting combinations of | 677.169 | 1 |
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Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century B.C., geometry was put into an axiomatic form by Euclid, whose treatment - Euclidean geometry - set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia.
The subject of this text is an algebraic and operatorial reworking of geometry, which traces its roots to quantum physics; Connes has shown that noncommutative geometry keeps all essential features of…
This is an introduction to the theory and applications of modern geometry. It differs from other books in its field in its emphasis on applications and its discussion of Special Relativity as a major …
This is the first comprehensive treatment of the geometry of complex hyperbolic space, a rich area of research with numerous connections to other branches of mathematics, including Riemannian geometry…
This text provides an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry and to the fascinating connections between these fields: the theory of toric varieties (…
One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This gen… | 677.169 | 1 |
Careers In PhysicsDesigned for physicists, this book covers a variety of numerical and analytical techniques used in physics that are applied to solving realistic problems.For a full book description and ordering information, please refer to
cThis lecture investigates the underlying linear algebra that forms the basis of MATLAB. Aly Kassam is the author of this lecture, it is being submitted under Matt McDonnell's name solely to fit it in with the rest of the lectures. The aim of this lecture series is to introduce students toStudents, here is an app for you! Much better than your calculator programs, Energy let you calculate everything about energy in physics class. ?Kinetic energy ?Potential energy ?Mechanical energy Everything is in it, from formulas to unit conversions! Of course, I still listen to if you have | 677.169 | 1 |
In this no prep fun activity, students determine reference angles for positive and negative angles in both degrees and radians, and also use reference angles to find the six trigonometric functions of angles. This can be used in Trigonometry, PreCalculus, Geometry, Algebra 2.
FREE Builds understanding of the Law of Sines and the Law of Cosines for Algebra 2 Honors, PreCalculus, Trigonometry, and College Algebra students by providing concentrated practice.Students will complete 11 questions related to mastery of the Law of Sines, the Law of Cosines, Herons Formula, and practical applications related to these concepts of upper level mathematics courses. 10-12
Biography Bottles! Simply have students fill an empty 2-liter bottle with either small rocks or sand. Then, have them choose a famous person to create using the bottle!! Super cute + super creative = SuCcESS in a Bottle!
Graphing the Sine and Cosine Flip Book with HW / Quiz for Trigonometry, PreCalculus, and some Algebra 2 classes. This Graphing the Sine and Cosine mini bundle reinforces and reviews Amplitude, Phase Shift, Period, Vertical Shift, finding the End Points, and graphing of sinusoidal functions. Over 16 examples, many for students to complete. Great Study Guide, Great activity.
PreCalculus, Trigonometry: Proving Trig Identities Cut & Paste ActivityEnjoy a small sample of a cut & paste activity for your PreCalculus or Trigonometry students. There are two identities with steps to cut and paste into the correct progression to satisfy the proof. | 677.169 | 1 |
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Confusing Textbooks? Missed Lectures? Not Enough Time?This Schaum's Outline gives you
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Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!
Schaum's Outlines-Problem Solved.show more
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This solvedproblem study guide helps you ace operations research courses! Completely worked problems shoe you each and every step to hundreds of solutions! Tackling the broad range of allocation problems that actually confront engineers, programmers, and analysts in today's business and industrial worlds--exactly the kind of problems that appear on operations research exams--this complete study guide gives you step-by-step guidance in all the mathematical programming techniques--including the trailblazing Karmarkar algorithm--you need to excel in any operations research course. Using illustrative word problems that reflect typical, real-world situations, this study guide eases you through the variety of mathematical choices and applications used in operations research. Clear, logical explanations and one-step-at-a-time procedures show you how to analyze and solve each type of problem. Step by step, and working at your own pace, you learn to choose and apply the appropriate method. So comprehensive that it can be used as a complete independent study course, this guide is also the perfect complement and reinforcement to any text. Hundreds of supplementary problems give you a chance to practice newly learned techniques. Professionals working in the field will also find this comprehensive study guide useful as a hands-on solutions manual that quickly supplies the variety of techniques needed every day on the job! Chapters include: Mathematical Programming; Linear Programming: Basic Concepts; Linear Programming: The Simplex and the Dual Simplex Methods; Linear Programming: Duality and Sensitivity Analysis; Linear Programming: Extensions, Including the Revised Simplex Method and Karmarkar'sAlgorithm; Integer Programming: Branch-and-Bound Algorithm; Integer Programming: Cut algorithms: Inter Programming: The Transportation Algorithm; Integer Programming: Scheduling Models Nonlinear Programming: Single-Variable Optimization; Nonlinear Programming: Single-Variable Optimization; Nonlinear Programming: Multivariable Optimization with Constraints; Network Analysis; Project Planning Using PERT/CPM; Inventory Models; Forecasting; Game Theory; Decision Theory; Dynamic Programming--Deterministic and Stochastic; Finite Markov Chains; Markovian Birth-Death Processes; Queuing Systems; M/M/1 Systems; Other Systems with Poisson-Type Input and Exponential-Type Service Times.show more
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Solving Inequalities Lesson for Algebra 2 or Algebra 2 Honors
Learning Scale is included with this lesson!
You can cut sections out if needed. Everything is editable.
Topics included are:
Writing an Inequality
Using an Inequality to Solve a Problem
Inequalities with no Solution and All Real Numbers as Solutions
Solving and and or Inequalities
This lesson includes PowerPoint presentations and coordinated guided notes with answers covering Solving Inequalities. The PowerPoint includes a warm up (do-now or bell ringer), key concepts, scales for students to track their understanding, and examples for students to follow.
I created this to accompany the Prentice Hall Pearson Algebra 2 Common Core textbook. This lesson covers Section 1.5. The lesson is 30 - 45 minutes. | 677.169 | 1 |
Lecture 5 typset notes - ME 218 ENGR COMPUTATIONAL METHODS...
ME 218: ENGR COMPUTATIONAL METHODS Lecture Notes 5 Matrix AlgebraA matrix is a rectangular array of numbers in which not only the value of the number is important, but also its position in the array. It is very easy to solve a system of linear equations using matrices. i)Matrix Addition Two matrices of the same size may be added or subtracted. Then, A+Cis not defined since Ais a and Cis a matrix! ii)Matrix Multiplication If Aand Bare two given matrices, then the product ABis only defined if the number of columns of Ais equal to the number of rows of B, and the product matrix has the number of rows of Aand the number of columns of B. For this reason, ABmay not be equal to BA!
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Example is not defined. Ais a matrix and Cis a matrix. The number of columns of Athe number of rows of C. iii)Transpose of a Matrix If the rows of a given matrix are interchanged with the columns of the same matrix, the resultant matrix is the transpose of the given matrix.
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IS&T SCV Tutorial - Introduction to MATLAB
Starts:
10:00 am on Monday, February 11, 2013
Ends:
12:00 pm on Monday, February 11, 2013
MATLAB (for MATrix LABoratory) is a numerical computing environment developed by MathWorks, Inc. MATLAB is essentially an interpretive high level language that does not require data type declaration or compilation. It can be used to implement mathematical computations such as matrix manipulations with existing linear algebra packages. Many plotting and visualization tools are available as an integral part of MATLAB. MATLAB operations are very intuitive, user-friendly, and are used primarily in an interactive environment to enable fast proto-typing of research activities and efficient software development. Many highly specialized applications, such as Mathematical Finance, Bioinformatics, and Image Processing, are also available as toolboxes.
In this tutorial, many of the basic MATLAB operations, including basic 2D and 3D graphics, will be introduced. You will learn many of these operations hands-on.
No prior programming experience in any languages is required to attend this course. However, basic knowledge of linear algebra, such as matrix operations, is required | 677.169 | 1 |
Text Size:
AAA
MFG135G - Technical Math
Credits: 3
This course will cover mathematic concepts frequently used in manufacturing, including some review of fractions, decimals, algebra, ratios, data visualization and statistical measures, plane and solid geometry and applications, Cartesian coordinate system, and right triangle trigonometry with applications including vectors. Students will be introduced to semi-precision measurement and precision measurement, using both Metric and English Standard systems. | 677.169 | 1 |
Linear Finite Element Analysis
Course Description:
This video series is a comprehensive course of study that presents effective finite element procedures for the linear analysis of solids and structures. The finite element method is the ideal tool for solving static and dynamic problems in engineering and the sciences. Linear analysis assumes linear elastic behavior and infinitesimally small displacements and strains. To establish appropriate models for analysis, it is necessary to become familiar with the finite element methods available. | 677.169 | 1 |
About this Book
This book enables readers to understand system identification and linear system modeling through 100 practical exercises without requiring complex theoretical knowledge. The contents encompass state-of-the-art system identification methods, with both time and frequency domain system identification methods covered, including the pros and cons of each. Each chapter features MATLAB exercises, discussions of the exercises, accompanying MATLAB downloads, and larger projects that serve as potential assignments in this learn-by-doing resource. | 677.169 | 1 |
201-NYC-05 Fall 2016
Homework
M. Hamel
August 17, 2016
Homework: Homework will be assigned weekly. There will be three types of homework assigned:
WeBWorK, written and collected assignments, and suggested exercises. You may not have all
types of homework
MAT- 703
Assignment 4
T. Zakon
Subspaces, basis and dimension
See posted Summary Document on Subspaces. It includes definitions and outline of procedures relevant to the
problems below. You may use the online Matrix Reducer for matrices involving numeric
MAT-703
STUDY GUIDE FOR TEST II
October, 2016
T. Zakon
Complex Numbers
Definitions (be able to state):
What is a complex number? What are the real part and imaginary part
of a complex number? What is the absolute value and the polar angle of a complex num
MAT-703
SUMMARY NOTE ON COMPLEX NUMBERS
T. Zakon
An overview of the evolution of number systems
Starting with counting numbers, our number system evolved gradually for a variety of purposes, both
computational and algebraic. At each stage of the evolution | 677.169 | 1 |
Free Consultation
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Algebra 1
Academic Year Mathematics
In Algebra 1, students learn how to work with various types of expressions both algebraically and geometrically.
They learn how to solve linear and quadratic equations and how to represent various expressions in the Cartesian plane.
We introduce students to the rich field of complex numbers, as well as to important common functions and concepts
in discrete math. We continue the emphasis on challenging word problems from Prealgebra,
so that students learn when and how to apply their new tools. Many of the more challenging problems of
Algebra 1 come from contests such as MATHCOUNTS, AMC 8, and AMC 10.
Students completing Algebra 1 are ready to apply their algebra skills to our Geometry course.
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Sample Problems
Below are examples of some of the types of problems that students will encounter in our
Algebra 1 course. | 677.169 | 1 |
Mathematics assignments At Your FingerTips
What is mathematics and why student have fear for this subject? Mathematics is an interesting and yet a mind sharpening subject. It provides knowledge to students for dealing with all the calculations and questions of the mathematical world. In this subject, there is no place for mugging up; rather students need to be smart enough to answer any questions even without practice. Yes, we have generally heard that math's is about practice, but it's not only about practice. It is also about logic and concentration. If any students stop practicing any topic, will they be weak in that portion of the subject? They will not, if they focus while solving problems and stick to the basic of those topics. But yes, if they only depend upon practice without clearly understanding the concept, they will surely face difficulty.
Hence, mathematics can be interesting to some and boring to the others. It can be tiresome job for some and a challenging task for the others. Likewise, assignments in mathematics can have similar impact on students mind. Some students may find assignments worth while other may find it a furious task. Therefore, in any case if students are disappointed or confused on the assigned work they can switch to myassignmenthelp.net. Once they get in touch with the teachers from this site, all fear and confusion will be vanished and interest and curiosity in the subjects will be generated. This assignment site is different from any other online sites in several ways. Students who have any problems in their mathematics work can get in contact with the teachers here for all the easy and effective solutions. This site function with all the advanced and well developed technologies, so students can understand any portion clearly. All the teachers have degrees from reputed colleges and universities and they are best in their respective field.
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5 Power 1: Represents patterns and relationships in a variety of ways to Assess a. Uses patterns and relationships to analyze mathematical situations b. Describes functions and their properties using function notations ITED * * ITED 1-3 items Patterns Term Prediction Sequence Function Domain Independent Variable Dependent Variable Range One-to-one correspondence Function notation Function rule Relation Vertical Line Test Knows patterns can be described in a variety of ways Knows how to use the pattern to predict change from one term to the next Knows for every value in the domain of a function, there is one and only one corresponding value in the range Understands the concept of a function as the correspondence between the elements of two sets Understands the definition of a function: domain, range, function, relation Describes patterns using words, tables, and graphs - Finds the terms in a sequence () Generates equations to describe linear patterns () Identifies the domain, range, and rule of a function - Describes functions and their properties using function notation - Recognizes the graphs of non-linear functions - Describes from a graph the relationship between two variables (ITED) Algebra Standards and s 5
6 Power 2: Represents and analyzes mathematical situations and structures using algebraic symbols to Assess a. Represents linear and quadratic functions in a variety of ways ITED * * ITED 1-3 items Standard Form Linear function Quadratic function Parabola Knows a function can be represented in a variety of ways Understands a family of functions has similar characteristics Describes functions using words, tables, graphs and equations Graphs linear and quadratic functions - Recognizes a linear and quadratic function from its graph (ITED) Writes the equation of a linear function given the table of values, graph, two points on the line, and slope and y-intercept - Algebra Standards and s 6
7 Power 2: Represents and analyzes mathematical situations and structures using algebraic symbols to Assess b. Interprets graphical representations of linear functions - Rate of change Slope Direct Variation Slope-Intercept Form Understands change can be described mathematically Understands how rate of change can be described numerically and graphically Knows a linear graph has a constant rate of change Interprets slope as the amount of one quantity (y) per unit of another quantity (x) - Defines slope as a rate of change - Finds the slope, x-intercept, and y-intercept of a line given its graph, equation, or two points on the line () Uses slope to differentiate between lines that are parallel, perpendicular, horizontal, or vertical - Compares the rates of change of two different graphs - Algebra Standards and s 7
10 Power 3: Uses expressions and symbols to represent mathematical relationships (con t) to Assess c. Solves multi-step equations ITED *** *** ITED more than 3 items Multi-step problems Literal equations Inverse Operations Recognizes like terms Understands rules for combining variables Knows the order of operations Knows the inverse relationship between addition and subtraction and multiplication and division Understands the process of working backwards to solve equations (strategy of undoing) Solves equations for a specified variable (ITED,,, ) Solves multi-step equations involving like variables on the same side of the equal sign, like variables on both sides of the equal sign, and the distributive property (ITED,,, ) Writes an equation for a specific problem (ITED,, ) d. Solves single variable inequalities Inequality Compound inequality Solution of an inequality Open dot Closed dot Understands the process of working backwards to solve equalities (strategy of undoing) Knows the meaning of the inequality symbols Solves multi-step single variable inequalities involving like variables on the same side of the equal sign, like variables on both sides of the equal sign, and distributive property (,, ) Graphs solutions of inequalities on a number line - Algebra Standards and s 10
11 Power 4: Analyze change in a variety of situations to Assess a. Solves systems of equations by graphing, substitution, and elimination b. Solves quadratic equations by quadratic formula, factoring, and square roots Systems of equations Graphing method Substitution method Elimination method No solution Infinitely many solutions One solution Perpendicular Lines Parallel Lines Quadratic equation Standard form Quadratic formula Factoring Square roots Zero-product property Parabola Knows how to graph a linear equation Knows how to solve a literal equation Knows order of operations Knows how to find the square root of a number Knows how to write the Standard Form of any polynomial Uses graphing to solve a systems of equations (,, ) Uses substitution to solve a systems of equations (,, ) Uses elimination to solve a systems of equations (,, ) Knows whether a systems of equations has one solution, no solution, or infinitely many solutions - Solves quadratic equations by quadratic formula, factoring, and square roots (,, ) Analyzes quadratic equations to decide which method of solving is appropriate - Analyzes quadratic equations through graphing - CT Algebra Standards and s 11
13 Problem Solving Standard: Understands and applies problem solving strategies Power 1: Uses a variety of strategies to solve problems Grade Level a. Applies and adapts a variety of appropriate strategies to solve problems Try, test, revise Make a model pertinent information irrelevant information insufficient information act it out make or use a model find or use a pattern draw a picture guess and check make a chart, table, graph, or organized list use logical reasoning Knows the general problem solving strategies Knows the same situation can often be represented in more than one way Knows different problems may be solved using the same method to Assess Chooses appropriate strategies to solve problems in the context of the problem situation (ITED) Uses previous learned strategies, skills, knowledge, and concepts to solve problems (ITED) Translates words to numbers to symbolic expressions (ITED) Identifies pertinent and irrelevant information (ITED) Generalizes solutions to new problem situations (ITED) CT Algebra Standards and s 13
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
Provider York County School Division Course Syllabus URL Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal
Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (EverythingAlabama Course of Study: Mathematics (Grades 9-12) NUMBER AND OPERATIONS 1. Simplify numerical expressions using properties of real numbers and order of operations, including those involving square rootsALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifiesMTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
Pacing for a Common Core Curriculum with Prentice Hall Algebra 1 This leveled Pacing Guide can help you transition to a Common Core-based curriculum with Pearson s Prentice Hall Algebra 1 2011. The first
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variableIn this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment, Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lessonA Correlation of Prentice Hall Mathematics Courses 1-3 Common Core Edition 2013 to the Topics & Lessons of Pearson A Correlation of Courses 1, 2 and 3, Common Core Introduction This document demonstrates
Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. TheseIntroduction to the Instructor TERM 1 This calendar of lessons was prepared as a textbook independent sequence of lessons and the order of topics can be modified based on the textbook selection. The columnsHigh School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices | 677.169 | 1 |
ISBN-10: 0840068573
ISBN-13: 9780840068576
Edition: 12 Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this book popular among students year after year. This latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains these features. The problems have been consistently praised for being at just the right level for precalculus students. The book also provides calculator examples, including specific keystrokes that show how to use various graphing calculators to solve problems more quickly. Perhaps most important--this book effectively prepares readers for further courses in mathematics.
Earl Swokowski authored multiple editions of numerous successful textbooks, including CALCULUS; CALCULUS OF A SINGLE VARIABLE; FUNDAMENTALS OF COLLEGE ALGEBRA; and PRECALCULUS: FUNCTIONS AND GRAPHS, all published by Cengage Learning Brooks/Cole.
Jeffery A. Cole has been teaching mathematics and computer science at Anoka-Ramsey Community College since fall 1981. He started working on the Swokowski series of precalculus texts in 1985 as an ancillary author, and has been a co-author since 1991. His contribution to the Swokowski texts also includes joining the revision team of the calculus text in 1989.
Each chapter ends with Review Exercises, Discussion Exercises, and a Chapter Test
Topics From Algebra
Real Numbers
Exponents and Radicals
Algebraic Expressions
Equations
Complex Numbers
Inequalities
Functions and Graphs
Rectangular Coordinate Systems
Graphs of Equations
Lines
Definition of Function
Graphs of Functions
Quadratic Functions
Operations on Functions
Polynomial and Rational Functions
Polynomial Functions of Degree Greater Than 2
Properties of Division
Zeros of Polynomials
Complex and Rational Zeros of Polynomials
Rational Functions
Variation
Inverse, Exponential, and Logarithmic Functions
Inverse Functions
Exponential Functions
The Natural Exponential Function
Logarithmic Functions
Properties of Logarithms
Exponential and Logarithmic Equations
Trigonometric Functions
Angles
Trigonometric Functions of Angles
Trigonometric Functions of Real Numbers
Values of the Trigonometric Functions
Trigonometric Graphs
Additional Trigonometric Graphs
Applied Problems
Analytic Trigonometry
Verifying Trigonometric Identities
Trigonometric Equations
The Additions and Subtraction of Formulas
Multiple-Angle Formulas
Product-To-Sum and Sum-To-Product Formulas
The Inverse Trigonometric Functions
Applications of Trigonometry
The Law of Sines
The Law of Cosines
Vectors
The Dot Product
Trigonometric Form for Complex Numbers
De Moivre's Theorem and nth Roots of Complex Numbers
Systems of Equations and Inequalities
Systems of Equations
Systems of Linear Equations in Two Variables
Systems of Inequalities
Linear Programming
Systems of Linear Equations in More Than Two Variables
The Algebra of Matrices
The Inverse of a Matrix
Determinants
Properties of Determinants
Partial Fractions
Sequences, Series, and Probability
Infinite Sequences and Summation Notation
Arithmetic Sequences
Geometric Sequences
Mathematical Induction
The Binomial Theorem
Permutations
Distinguishable Permutations and Combinations
Probability
Topics From Analytical Geometry
Parabolas
Ellipses
Hyperbolas
Plane Curves and Parametric Equations
Polar Coordinates
Polar Equations of Conics
Common Graphs and Their Equations
A Summary of Graph Transformations
Graphs of the Trigonometric Functions and Their Inverses
Values of the Trigonometric Functions of Special Angles on a Unit Circle | 677.169 | 1 |
Written by a computer scientist for computer scientists, this book teaches topology from a computational point of view, and shows how to solve real problems that have topological aspects involving computers. Such problems arise in many areas, such as computer graphics, robotics, structural biology, and chemistry. The author starts from the basics of topology, assuming no prior exposure to the subject, and moves rapidly up to recent advances in the area, including topological persistence and hierarchical Morse complexes. Algorithms and data structures are presented when appropriate.
Afra J. Zomorodian is an assistant professor in the Department of Computer Science at Dartmouth College. He is a former Post-Doctoral Fellow in Bio-X and the Department of Computer Science at Stanford University, California and has been a researcher at the Max-Planck-Institut für Informatik, Saarbrücken. Using the material contained in this book, he has organized classes on computational topology at Stanford University, the MPI, and Dartmouth College.
Review:
"In my knowledge, it is the first book covering these topics." Numerical Algorithms
"This authoritative, well-written, and highly focused book will explain to the reader the considerable power of topology. It is an eye-opener that I highly recommend." George Hacken, reviews.com
Descrizione libro CAMBRIDGE UNIVERSITY PRESS, United Kingdom, 2009. Paperback. Condizione libro: New. Language: English . Brand New Book ***** Print on Demand *****.. Paperback. Condizione libro: new. BRAND NEW PRINT ON DEMAND., Topology for Computing, Afra J. Zomorodian,'s B9780521136099
Descrizione libro CAMBRIDGE UNIVERSITY PRESS, United Kingdom, 2009. Paperback. Condizione libro: New. Language: English . Brand New Book ***** Print on Demand *****.36099
Descrizione libro 2009. Paperback. Condizione libro: NEW. 9780521136099 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0447881 | 677.169 | 1 |
Students' use of variables and multiple representations in generalizing functional relationships prior to secondary school
Abstract
Algebra has been explicit in many school curriculum programs from the early years but there are competing views on what content and approaches are appropriate for different levels of schooling. This study investigated 12–13-year-old Australian students' algebraic thinking in a hybrid environment of functional and equation-based approaches to learning algebra. This article reports on a survey of 102 students examining their generalization ability and knowledge of multiple representations and variables prior to formal study of algebraic expressions and equations at secondary school. Nearly half of the students demonstrated the ability to generalize explicitly with one fifth able to construct a symbolic functional rule. Nearly half were able to represent a real-life scenario of a linear relationship algebraically. There was little evidence yet of connecting a growing pattern or real-life scenario to an appropriate graphical representation. Level of pattern generalization ability was found to be associated with flexible thinking for exploring functional relationships in reverse and with explaining the inappropriateness of proportional reasoning for linear functions with a constant. Implications for the teaching and learning of algebra are presented.
Bardini, C., Pierce, R., & Vincent, J. L. (2013). First year university students' understanding of functions: Over a decade after the introduction of CAS in Australian high schools, what is new? Paper presented at the Lighthouse Delta 2013: The 9th Delta Conference on teaching and learning of undergraduate mathematics and statistics, Kiama, New South Wales.Google Scholar
Chazan, D., & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: research on algebra learning and directions of curricular change. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 123–135). Reston: National Council of Teachers of Mathematics.Google Scholar
Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. Curcio (Eds.), The roles of representation in school mathematics (pp. 1–23). Reston: National Council of Teachers of Mathematics.Google Scholar
Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator, 8(1), 139–151.Google Scholar
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 707–762). Charlotte: National Council of Teachers of Mathematics, Information Age Publishing.Google Scholar | 677.169 | 1 |
Course Description: This course reviews the concepts of Algebra 1 and more advanced methods of solving equations and inequalities, the complex number system and all of the important subsets, techniques of factoring, graphing functions and relations. The conic sections and matrices are introduced. A scientific or graphing calculator is recommended. This class does not fill the prerequisite for Precalculus Honors. | 677.169 | 1 |
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This Presentation was designed around Unit 7 of the Year 8 C2C Maths Queensland Curriculum. The PowerPoint introduces students to Algebra and using algebraic expressions. This presentation comes with questions for the students to work on as you proceed through the PowerPoint. It has a hyperlink to a quiz that the students can then do to reinforce their understanding of how to expand algebraic equations.
I also have a specific PowerPoint Quiz titled 'Intro to Algebra Quiz' that is available on my site.
I have designed the Quiz to follow directly on from this PowerPoint and it is actually more appropriate to have students do this Quiz before you guide them to the hyperlinked Quiz on the internet.
The Presentation provides good visual steps rather than students needing to rely on the teacher just talking. Presentation designed to provide students with visual examples. Caters for students who are visual learners or those with Hearing Impairments and Auditory Processing Disorders. This Presentation was designed to cater for Year 8 Maths Queensland Curriculum but could be used for any year level from year 6 to Year 9.
(Australia, Queensland, Mathematics, Special Education, Year 8, C2C) PowerPoint Queen Australia | 677.169 | 1 |
The classic Heath translation, in a completely new layout with plenty of space and generous margins. An affordable but sturdy student and teacher sewn softcover edition in one volume, with minimal notes and a new index/glossary.
The quadrivium-the classical curriculum- Number, music, and geometry are metaphysical truths: life across the universe investigates them; they foreshadow the physical sciences.
Quadrivium is the first volume to bring together these four subjects in many hundreds of years. Composed of six successful titles in the Wooden Books series-Sacred Geometry, Sacred Number, Harmonograph, The Elements of Music, Platonic & Archimedean Solids, and A Little Book of Coincidence-it makes ancient wisdom and its astonishing interconnectedness accessible to us today.
Beautifully produced in six different colors of ink, Quadrivium will appeal to anyone interested in mathematics, music, astronomy, and how the universe works.
A review of the most important test-taking strategies students need to know to succeed on this exam
Topic review chapters covering all the math students need to know for this test, including: arithmetic, algebra, plane geometry, solid and coordinate geometry, trigonometry, functions and their graphs, probability and statistics, real and imaginary numbers, and logic
Three full-length model tests with complete solutions for every problem
The manual can be purchased alone or with an optional CD-ROM that presents two additional full-length practice tests with answers, explanations, and automatic scoring.
Developed for test-takers who need a refresher, GMAT Foundations of Math provides a user-friendly review of basic math concepts crucial for GMAT success.
Designed to be user-friendly for all students, GMAT Foundations of Math provides: * Easy-to-follow explanations of fundamental math concepts * Step-by-step application of concepts to example problems * 700+ practice problems and 10 chapters * Purchase of this book includes one year of online access to the Foundations of Math Homework Banks with 400+ extra practice questions and detailed explanations not included in the book.
GMAT Foundations of Math is an invaluable resource for any student who wants to cement their understanding and build their basic math skills for the GMAT.
"If Hidden Figures has you itching to learn more about the women who worked in the space program, pick up Nathalia Holt's lively, immensely readable history, Rise of the Rocket Girls." --Entertainment Weekly
The riveting true story of the women who launched America into space.
InFor the first time, Rise of the Rocket Girls tells the stories of these women--known as "human computers go-to study guide provides the concepts, study strategies, and practice you need to dramatically raise your ACT scores in English, Reading, and Science areas of the exam.
McGraw-Hill Education: Top 50 ACT Math Skills for a Top Score, 2 edition presents the heart of each of the 50 most important concepts in this subject area. Within each topic, exercises help you build critical skills and illustrate how each concept is tested on the ACT. Written by a test-prep expert with years of experience with the SAT and ACT preparation, this easy-to-use book features:
A pretest designed to identify your weaknesses in the Math areas ACT traditionally focuses on | 677.169 | 1 |
Instructor's Edition for Thomas' Calculus (12th Edition)
Instructor's Edition for Thomas' Calculus (12th Edition)
This text is designed for a three-semester or four-quarter calculus course (math, engineering, and science majors).
Calculus hasn't changed, but your students have. Today's students have been raised on immediacy and the desire for relevance, and they come to calculus with varied mathematical backgrounds. Thomas' Calculus, Twelfth Edition, | 677.169 | 1 |
This book covers dimension theory, ANR theory (theory of retracts) and related topics, connecting with various fields in geometric and general topology. Many proofs are illustrated by figures or diagrams, making it easier to understand the underlying concepts. more...
This volume's historical focus, with fully worked solutions to all the famous problems in classical geometry, demonstrates the profound influence of axiomatic geometry, over more than three millennia, on the evolution of mathematics as an academic discipline. more...
As a result of the editors' collaborative teaching at Harvard in the late 1960s, they produced a ground-breaking work -- The Art Of Problem Posing -- which related problem posing strategies to the already popular activity of problem solving. It took the concept of problem posing and created strategies for engaging in that activity as a central theme... more...
Derived from the author's course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topology,... more...
The authors show that there are underlying mathematical reasons for why games and puzzles are challenging (and perhaps why they are so much fun). They also show that games and puzzles can serve as powerful models of computation—quite different from the usual models of automata and circuits—offering a new way of thinking about computation. The appendices... more... | 677.169 | 1 |
Product Overview
The fast and easy way to learn MathematicaAndnbsp;Mathematica, produced by Wolfram Research, is one of the most successful and widely used software packages for scientists, mathematicians, engineers, students and instructors for doing all sorts of mathematical computations and programming. This book will teach beginners how to use Mathematica and acquaint veteran users with the new Version 6, which is a significant upgrade from the previous version. Mathematica Demystified includes hundreds of examples plus end-of-chapter quizzes and a final exam. | 677.169 | 1 |
Publicon in Action
Extracts from Students' Work
This chapter presents examples of work by students which illustrates the appearance of basic elements in a Publicon document. Authors of the extracts are the students of Slovak University of Technology, Faculty of Mechanical Engineering.
1. Numerical Methods and Mechanical Engineering
CALCULATING THE INTERNAL FORCES OF A TRUSS BRIDGE CONSTRUCTION
1.1. Introduction to the technical background
In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight, slender members whose ends are connected at joints.
Figure 1.
A plane truss is one where all the members and joints lie within a 2-dimensional plane, while a space truss has members and joints extending into 3 dimensions. A necessary (but not sufficiant) condition for stability is: m≥2j–r (m - number of trusses, j - number of joints, r- number of reactions). When m=2j−3, the truss is said to be statically determinate. In order for a truss with pin-connected members to be stable, it must be entirely composed of triangles. When the external loads and the geometry of a truss are known, we can create the equilibrium equations. This method of solution is called the joints method and more in detail can be seen in the solution of the example... [1]
Each member represents an internal bond between two point masses. The bond removes one degree of freedom from the mass points, in the direction of the member axis. Then, the bonds are substituted by internal bond reactions, which are called the axial forces in members. They are noted as Pi, where i is the number of the member. [2]
Figure 3.
Notation and sign convention for trusses
The amplitude and orientation of internal forces in a truss depends on the geometry of the structure, loaded by external forces and the positioning of supports. Positive (tensile) axial force is oriented out of an imaginary cut in a member or out of a mass point. A negative (compressive) axial force is oriented into an imaginary cut or mass point. [2]
To calculate the forces in a structure we have to derive equilibrium equations for every joint. [3]
A system of 10 linear equations in 10 variables has to be solved. Assuming that angle is 45˚, we obtain the coefficients for the tables and A matrix in the formula A.x=b, where x={P1,P2,P3,P4,P5,P6,P7,Ex,Ey,A}[3].
1.2.2. Iteration Methods
If we want to use the iteration methods to solve the problem we have to keep on these rules:
The matrix A must be a square matrix
The matrix A must be strictly diagonally dominant
The matrix must be non-singular (i.e. have a non-zero determinant)
The matrix A isn't strictly diagonally dominant so we have to apply linear modification to achieve this. The form of strictly diagonally dominant matrix A will be... [3]
2. Statistics
ANALYZING THE WEIGHT DISTRIBUTION OF PORTORICO CIGARS
2.1. Introduction
.... In this section we use data that has been collected about the weight of cigars produced by Portorico. A sample of 300 cigars has been used in the analysis. Our main task is to organize and analyze the data using the method of Shewhart's control chart. This chart contains the central line (CL - main line), the upper control limit line (UCL), and the lower control limit line (LCL). These parameters are calculated using the formulas:
First analysis: Nonexcluded data
In the first case we divided data into 30 subgroups of subgroup size 5 and basic parameters were computed .
X-bar and Range - Initial Study for weight with the subgroup size 5
... Outliers are colored by red stars in the graph. As one can see, there is 1 outlier in the X-bar chart (Figure 5)2 and none in the Range chart (Figure 6)2 . The outlier is excluded in the next process and ...
Figure 5.
Figure 6.
TRAJECTORY OF THE CENTRE OF GRAVITY
Figure 72shows the trajectory of the centre of gravity measured on ...
Figure 7. 7
3. Summary
Publicon belongs to the new editors focussed mainly on web publishing. Its main advantage consists in user friendly button click interface and MathML coding used for mathematical and chemical notation. Although some aspects of its design are less than perfect, Publicon is easy to be use allowing the importing and exporting of objects from and to other software products withot the loss of information.
This document has been written under the Article2 style sheet, and demonstrates | 677.169 | 1 |
Good overview for Calculus 1 properties. It includes constant functions, linear functions, polynomial functions, exponential and logarithmic functions. Additionally, it has the derivative rules defined along with how to read sine graphs. Good reference & cheat sheet for those in Calculus.
What is Domain Authority? How we can increase the Domain Authority? Domain Authority is a logarithmic calculation which defines the score of your domain. #domain #enest #domainauthority #whatisdomainauthority #digitalmarketing | 677.169 | 1 |
Abstract Algebra An Introduction, 3rd Edition
Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. New to this edition is a "groups first" option that enables those who prefer to cover groups before rings to do so easily | 677.169 | 1 |
$160Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until w Voroi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincare conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also features numerous exercises and unsolved problems. * The essential introduction to discrete and computational geometry * Covers traditional topics as well as new and advanced material * Features numerous full-color illustrations, exercises, and unsolved problems * Suitable for sophomores in mathematics, computer science, engineering, or physics * Rigorous but accessible * An online solutions manual is available (for teachers only). To obtain access, please e-mail: Vickie_Kearn@press.princeton.edu | 677.169 | 1 |
Maths from Scratch for Biologists
Alan J Cann
Wiley
2002
Hardcover 240 pp ISBN 9780471498346 £145.00
Paperback 240 pp ISBN 9780471498353 £20.00
This refreshing title is written by a lecturer who saw the lack of material available
on basic maths for students studying biology, who have not studied maths at higher level. Alan
Cann provides an essential introduction to maths in this book, which will provide a strong
foundation for those studyiong or working in the industry. | 677.169 | 1 |
COURSE INFORMATION
CMA introduces Vedic Mathematics the name given to the ancient system of Indian Mathematics which was rediscovered from the vedas between 1911 and 1918 by respected Late Shri Bharati Krishna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras, or word formulae. VEDIC MATHS has been a proven course to perform better in Competitive examinations. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution. In the Vedic system difficult problems or huge sums can often be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of mathematics which is far more systematic than the modern system. Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, direct and easy.
Mathematics is always a practical subject if you think you can learn math which is not possible. Maths plays an important role into every human life. As mathematics accompany us throughout our lives, it is always useful to be able to solve mathematical problems easily and quickly. Vedic Maths empower us to do so, whether the beneficiary is a child or an adult, it helps both. In children, it helps in developing calculation speed, memory power, concentration and confidence. With these, every child will be able to excel in the respective field not only for the time being but for future purposes too.
In adults, as there is an ever increasing level of competitive exams there is an equal level of competitors, Vedic Maths help to develop skills to solve tough mathematical calculations easily so that the candidate can easily tackle the mathematics section and move on to the next sections, this saves a lot of time. Vedic Maths Program is the set of more than 34 Formula including Addition, Subtraction, Multiplication, Division, Square , Cube, Square root, Cube Root, Fraction, Result Verification technique, Tables Making method, speed, Average and time calculation Formulas. | 677.169 | 1 |
CAS for All
Every student should have access to a CAS (Computer Algebra System) in a handheld and/or computer-based format at least as early as he or she begins learning algebraic concepts.
Used properly, a CAS creates a dynamic laboratory environment for a student in which he or she can explore algebraic relationships, receive instantaneous confirmation of the validity of algebraic manipulations, and scaffolding for deeper exploration and understanding of mathematics. In short, a CAS enables a student to have a mathematical solving expert available at all times in all places. Most importantly, students get the opportunity to explore mathematics without needing
Of course, to use a CAS, one needs to learn how to ask questions and how to interpret the solutions. A CAS will always provide an answer to the question asked. Users must know precisely what is being asked so that they can interpret their results.
I'm always open to any tool students could use to enhance learning, and commend what I see in a cursory glance at Microsoft Maths. That said, I still believe that nothing comes close to the power, flexibility, and utility of Wolfram Alpha ( | 677.169 | 1 |
This is the fifth edition of a work which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities. It is neither a systematic treatise on the theory of numbers nor a 'popular' book for non-mathematical readers.
This is the fourth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities.
This text is a rigorous introduction on an elementary level to the theory of analytic functions of one complex variable. At American universities it is intended to be used by first-year graduate and advanced undergraduate students.
Designed for use in a two-semester course on abstract analysis, Real Analysis: An Introduction to the Theory of Real Functions and Integration illuminates the principle topics that constitute real analysis. | 677.169 | 1 |
Bob Blitzer has inspired thousands of students with his engaging approach to mathematics, making this beloved series the #1 in the market. Blitzer draws on his unique background in mathematics and behavioral science to present the full scope of mathematics with vivid applications in real-life situations. Students stay engaged because Blitzer often uses pop-culture and up-to-date references to connect math to students' lives, showing that their world is profoundly mathematical. With the Fifth Edition, Blitzer takes student engagement to a whole new level. In addition to the multitude of exciting updates to the text and MyMathLab(R) course, new application-based MathTalk videos allow students to think about and understand the mathematical world in a fun, yet practical way. Assessment exercises allow instructors to assign the videos and check for understanding of the mathematical concepts presented.
Key Features
Author: Robert F. Blitzer
Publisher: Pearson
Copyright: 2014
Language: English
ISBN13 : 9781447964445
Number Of Pages: N/A
Edition: 5
Specifications of Algebra And Trigonometry | 677.169 | 1 |
Help with algebra 1
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PERCENTAGES -Definition -Importance of Base for Calculations -Concept of Percentage Change -Difference between the Percentage Point Change and the Percentage Change -Calculating Percentage Values through additions -Percentage Change Graphic -Effect of a Change in Both Numerator and Denominator on the Ratio -Practice ProblemsEurope, I wanted something that I could get my hands on instantly, because I only had two weeks to go.
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Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life [Paperback]
Item description for Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life by Catalin Barboianu...
Man's daily life is full of decisional situations. Whether we have math skills or not, we frequently estimate and compare probabilities, sometimes without realizing it, especially when making decisions. But probabilities are not just simple numbers attached objectively or subjectively to events, as they perhaps look, and their calculus and usage is highly predisposed to qualitative or quantitative errors in the absence of proper knowledge. That is why a book explaining the probability concept and its interpretations and applications for non-mathematicians is a necessity. This is an enlightening journey through the world of probability theory. Its multiple goals are to help the reader understand what probability really means, to teach the reader how to rigorously perform and apply the probability calculus, even without a solid mathematical background, and to stimulate the reader to go deeper into the notions involved. In the first part, the author tries to build a clear image of the probability concept by reconstructing its mathematical definition step by step through its constituent notions. It starts with a general presentation of the conceptual ensemble word - definition - notion - model any theory is based on when trying to reproduce reality. Then, the probability notion is defined and explained starting from the classical definition to the definition for the countable case; then probability is presented as a limit and as a measure. This book presents not only the mathematical concept of probability, but also its philosophical aspects, the relativity of probability and its applications and even the psychology of probability. All explanations are made in a comprehensible manner and are supported with suggestive examples from nature and daily life and even with challenging math paradoxes. After these points are laid out the math chapter follows. It contains all the notions and principal theoretical results that ground Probability Theory, starting with fundamental notions like Sets, Functions, Boole algebras, and Sequences, and continuing with Measure Theory Basics - Tribes, Borel sets, Measurable spaces, and Measure, ending with Field of events, Sigma-fields, Probability, Conditional probability, Discrete random variables, Classical probability distributions, and Convergence. And, of course, it includes all important theorems and results dealing with them. A special section is dedicated to Combinatorics and combinatorial calculus. Readers with no minimal mathematical background may choose to skip this chapter because the teaching material is structured for developing probability calculus skills based on algorithmic procedures. This is the subject of the chapter titled Beginners' Calculus Guide, in which the reader is taught to apply the properties of probability and to perform calculations in practical applications. The skills acquired here can be practiced on the more than 200 solved and unsolved problems and exercises in the book. Gamblers have a dedicated chapter, which is a large collection of applied probability results covering most of the gaming situations from the major games of chance: Slots, Roulette, Blackjack and Texas Hold'em Poker. The author brings important contributions to the decision theory, offering in the last chapter a mathematical model on which to base proof that a probability-based strategy is theoretically optimum, which has immediate applications in games as well as decisional situations from daily life. Everyone should find something of interest here: philosophers and mathematicians may focus on the sections on philosophical matters of the probability model and decisional matters, students and non-mathematicians can find solid A to Z teaching material about Probability Theory and the practical person can find all the tools needed to apply and perform probability calculus without a teacher Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life?
Big problems in Roulette City Nov 1, 2007
The math appears correct dispite many typos which become magnified because the english usage is awkward. HOWEVER, the section concerning roulette is completely off base. The author uses an incorrect payout of 40 to 1 for single number wagers rather than the correct payoff of 35 to 1. This is an unforgiveable error and yields totally incorrect results. This casts a pall over the other gaming analyses which I haven't the patience to evaluate. Someone must correct this (and possibly others) for the book to be useful; otherwise it must be discarded.
Average introduction to probability Feb 9, 2007
This book covers the basics in probability including boole algebras, measure spaces, law of large numbers, combinatorics, discrete random variables and bayesian probability.
The first 118 pages is basically a short story on probability summarizing the next 200 pages. The problem is the author uses an unnecessarily complex writing style enough to frustrate any individual whose first language is english what to say of others. Regardless, after a few pages you get used to it.
The mathematical chapter is really not mathematical at all. It is simply a statement of definitions, properties and theorems with no justification ( proofs ) and only a spattering of justification.
The prize however, is the problems and algorithms with plenty of solved examples. This has it's merits but at some points the author solves problems by a circuitous and often confusing route. There are other much simpler ways to solve these problems.
Regardless, I relished the fact that the author gives a general problem solving method rather than present formulas to memorize. This enhances intuition and understanding.
Finally a readable math book! Jul 4, 2006
It covers fundamentals well, easy to read, good variety of problems, nice examples, philosophic sections are interesting. It has everything. I've never seen a probability book with such good examples and algorithms of solving. Most books on probability give you all of the equations, but they don't really tell you how to apply them to real situations. It is the book on probability that I have been looking for. Great.
Excellent primer and not only Jun 23, 2006
This is the book to start with and to finish with. In this big book (in size and amount of information) you will find all you need to know about both theoretical and applied probability. The author made the text comprehensible for all categories of readers: laymen, students and advanced. And the result is a masterly exposition with amazing clarity. I read a lot of textbooks on this subject, but I found no one to touch on all basic issues in such a teachable manner. Each subject has a short, concise and to the point exposition. Great job and a very useful product.
Ask A Question or Provide Feedback regarding Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Spam | 677.169 | 1 |
2: Eulers Formula ib a comple
Math 121 Trigonometry
Worksheet 2: Eulers Formula
SOLUTIONS i
Math 121 Trigonometry
Worksheet 1: Modeling a spring
Problem 1: Modeling a spring.
Suppose a mass is placed at the end of a spring and allowed to attain equilibrium position. We can
measure the displacement x of the mass from equilibrium by the following
Math 121 Trigonometry
Worksheet 1: Modeling a spring
SOLUTIONS
Problem 1: Modeling a spring.
Suppose a mass is placed at the end of a spring and allowed to attain equilibrium position. We can
measure the displacement x of the mass from equilibrium by the
Math 121 (Trigonometry)
Winter 2011
Practice Midterm
Seattle University
Department of Mathematics
Purpose: The following questions are intended only to provide you with practice problems when
studying for the exam. There will be signicant dierences betwee
Math 121 (Trigonometry)
Winter 2011
Practice Midterm (Solutions)
Seattle University
Department of Mathematics
Purpose: The following questions are intended only to provide you with practice problems when
studying for the exam. There will be signicant dier
Math 121
Final Exam Practice (Solutions)
Seattle University
Department of Mathematics
Show all work. Without supporting work, even correct answers may be marked wrong. All answers must be
exact, except for the Solve the Triangle problems. For other proble
Math 121
Winter Quarter 2011
Final Exam Practice
Seattle University
Department of Mathematics
Show all work. Without supporting work, even correct answers may be marked wrong. All answers must be
exact, except for the Solve the Triangle problems. For othe | 677.169 | 1 |
Holt McDougal Geometry Textbook Answers
Holt McDougal Littell Larson, et al California Math: Algebra.holt mcdougal online homework help CliCK GO.Test and improve your knowledge of Holt McDougal Earth Science: Online Textbook Help with fun multiple choice exams you can take online with Study.com.Holt Mcdougal Geometry Homework Help Holt McDougal Online provides access to online books,assessments, and resources for students and teachers.Fulsome charged Pietro deep-freezes salt heist funning arrantly.Parents and students if you are having trouble logging into My Math Universe and it appears to be down try the following sites: Digits Dashboard.
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ISBN-10: 0071435336
ISBN-13: 9780071435338 revisits the stumbling blocks encountered in studying mathematics with an array of techniques for overcoming the anxiety and allowing people to use the math they know and learn the math they need as they need it.
Britta Immergut, Ed.D. is professor of mathematics at Fiorello LaGuardia Community College, in New York City. She has designed and taught courses for math-anxious adults for many years. Jean Burr Smith taught mathematics for 22 years at Middlesex Community College in Connecticut. One of the originators of the concept of math anxiety, she has received numerous awards for her math teaching | 677.169 | 1 |
ISBN-10: 0323170560
ISBN-13: 9780323170567 Incorporating the ratio and proportion, formula, and dimensional analysis methods, this online course presents a step-by-step approach to the calculation and administration of drug dosages. This Drug Calculations Online course is designed to be used with the 5th edition of Gray: Calculate with Confidence. Once you have read topics in the text, the online course provides you with an opportunity for application and practice. Animations, voice-overs, and interactive self-assessment activities are used to provide an engaging and interactive course platform. This course includes practice problems to promote active learning and quizzes that can be used to evaluate your understanding of content presented in the course.Includes three major drug calculation methods (ratio and proportion, formula, and dimensional analysis) to give students the option of choosing the method which works best for them.Each module is organized by topic sections that include an overview, objectives, a reading assignment for the topic being covered, example problems, practice problems, and one or more quizzes.Many of the math practice problems include a tutorial for each of the three drug calculation methods. When one of the solution buttons is chosen, a step-by-step tutorial to solving the problem in the method chosen is initiated for the student to view.Animations are used throughout this course to demonstrate various concepts related to dosage calculation and drug administration. Some animations will require student participation such as using the mouse to move the plunger on a syringe.Interactive self-assessment activities related to various topic areas are incorporated throughout the course to allow the student to apply their knowledge in context. Voice-overs are used throughout the course to enhance the step-by-step explanation of medication administration procedures and the drug calculation methods demonstrated throughout the course.Quizzes are included within each module that instructors can use to evaluate students' understanding of all the major topics covered in that particular module.A comprehensive test bank of approximately 350questions, organized by module, will be provided for instructor's to build quizzes and tests.Terminology is defined within the content for easy reference.Provides the latest drug administration techniques and devices and detailed explanation of the various forms of administering drugs, including oral, intravenous, intra-muscular, subcutaneous and other routes used in drug administration. This allows the student to become more knowledgeable about the specifics of each technique.Includes the most up-to-date, commonly used drugs so students have exposure to what is being used in the "real world" of clinical practice.Presents information on infusion pumps (enteral, single, multi-channel, PCA and insulin) to help students understand their increased use in drug administration.Interactive case studies of real life patient scenarios are integrated within the modules providing real world simulations for you to learn.Audio glossary defines pertinent terms with the course serving as a quick reference to key definitions.Includes an extensive collection of drug labels and has its own glossary.Caution boxes, Tips for Clinical Practice, and Rule boxes are incorporated throughout to create a more uniform correlation between the course and text.Additional practice problems with the most current drug labels are included.The most current guidelines for safe medication practice the The Joint Commission and the Institute for Safe Medication Practice will be incorporated throughout | 677.169 | 1 |
MCDOUGAL LITTELL GEOMETRY ANSWERS CHAPTER 10
Mcdougal littell geometry answers chapter 10
Mcdougal littell geometry answers chapter 10 Versions Hardcopy
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Focus in High School Mathematics: Reasoning and Sense Making
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"Reasoning" and "sense making" - or "understanding", "fluency", "problem solving" and "reasoning" in the Australian Curriculum: Mathematics - are critical complements to the content knowledge we expect students to learn as they complete their schooling in mathematics. This book provides a broad framework for achieving a focus on reasoning and sense making in the secondary mathematics classroom. It goes on to illustrate this through some practical examples in each of the key contents areas of the curriculum. An essential addition to your personal or faculty library as you strive to realise the important emphasis on the "how" of doing mathematics in the Australian Curriculum at the secondary level. | 677.169 | 1 |
9781441972873
1441972870Marketplace
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Summary
This unusually lively textbook on complex variables introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. "Complex Analysis" offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. Stressing motivation and technique, and complete with exercise sets, this volume may be used both as a basic text and as a reference. For this second edition, the authors have revised some of the existing material and have provided new exercises and solutions.
Table of Contents
Preface to the Third Edition
p. v
Preface to the Second Edition
p. vii
The Complex Numbers
p. 1
Introduction
p. 1
The Field of Complex Numbers
p. 1
The Complex Plane
p. 4
The Solution of the Cubic Equation
p. 9
Topological Aspects of the Complex Plane
p. 12
Stereographic Projection; The Point at Infinity
p. 16
Exercises
p. 18
Functions of the Complex Variable z
p. 21
Introduction
p. 21
Analytic Polynomials
p. 21
Power Series
p. 25
Differentiability and Uniqueness of Power Series
p. 28
Exercises
p. 32
Analytic Functions
p. 35
Analyticity and the Cauchy-Riemann Equations
p. 35
The Functions ez, sin z, cos z
p. 40
Exercises
p. 41
Line Integrals and Entire Functions
p. 45
Introduction
p. 45
Properties of the Line Integral
p. 45
The Closed Curve Theorem for Entire Functions
p. 52
Exercises
p. 56
Properties of Entire Functions
p. 59
The Cauchy Integral Formula and Taylor Expansion for Entire Functions
p. 59
Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem | 677.169 | 1 |
Essential Advanced General Mathematics Third Edition Enhanced TIM/CP Version
Essential Mathematics Essential VCE Mathematics series has a reputation for mathematical excellence, with an approach developed over many years by a highly regarded author team of practising teachers and mathematicians. This approach encourages understanding through a wealth of examples and exercises, with an emphasis on VCE examination-style questions. New in the enhanced versions: * TI-Nspire OS3 and Casio ClassPad calculator explanations, examples and problems are integrated into the text. * Page numbers in the printed text reflect the previous TI-nspire and Casio ClassPad version allowing for continuity and compatibility. * Digital versions of the student text are available in Interactive HTML and PDF formats through Cambridge GO. | 677.169 | 1 |
MATH 121: Mathematical Analysis for Business and Economics
Description
An introduction to mathematics for business and economics students using examples from business to motivate mathematical techniques. Necessary mathematical terms and concepts are developed, but emphasis is on applications to business with sufficient theory to support applications. Topics: algebraic functions, mathematics of finance, analysis of functions, differential and integral calculus.
Students may have credit for only one of MATH 110, 121, 123, or 125. Arts & Science students needing 6 credit units of 100-level calculus should take MATH 110 followed by MATH 116. Students who score below 60% on the Math Placement Test ( are advised to take MATH 102 to review their basis precalculus skills. | 677.169 | 1 |
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Arithmetic and Algebraic Structures
Final Exam Study Guide
The final exam will be on Monday, December 8th from 6-8pm in Lecture Hall F6.
To prepare for the midterm:
1. Your notebook will be an important resource during the final exam. Read
through your
notebook, especially the entries related to the content listed below. If there
are ideas or
concepts you have questions about, please be sure to seek clarification from a
classmate or one
of us before the exam.
2. Review all of the homework assignments. If you did not receive full credit for
a problem, make
sure you are now able to solve the problem correctly. You may want to read other
peoples'
assignments to get an idea of different solutions or other ways to explain your
thinking.
• Remember, there is no collaboration on the final. You are allowed to use your
notebooks, but
not the textbook. All of your work must be your own.
Content that could be covered on the final exam:
Number Theory:
• List all of the factors of a given number and explain how you know you have
them all
• Identify and define prime numbers, square numbers, multiples of a number,
factors of a
number, even numbers, odd numbers
Fractions and Division:
• Identify the different interpretations of fractions and division
• Determine whether a picture represents a particular fraction
• Write and evaluate word problems that represent each of the two
interpretations of division
(including division of fractions)
Divisibility Rules:
• Explain why the divisibility rules for 2, 4, and 9 works for any numbers
Definitions
• Evaluate definitions, using the criteria of precision and usability (i.e., is
the definition correct
mathematically? Does it rely on terms known to the users?)
• Determine whether specific examples are included or excluded by a particular
definition (e.g.,
given a definition of "even number," decide whether a particular number is even)
Proofs
• Prove statements about adding, subtracting, or multiplying odd and even
numbers
• Analyze a given proof –– is it convincing? On what is it based–– that is, what
does it use as
its basic definition or assumption, from which the argument grows?
Mathematical practices and sensibilities
• Giving mathematical explanations: Evaluate the adequacy of an explanation, or
give an
explanation, for a solution to a problem.
• Careful attention to language: Examine a mathematical statement to determine
whether it is
ambiguous or precise | 677.169 | 1 |
Will I Ever Use This in the Real World?
This is probably one of the most common questions students have; whether they ask it or not, they still can't stop thinking about it. Many students come up with the conclusion on their own that quadratic equations are useless in the real world and therefore lack the motivation to really put time and effort into learning it. Unfortunately, they are only half right. The examples I am about to give might be more focused on math but they apply to any of the physical sciences.
The answer to this question could be yes and no, depending on how you look at it. No, if you are talking about quadratic equations, hyperbolas, anti-derivatives, and so on. Unless you are going into the few occupations where these knowledge might matter, chances are, you will not solve a quadratic equation again once you are done with school. I understand the text book does provide some word problems that seem like the real world, but let's face it, if I needed to build a fence that needs the length to be 4 feet more than the width with a certain area, I personally would whip out my pencil and paper to set up and equation. However, most people would just go to the hardware store, buy a bunch of fencing material and return the leftovers. The truth is, It is not that big of a deal if you forget factoring and finding x.
So what is the "yes" part of the answer? What we often lose sight of and what is hard for students to see is that there are two parts of an education: knowledge and skill. Math and science courses aren't just there to teach us the knowledge of math and science, they are also there to teach us how to THINK like a scientist. When we are being taught how to solve for x, we are not just being taught the mathematical process of finding x, but more importantly how to "solve."
Once you finally leave school and enter the real world, chances are, you will rarely have to find x ever again, but you will always be asked to "solve." Approaching most problems in life is very similar to solving math problems. Step one is to find out what you know; followed by some reasonable assumptions and hypotheses, which will eventually l lead you to a conclusion or solution. Math is the foundation of any type problem solving. The physical sciences takes math and adds some applicability.
At the end the of the day, once you are in the real world, whatever you learned from math becomes your analytical, critical thinking, and problem solving skills; whatever you learned from science becomes your common sense. So next time when you are working a math problem and wondering "why" you are doing it, pay more attention to "how" you are doing it. In other words, how you are approaching the problem.
Comments
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Numerical Methods for Mathematics, Science and Engineering
A modern, computer-oriented approach to numerical analysis that shows how the mathematics of calculus and linear algebra are implemented in computer algorithms. Computer output is displayed in tables and used to develop topics of computer accuracy, pitfalls in computational methods and error estimation. | 677.169 | 1 |
...a tool in many field like: Engineering, medicine, science and Social Science. Mathematics is also used other than in calculation like it is used in logical reasoning... Learn about: Mathematics Algebra...
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Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problems.
Geared toward upper-level undergraduates and graduate students, this text surveys fundamental algebraic structures and maps between these structures. Its techniques are used in many areas of mathematics, with applications to physics, engineering, and computer science as well. Author Robert B. Ash, a Professor of Mathematics at the University of Illinois, focuses on intuitive thinking. He also conveys the intrinsic beauty of abstract algebra while keeping the proofs as brief and clear as possible. The early chapters provide students with background by investigating the basic properties of groups, rings, fields, and modules. Later chapters examine the relations between groups and sets, the fundamental theorem of Galois theory, and the results and methods of abstract algebra in terms of algebraic number theory, algebraic geometry, noncommutative algebra, and homological algebra, including categories and functors. An extensive supplement to the text delves much further into homological algebra than most introductory texts, offering applications-oriented results. Solutions to all problems appear in the text.
"This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--
Acclaimed by American Mathematical Monthly as "an excellent introduction,"this treatment ranges from basic definitions to important results and applications, introducing both the spirit and techniques of abstract algebra. It develops the elementary properties of rings and fields, explores extension fields and Galois theory, and examines numerous applications. 1982 edition.
Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. New to this edition is a groups first option that enables those who prefer to cover groups before rings to do so easily. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-referencesTo learn and understand mathematics, students must engage in the process of doing mathematics. Emphasizing active learning, Abstract Algebra: An Inquiry-Based Approach not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The book can be used in both rings-first and groups-first abstract algebra courses. Numerous activities, examples, and exercises illustrate the definitions, theorems, and concepts. Through this engaging learning process, students discover new ideas and develop the necessary communication skills and rigor to understand and apply concepts from abstract algebra. In addition to the activities and exercises, each chapter includes a short discussion of the connections among topics in ring theory and group theory. These discussions help students see the relationships between the two main types of algebraic objects studied throughout the text. Encouraging students to do mathematics and be more than passive learners, this text shows students that the way mathematics is developed is often different than how it is presented; that definitions, theorems, and proofs do not simply appear fully formed in the minds of mathematicians; that mathematical ideas are highly interconnected; and that even in a field like abstract algebra, there is a considerable amount of intuition to be found.
Lucid coverage of the major theories of abstract algebra, with helpful illustrations and exercises included throughout. Unabridged, corrected republication of the work originally published 1971. Bibliography. Index. Includes 24 tables and figures.
Presents a systematic approach to one of math's most intimidating concepts. Avoiding the pitfalls common in the standard textbooks, this title begins with familiar topics such as rings, numbers, and groups before introducing more difficult concepts.
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts of groups, sets, subsets, topology, Boolean algebra, and other mathematical subjects. 200 illustrations. | 677.169 | 1 |
Scientific Notation provides practice for students in grades five through nine. Students may choose to change numbers from standard form to scientific notation, or the reverse. An on-screen pencil allows students and teachers to show the process. Twenty examples give teachers pre-determinedAbout CalculatorMaX 2
CalculatorMaX 2 is an algebraic and RPN calculator with other capabilites such as: store and recall, trigonometric funtions (in Radians, Degrees, as well as Hyperbolic), scientific notation, and more. CalculatorMaX 2 contains tools that add power to the application: -... | 677.169 | 1 |
Practical Optimization: A Gentle
Introduction
The chapters appearing
below are draft chapters of an introductory textbook on optimization. The
ultimate objective is the creation of a complete, yet compact, introductory
survey text on the major topics in optimization. The material is derived from
the lecture notes used by the author in engineering courses at Carleton
University, and reflects the design considerations for those courses:
Students need to have a solid intuitive understanding of how
and why optimization methods work. This enables them to recognize when
things have gone wrong, and to diagnose the source of the difficulty and
take appropriate action. It also permits students to see how methods can
be combined or modified to solve non-standard problems.
Explanation and diagrams are more effective in transmitting a real
understanding than a total reliance on mathematics.
There should be some exposure to how things are done in practice,
which can be significantly different than how they are usually done in
textbooks.
The goal is for students to be able to recognize when problems that
they encounter in their jobs or in thesis work can be successfully tackled
by optimization methods. Students should be able to abstract a useful
formulation of a problem from a messy real world description.
The material is written at
the introductory level, assuming no more knowledge than high school algebra.
Most concepts are developed from scratch. I intend this to be a gentle
introduction.
You will need an Adobe Acrobat
reader to read the files. A free reader is downloadable from
Comments and suggestions
are actively sought. Please contact the author at chinneck@sce.carleton.ca. This is a draft,
so some diagrams are hand-drawn, and there may be typos, etc. Further chapters
will be added as time permits. Please let me know of any corrections that are
needed.
Browse the algorithm
animations. These provide animated illustrations of many of the key
concepts. The student developers of these animations were funded through
an IBM Faculty Award to Prof. Chinneck. Last revision: August 28,
2007.
Printing out the whole
volume? Then add the front matter
for a better look. Last revision: June 23, 2015.
Chapter 1:
Introduction. An introduction to the process of optimization and an
overview of the major topics covered in the course. Last revision: December 12,
2010.
Chapter 2:
Introduction to Linear Programming. The basic notions of linear programming
and the simplex method. The simplex method is the easiest way to provide a
beginner with a solid understanding of linear programming. Last revision:
September 19, 2007.
Chapter 7:
Linear Programming in Practice. Mention of other solution methods such as
revised simplex method and interior point methods. Mention of advanced
techniques used in practice such as advanced and crash start methods,
infeasibility analysis, and modelling systems. Last revision: August 18, 2006.
Chapter 10:
Network Flow Programming. A surprising range of problems can be
solved using minimum cost network flow programming, including shortest route,
maximum flow and minimum cut, etc. Variations such as generalized and
processing networks are also briefly introduced. Last revision: October 23, 2012.
Chapter 11:
PERT for Project Planning and Scheduling. PERT is a network-based aid
for project planning and scheduling. Many optimization problems involve
some aspect of the timing of activities that may run sequentially or in
parallel, or the timing of resource use. PERT diagrams help you to
understand and formulate such problems. Last revision: November 3, 2016.
Chapter
13: Binary and Mixed-Integer Programming. These are specialized
versions of branch and bound. A binary program has only binary variables
(0 or 1 only). A mixed-integer program looks like a linear program,
except that some or all of the variables are integer-valued (or binary-valued),
while others might be real-valued. Last revision: November 22, 2016.
Chapter
14: Heuristics for Discrete Search: Genetic Algorithms and Simulated Annealing.
Some problems are just too big for branch and bound, in which case you must
abandon the guarantee of finding the optimum solution and instead opt for
heuristic methods which can only guarantee to do fairly well most of the
time. Genetic Algorithms and Simulated Annealing are two popular
heuristic methods for use on very large problems. Last revision:August 22, 2006.
Chapter
15: Dynamic Programming. This optimization technique builds towards a
solution by first solving a small part of the whole problem, and then gradually
incrementing the size in a series of stages until the whole problem is
solved. Efficiency results from combining the local solution for a stage
with the optimum found for a previous stage. We look at the simplest
deterministic discrete cases. Last revision: December 16, 2015.
Chapter
16: Introduction to Nonlinear Programming (NLP). NLP is a lot harder
than linear programming. We start by looking at the reasons for this. Next we
look at the simplest method for solving the simplest type of NLP: unconstrained
problems that consist only of a nonlinear objective function. The method of
steepest ascent/descent is described. Last revision: December 16, 2015.
Chapter 17: Pattern Search for Unconstrained NLP.
What do you do if you don't have access to gradient information? In that case
you can use pattern search techniques (also known as derivative-free, direct
search, or black box methods). We look at the classic Hooke and Jeeves pattern
search method. Last revision: April 29, 2014.
Chapter 18: Constrained Nonlinear Programming. Now
that we have some idea of how to solve unconstrained NLPs, how do we deal with
constrained NLPs? The first idea is to turn them into unconstrained NLPs of
course! This is done by using penalty and barrier methods which replace
or modify the original objective function in ways that make feasible points
attractive in the resulting unconstrained problem. Last revision: April 29, 2014.
Chapter 19: Handling Equality Constraints in NLP.
Equality constraints are the hardest to handle in nonlinear programming. We
look at two ways of dealing with them: (i) the method
of Lagrange, and (ii) the Generalized Reduced Gradient (GRG) method. And we
take a look at making linear approximations to nonlinear functions because we
need that for the GRG method. Last revision: December 2, 2015.
Different
nonlinear local solvers can give quite different solution trajectories,
i.e. the sequence of intermediate solutions reached before the final
solution. You can see this in the next few animations. | 677.169 | 1 |
Problem-Solving Sessions for Sixth Formers 2016/17
Description
Year 12 and Year 13 students are invited to join a series of maths problem-solving sessions which offer students the opportunity to develop their problem-solving skills by trying out challenging problems that require deep mathematical thinking, and so help them meet the challenges they may face in taking a maths-rich university degree or career. The problems students will tackle are fun and rewarding. Attending the sessions will enrich students' mathematical experience. Students will look at problems from a range of sources including Sixth Term Examination Papers used by Cambridge and Warwick and the Mathematics Admissions Test used by Oxford and Imperial.
Suitability: The sessions are not just for students who intend to sit any of these exams or even just for those planning to study maths at university. They are for anyone who is studying A-level mathematics and who enjoys solving challenging problems.
Cost: The series of 12 sessions costs £20 per student. Students will be offered refreshment during each session and a problem-solving book.
Please note that we can only take online bookings (using the booking form links above) and payment must be made by credit card at the time of booking.
If you have a query about payment, please contact Margaret Bull on 01223 764777 or mab93@cam.ac.uk. | 677.169 | 1 |
Mathematics: A Very Short Introduction
£3.99
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In teaching first courses in real analysis over the years, we have found increasingly that the classes form rather heterogeneous groups. It is no longer true that most of the students are first-year graduate students in mathematics, presenting more or less
common backgrounds for the course. Indeed, nowadays we find diverse
backgrounds and diverse objectives among students in such classes. Some
students are undergraduates, others are more advanced. Many students are
in other departments, such as statistics or engineering. Some students
are seeking terminal master's degrees; others wish to become research
mathematicians, not necessarily in analysis.
We have tried to write a book that is suitable for students with minimal backgrounds, one that does not presuppose that most students will eventually specialize in analysis. We have pursued two goals. First, we would like all students to have an opportunity to obtain an appreciation of the tools, methods, and history of the subject and a sense of how the various topics we cover develop naturally. Our second objective is to provide those who will study analysis further with the necessary background in measure, integration, differentiation, metric space theory, and functional analysis.
To meet our first goal, we do several things. We provide a certain
amount of historical perspective that may enable a reader to see why a
theory was needed and sometimes, why the researchers of the time had
difficulty obtaining the "right" theory. We try to motivate topics
before we develop them and try to motivate the proofs of some of the
important theorems that students often find difficult. We usually avoid
proofs that may appear "magical" to students in favor of more revealing
proofs that may be a bit longer. We describe the interplay of various
subjects—measure, variation, integration, and differentiation. Finally,
we indicate applications of abstract theorems such as the contraction
mapping principle, the Baire category theorem, Ascoli's theorem,
Hahn-Banach theorem, and the open mapping theorem, to concrete settings
of various sorts.
We consider the exercise sections an important part of the book. Some of
the exercises do no more than ask the reader to complete a proof given
in the text, or to prove an easy result that we merely state. Others
involve simple applications of the theorems. A number are more
ambitious. Some of these exercises extend the theory that we developed
or present some related material. Others provide examples that we
believe are interesting and revealing, but may not be well known. In
general, the problems at the ends of the chapters are more substantial. A
few of these problems can form the basis of projects for further study.
We have marked exercises that are referenced in later parts of the book
with a ♦ to indicate this fact.
When we poll our students at the beginning of the course, we find there
are a number of topics that some students have seen before, but many
others have not. Examples are the rudiments of metric space theory,
Lebesgue measure in ℜ1, Riemann–Stieltjes integration,
bounded variation and the elements of set theory (Zorn's lemma,
well-ordering, and others). In Chapter 1, we sketch some of this
material. These sections can be picked up as needed, rather than covered
at the beginning of the course. We do suggest that the reader browse
through Chapter 1 at the beginning, however, as it provides some
historical perspective.
Text Organization
Many graduate textbooks are finely crafted works as intricate as a
fabric. If some thread is pulled too severely, the whole structure
begins to unravel. We have hoped to avoid this. It is reasonably safe to
skip over many sections (within obvious limitations) and construct a
course that covers your own choice of topics, with little fear that the
student will be forced to cross reference back through a maze of earlier
skipped sections.
A word about the order of the chapters. The first chapter is intended as
background reading. Some topics are included to help motivate ideas
that reappear later in a more abstract setting. Zorn's lemma and the
axiom of choice will be needed soon enough, and a classroom reference to
Sections 1.3, 1.5 and 1.11 can be used.
The course can easily start with the measure theory of Chapter 2 and
proceed from there. We chose to cover measure and integration before
metric space theory because so many important metric spaces involve
measurable or integrable functions. The rudiments of metric space theory
are needed in Chapter 3, however, so we begin that chapter with a short
section containing the necessary terminology.
Instructors who wish to emphasize functional analysis and reach Chapter 9
quickly can do so by omitting much of the material in the earlier
chapters. One possibility is to cover Sections 2.1 to 2.6, 4.1, 4.2, and
Chapter 5 and then proceed directly to Chapter 9. This will provide
enough background in measure and integration to prepare the student for
the later chapters.
Chapter 6 on the Fubini and Tonelli theorems is used only occasionally
in the sequel (Sections 8.4 and 13.9). This is presented from the outer
measure point of view because it fits better with the philosophy
developed in Chapters 2 and 3. One can substitute any treatment in its
place. Chapter 11 on analytic sets is not needed for the later chapters,
and is presented as a subject of interest on its own merits.
Chapter 13 on the Lp–spaces can be
bypassed in favor of Chapter 14 or 15 except for a few points. Chapter
14 on Hilbert space could be undertaken without covering Chapters 12 and
13 since all material on the spaces ℓ2and L2 is repeated as needed. Chapter 15 on Fourier series does not need the Hilbert space
material in order to work, but, since it is intended as a showplace for
many of the methods, it does draw on many other chapters for ideas and
techniques. | 677.169 | 1 |
Print Edition India
Print Edition International
E-Book
Overview
Summary of the Book
Permutation and Combinations has always been a dreaded
chapter in every student's life and they usually have confusion as to which
book to follow. There is a large gap between the student's understanding and
the presentation of the numerous books available in the market today. To bridge
this gap, Ramesh Chandra has come up with this book on P&C written in the
best possible sequence. The book is compiled by the author to cover up the
difficulties faced by the students. A No.1 success tool that includes all the
concepts, samples, subjective & objective exercises, puzzles, IQ tests,
easy learning methods and analysis to bring out the best in every student. This is a must-have for
students aspiring to make it to the IIT's.
About the Author
Ramesh
Chandra, a proud alumnus of IIT Kanpur has always had a consistent academic
record. His love towards science and passion towards teaching made him join
FIITJEE as a teacher where he worked for almost 3years and 10 months during
which he presented several papers for their A.I.T.S. in various groups, i.e.
IIT JEE – mains, IIT JEE – Advanced, BITSAT, KVPY, and NTSE. Mr. Chandra
currently works in a reputed education industry in Pune, Maharashtra. | 677.169 | 1 |
MTH148: Integrated Mathematics IThis first-year high school integrated math course focuses on linear and simple exponential models. The course contrasts linear behavior with exponential behavior, and it uses both linear and simple exponential equations as models. Students learn about and work extensively with functions—analyzing function properties and behavior, creating new functions from known functions, and applying functions to various continuous and discrete situations. The statistics in the course focus on modeling. Topics in geometry include constructions, transformations, similarity, and congruence, and students use the Pythagorean theorem in analytic geometry contexts.
Course Length
Prerequisites
Course Outline
SEMESTER 1
Semester A, Unit 1: Expressions and Problem Solving
This unit focuses on variables and measurement. Students practice translating real-world situations into mathematical expressions and equations, and they use units to understand problems. In addition, students look at what the structure of a mathematical expression can say about the relationships between quantities in a real-world context it models.
Semester 1 Introduction
Foundations for Unit 1
Expressions
Variables
Translating Words into Variable Expressions
Equations
Translating Words into Equations
Problem Solving
Dimensional Analysis
Precision and Accuracy
Core Focus: Structure and Meaning
Unit Review
Unit Test
Semester A, Unit 2: Solving Linear Equations and Inequalities
This unit is all about equations and inequalities. Students see how to solve equations, and they extend this skill to solving inequalities and real-world applications involving inequalities. Students also use mathematical reasoning to justify each step when solving an equation or inequality.
Foundations for Unit 2
Addition and Subtraction Equations
Multiplication and Division Equations 1
Multiplication and Division Equations 2
Multiple Transformations
Variables on Both Sides of an Equation
Discuss: Linear Equations
Transforming Formulas
Inequalities
Solving Inequalities
Applications of Inequalities
Core Focus: Reasoning
Unit Review
Unit Test
Semester A, Unit 3: Linear Equations and Inequalities
In this unit, students graph linear equations in two variables and use graphs to solve real-world problems. Students also graph linear inequalities in two variables and use inequalities to model constraints in real-world contexts.
Foundations for Unit 3
Graphs of Lines
Forms of Linear Equations
Writing Equations of Lines
Applications: Linear Equations
Graphing Linear Inequalities
Systems of Linear Inequalities
Constraints
Core Focus: Linear Modeling
Unit Review
Unit Test
Semester A, Unit 4: Introduction to Functions
In this unit, students learn about mathematical relations, functions, and function notation. They will work with function equations and find intercepts and average rate of change for functions. In addition, they'll learn how the domain and range of a function can relate to the situation the function models.
Foundations for Unit 4
Relations
Functions
Function Equations 1
Function Equations 2
Discuss: Function Applications
Linear Functions
Intercepts
Average Rate of Change
Core Focus: Thinking About Domain and Range
Unit Review
Unit Test
Semester A, Unit 5: Exponential Equations and Functions
This unit is all about exponents. Students start with reviewing properties of exponents, and then they move on to solving simple exponential equations and graphing exponential functions. Then, students see how exponential models can fit situations involving growth and decay.
Foundations for Unit 5
Properties of Exponents
Exponential Expressions and Equations I
Exponential Expressions and Equations II
Graphing Exponential Functions
Applications: Growth and Decay
Features of Exponential Functions
Core Focus: Multiple Representations
Unit Review
Unit Test
Semester A, Unit 6: Sequences and Modeling with Functions
A sequence is a function with integer domain. In this unit, students learn about arithmetic, geometric, and other types of sequences. Then, students work with other types of functions as models for real-world situations.
Foundations for Unit 6
Sequences and Patterns
Arithmetic Sequences
Geometric Sequences
Discuss: Sequences
Applications: Sequences
Function Parameters
Combining Functions
Core Focus: Comparing Models
Unit Review
Unit Test
Semester A, Unit 7: Semester Review and Test
Semester Review
Semester Test
SEMESTER 2
Semester B, Unit 1: Systems of Equations
When two equations have to be true same at the same time, they are called a system of equations. In this unit, students use graphs, substitution, and linear combination to solve systems of equations. Students also use systems to solve real-world problems.
Semester 2 Introduction
Foundations for Unit 1
Systems of Equations
Approximating Solutions with Graphs
Substitution Method
Linear Combination
Linear Combination with Multiplication
Applications: Systems of Linear Equations
Core Focus: Justifying Linear Combination
Unit Review
Unit Test
Semester B, Unit 2: Describing Data
This unit focuses on data. First, students calculate measures of center (mean, media, and mode) and variability (variance and standard deviation). Then, students use graphs and tables to make sense of data and identify outliers.
Foundations for Unit 2
Measures of Center
Variability
Graphs of Univariate Data
Selecting Measures of Center
Frequency Distributions
Discuss: Comparing Data Sets
Core Focus: Outliers
Unit Review
Unit Test
Semester B, Unit 3: Linear Models for Data
Sometimes, data come in ordered pairs. Ordered pairs can be plotted on a graph, and then you can start to see patterns. In this unit, students graph two-dimensional data and describe how the input values (the x-values) are related to the output values (the y-values). Then, students use correlation and linear regression to describe the relationship between the values.
Foundations for Unit 3
Two-Way Tables
Scatterplots
Association
The Correlation Coefficient
Fitting a Line to Data
Least Squares Regression
Core Focus: Parameters and Residuals
Unit Review
Unit Test
Semester B, Unit 4: Transformations
This unit covers essential geometric terms such as point, line, plane, and angle. Once students have mastered those definitions, they'll be ready to learn about polygons, transformations, and symmetry. Then, students will see how to use coordinates for transformations.
Foundations for Unit 4
Basic Geometric Terms and Definitions 1
Basic Geometric Terms and Definitions 2
Measuring Length
Measuring Angles
Transformations 1
Transformations 2
Discuss: Transformations
Using Algebra to Describe Geometry 1
Using Algebra to Describe Geometry 2
Polygons and Symmetry 1
Polygons and Symmetry 2
Core Focus: Looking at Transformations
Unit Review
Unit Test
Semester B, Unit 5: Constructions and Congruence
In this unit, students learn about congruence, with a focus on reasoning about the congruence of polygons and triangles. Then, students see how to perform geometric constructions including segment bisectors and simple regular polygons. Finally, they see how congruence and transformations are related to each other.
Foundations for Unit 5
Congruent Polygons and Their Corresponding Parts 1
Congruent Polygons and Their Corresponding Parts 2
Triangle Congruence: SSS, SAS, and ASA 1
Triangle Congruence: SSS, SAS, and ASA 2
Bisectors and Line Relationships 1
Bisectors and Line Relationships 2
Constructions with Polygons 1
Constructions with Polygons 2
Core Focus: Congruence and Rigid Motions
Unit Review
Unit Test
Semester B, Unit 6: Analytic Geometry
When the tools of algebra are applied to geometry, the result is called analytic geometry. In this unit, students see how to use coordinates and algebra to solve geometric problems, prove theorems, and describe geometric relationships. | 677.169 | 1 |
Calculus Chain Rule Derivatives Foldable plus HW
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this file type before downloading and/or purchasing.
1 MB|5 pages
Product Description
Calculus Chain Rule for Derivatives Foldable plus Homework/ Quiz.
This activity is designed for AP Calculus AB, AP Calculus BC, Honors Calculus, and College Calculus 1. Great Organizer!!
This fun activity will help your students better understand the chain rule and all the steps involved. Some of the examples and HW problems involve transcendentals. Great for Interactive Notebooks!
Included in this resource are::
✓ 2 Foldables showing step by step solutions for two different problems.
✓ 2 Fill-In Foldables for students to complete with two more problems.
Students can create their foldables alone or in small groups helping each other to make sure they don't "break the chain". A teacher version with full step by step solutions is included for all problems.
✓ Handout with five problems. The handout can be used as enrichment, homework, group work, or as an assessment. Full typed solutions included | 677.169 | 1 |
Quadratic Formula Introduction Lesson
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Product Description
The Quadratic Formula can be a tricky Algebra lesson, however this handout is a great introduction! It includes a review of standard form (a, b & c coefficients) and imaginary numbers. It also has fill-in the blank style practice problems to get students comfortable with using the quadratic formula, and it includes a quadratic formula card sort that can be used as a recap the next day! | 677.169 | 1 |
Mathematics - AllConferenceAlert
The main idea behind these conferences on Mathematics is to bring along the scholars, engineers, researchers, scientists, and practitioners from across the globe in order to present and exchange ideas on ongoing researches on this particular field of Mathematics. Besides that these conference also provide a scope for the dignitaries to exchange their own ideas and methodologies as well as their application experiences directly form face to face, so as to establish further business/ research relations for future collaborations. | 677.169 | 1 |
Description
Book 3 in the Princeton Mathematical Series.
Originally published in 1950SimilarThis classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.
Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.
Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there.
The book will provide an invaluable resource to all those taking a first course in differential geometry, for their lecturers, and for all others interested in the subject.
Andrew Pressley is Professor of Mathematics at King's College London, UK.
The Springer Undergraduate Mathematics Series (SUMS) is a series designed for undergraduates in mathematics and the sciences worldwide. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully worked solutions.
This book is an introduction to hyperbolic and differential geometry that provides material in the early chapters that can serve as a textbook for a standard upper division course on hyperbolic geometry. For that material, the students need to be familiar with calculus and linear algebra and willing to accept one advanced theorem from analysis without proof. The book goes well beyond the standard course in later chapters, and there is enough material for an honors course, or for supplementary reading. Indeed, parts of the book have been used for both kinds of courses. Even some of what is in the early chapters would surely not be nec essary for a standard course. For example, detailed proofs are given of the Jordan Curve Theorem for Polygons and of the decomposability of poly gons into triangles, These proofs are included for the sake of completeness, but the results themselves are so believable that most students should skip the proofs on a first reading. The axioms used are modern in character and more "user friendly" than the traditional ones. The familiar real number system is used as an in gredient rather than appearing as a result of the axioms. However, it should not be thought that the geometric treatment is in terms of models: this is an axiomatic approach that is just more convenient than the traditional ones.
curvilinear coordinates. This treatment includes in particular a direct proof of the three-dimensional Korn inequality in curvilinear coordinates. The fourth and last chapter, which heavily relies on Chapter 2, begins by a detailed description of the nonlinear and linear equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are "two-dimensional", in the sense that they are expressed in terms of two curvilinear coordinates used for de?ning the middle surface of the shell. The existence, uniqueness, and regularity of solutions to the linear Koiter equations is then established, thanks this time to a fundamental "Korn inequality on a surface" and to an "in?nit- imal rigid displacement lemma on a surface". This chapter also includes a brief introduction to other two-dimensional shell equations. Interestingly, notions that pertain to di?erential geometry per se,suchas covariant derivatives of tensor ?elds, are also introduced in Chapters 3 and 4, where they appear most naturally in the derivation of the basic boundary value problems of three-dimensional elasticity and shell theory. Occasionally, portions of the material covered here are adapted from - cerpts from my book "Mathematical Elasticity, Volume III: Theory of Shells", published in 2000by North-Holland, Amsterdam; in this respect, I am indebted to Arjen Sevenster for his kind permission to rely on such excerpts. Oth- wise, the bulk of this work was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].
Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in I\!\!R^3 that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas.
With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces.
The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces.
The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces.
The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.
Differential Geometry offers a concise introduction to some basic notions of modern differential geometry and their applications to solid mechanics and physics.
Concepts such as manifolds, groups, fibre bundles and groupoids are first introduced within a purely topological framework. They are shown to be relevant to the description of space-time, configuration spaces of mechanical systems, symmetries in general, microstructure and local and distant symmetries of the constitutive response of continuous media.
Once these ideas have been grasped at the topological level, the differential structure needed for the description of physical fields is introduced in terms of differentiable manifolds and principal frame bundles. These mathematical concepts are then illustrated with examples from continuum kinematics, Lagrangian and Hamiltonian mechanics, Cauchy fluxes and dislocation theory.
This book will be useful for researchers and graduate students in science and engineering.
Non-Riemannian Geometry deals basically with manifolds dominated by the geometry of paths co-developed by the distinguished mathematician Luther Pfahler Eisenhart, the author of this text. He begins with a consideration of asymmetric connections, and then proceeds to a contrasting survey of symmetric connections. Discusses projective geometry of paths and the geometry of sub-spaces. 1927 editionCreated especially for graduate students, this introductory treatise on differential geometry has been a highly successful textbook for many years. Its unusually detailed and concrete approach includes a thorough explanation of the geometry of curves and surfaces, concentrating on problems that will be most helpful to students. 1909 | 677.169 | 1 |
a mind for numbers how to excel at math and science even if you flunked algebra
Start Reading or Download A Mind For Numbers How To Excel At Math And Science Even If You Flunked Algebra in PDF EPUB format. You can Read Online A Mind For Numbers How To Excel At Math And Science Even If You Flunked Algebra here in PDF, EPUB, Mobi or Docx formatsAn engineering professor who started out doing poorly in mathematical and technical subjects in school offers tools, tips and techniques to learning the creative and analytical thought processes that will lead to achievement in math and science. Original the tradition of The Power of Habit and Thinking, Fast and Slow comes a practical, playful, and endlessly fascinating guide to what we really know about learning and memory today—and how we can apply it to our own lives. From And what if there was a way to achieve more with less effort? In How By The Praise for How We Learn "This book is a revelation. I feel as if I've owned a brain for fifty-four years and only now discovered the operating manual."—Mary Roach, bestselling author of Stiff and Gulp "A welcome rejoinder to the faddish notion that learning is all about the hours put in."—The New York Times Book Review "A valuable, entertaining tool for educators, students and parents."—Shelf Awareness "How We Learn is more than a new approach to learning; it is a guide to making the most out of life. Who wouldn't be interested in that?"—Scientific American "I know of no other source that pulls together so much of what we know about the science of memory and couples it with practical, practicable advice."—Daniel T. Willingham, professor of psychology at the University of Virginia From the Hardcover edition.
Discusses the best methods of learning, describing how rereading and rote repetition are counterproductive and how such techniques as self-testing, spaced retrieval, and finding additional layers of information in new material can enhance learning techniquesHave you ever heard of a person who left you wondering, "How could someone be so twisted? So evil?" "evil" people act the way they do mainly as the result of a dysfunction. In fact, some deceitful, manipulative, and even sadistic behavior appears to be programmed genetically—suggesting that some people really are born to be bad. Oakley links the latest findings of molecular research to a wide array of seemingly unrelated historical and current phenomena, from the harems of the Ottomans and the chummy jokes of "Uncle Joe" Stalin, to the remarkable memory of investor Warren Buffet. Throughout, she never loses sight of the personal cost of evil genes as she unravels the mystery surrounding her sister's enigmatic life—and death. Evil Genes is a tour-de-force of popular science writing that brilliantly melds scientific research with intriguing family history and puts both a human and scientific face to evil. From the Hardcover edition. | 677.169 | 1 |
ALEX Lesson Plan Resources
Subject: Mathematics (9 - 12) Title: Solving Formulas for the Given Variable Description: This lesson will use the process of inverse operations to solve formulas for a given variable. Some formulas will not be recognized by the student. The actual formula is not important, but the variables are.
This lesson results from the ALEX Resource Gap Project. | 677.169 | 1 |
Yasha2012-06-26 19:03:31 My name is Yasha. I am a former student and counselor at Mathcamp, and I helped grade this year's quiz. As Richard mentioned, I'm also an instructor here at AoPS.
Yasha2012-06-26 19:03:34 I'll be visiting camp in week 4 this summer, so I'll see some of you there.
Yasha2012-06-26 19:03:50 For those of you who aren't familiar with it, the Mathcamp Qualifying Quiz is a seven-question untimed set of problems which all applicants to Canada/USA Mathcamp must solve for admission. You don't need to solve all of the problems, but you need to make good progress! It's one of several factors that impacts your admission. I'm expecting that many of you were indeed applicants this year. To follow along, you should all have the quiz open in another window:
Yasha2012-06-26 19:04:03 Additionally, if you did apply this year, I recommend having your solutions open. That way, you can reply more quickly to the questions I ask of the room. I like to have a very interactive session, so I'm expecting you all to pitch in to the solutions!
Yasha2012-06-26 19:04:24 We are going to go through all seven problems on the quiz. I'm going to try to show you more than just fully correct answers: I also want to show you how you could come up with those answers, and how to write them up in a way that really communicates the mathematics you're doing.
Yasha2012-06-26 19:04:35 This whole session is rather experimental --- we've only done it once before --- so I'd love to get your comments during or after the session on how it went and how useful it was (or wasn't).
Yasha2012-06-26 19:04:44 We're going to start relatively slowly, but accelerate as we go along. The idea is for everyone to be able to follow at first, but be warned that things will get more and more difficult as we go on.
Yasha2012-06-26 19:05:22 As Richard mentioned, this chat room is moderated. That means your messages go only to me, and I will choose which to pass on, so feel free to contribute to problems and ask questions.
Yasha2012-06-26 19:05:46 If you have any questions about how we select problems or how we grade solutions, save them and I'll try to leave time to answer them at the end of the Math Jam. If I don't end up getting to them, feel free to post them on the Mathcamp forum on AoPS at
Yasha2012-06-26 19:07:07 First off, there's some ambiguity in the wording of the problem: When you make your first guess, it's not clear whether the problem intends for the frog to have had time for one jump or two. We accepted solutions based on either interpretation, but for the sake of staying on the same page in this Math Jam, let's interpret the problem to mean that the frog had time for one jump when you make your first guess. So, for example, if n=17, then when you make your first guess the frog is at 17, when you make your second guess the frog is at 34, and so forth.
Yasha2012-06-26 19:07:28 So, how can we catch the frog without knowing what n is?
Watermelon8762012-06-26 19:07:50 we can guess a value of n equal ot the elapsed seconds.
viperstrike2012-06-26 19:07:55 after the first second the frog is on n and we begin guessing its location. so after s seconds the frog is on the integer x=ns
CDerwin12012-06-26 19:08:00 we can systematically check each n
Yasha2012-06-26 19:08:14 We can systematically go through all the possibilities for n, starting with n=1, then n=2, then n=3, and so forth. Each time we guess a value for n, we search the integer where we'd expect the frog to be if our guess were correct. Eventually, we'll guess the right value and find the frog.
Yasha2012-06-26 19:08:24 For our first guess, we guess that n=1. Where should we search for the frog?
jared4292012-06-26 19:08:40 1
dantx52012-06-26 19:08:40 at 1
numbertheorist172012-06-26 19:08:40 at 1
Watermelon8762012-06-26 19:08:40 n*s=1*1=1
viperstrike2012-06-26 19:08:40 1
Yasha2012-06-26 19:08:47 The frog has had time to jump once, so, assuming n=1, it will be at the integer 1. We should search there. If we don't find it, we guess that n=2. Where should we search for the frog for our second guess?
theGoodGuy2012-06-26 19:08:58 4
yingted2012-06-26 19:08:58 4
numbertheorist172012-06-26 19:08:58 at 4
nemarci2012-06-26 19:09:07 4
dantx52012-06-26 19:09:08 4
jjchoi52012-06-26 19:09:08 4
MA7HL0V3R2012-06-26 19:09:08 4
spacebacon992012-06-26 19:09:08 4
viperstrike2012-06-26 19:09:08 4
Yasha2012-06-26 19:09:14 By the time we make our second guess, the frog has had time to jump twice. Assuming n=2, the frog jumped 2 to the right each time, so it will end up at the integer 4. We search there. Where do we search for our third guess?
yingted2012-06-26 19:09:31 9
theGoodGuy2012-06-26 19:09:31 9
CDerwin12012-06-26 19:09:31 9
dantx52012-06-26 19:09:31 9
numbertheorist172012-06-26 19:09:31 9
frtennis12012-06-26 19:09:31 9
nemarci2012-06-26 19:09:31 9
spacebacon992012-06-26 19:09:31 6
jjchoi52012-06-26 19:09:31 9
numbertheorist172012-06-26 19:09:31 9
Watermelon8762012-06-26 19:09:31
Mathpather2012-06-26 19:09:31 3^2=9
MA7HL0V3R2012-06-26 19:09:31 9
Yasha2012-06-26 19:09:39 For our third guess, we guess that n=3. Since the frog has had time to jump three times, it will be at the integer 9, so we search there.
dantx52012-06-26 19:09:54 perfect square integers?
melikababadi2012-06-26 19:09:54 n^2?
Mathpather2012-06-26 19:09:54 so we check n^2 for the frog at each n
numbertheorist172012-06-26 19:09:54 basically at every perfect square
viperstrike2012-06-26 19:09:54 since n=s x is a perfect sqaure
Yasha2012-06-26 19:10:12 On our kth guess, we guess that n=k. By then, the frog has had time to jump k times going k to the right each time, landing on the integer k^2.
19oshawott982012-06-26 19:11:02 but what about the second in the beginning
Yasha2012-06-26 19:11:05 The problem is a bit ambiguous about whether the frog got to jump once or twice when you make your first guess. We accepted both interpretations, but for the MathJam we're sticking with once.
Yasha2012-06-26 19:11:23 OK, so our strategy is to search the integer k^2 on guess number k. How long will it take us to catch the frog?
frtennis12012-06-26 19:11:42 n times
Mathpather2012-06-26 19:11:42 n seconds
koel172012-06-26 19:11:45 n seconds
Yasha2012-06-26 19:11:52 We will catch the frog on guess number n. On that guess, we will search integer n^2, which is where the frog will be after n seconds.
Yasha2012-06-26 19:11:59 Any questions about this part?
Yasha2012-06-26 19:12:16
Yasha2012-06-26 19:12:27 How can we modify our reasoning in part (a) to solve this problem?
Yasha2012-06-26 19:12:48 In part (b), we need to not only go through all of the possibilities for n, but also both possibilities for the frog's direction.
Yasha2012-06-26 19:12:53 How can we take into account all of the possibilities in our search for the frog?
Mathpather2012-06-26 19:13:06 we check n=1, n=-1, n=2, n=-2, and so on
spacebacon992012-06-26 19:13:06 we first try a positive n, and then a negative n
dantx52012-06-26 19:13:06 find if the frog's jumps are -1, then 1, then -2, then 2, and so on
Yasha2012-06-26 19:13:15 Here's one way to do it: On our first guess, we assume that n=1 and the frog goes right. On the second guess, we assume n=1 and left, on the third guess, we assume n=2 and right, on the fourth guess, we assume n=2 and left, and so forth.
Yasha2012-06-26 19:13:21 To make matters easier, we can use negative values of n to denote motion to the left, and positive values to denote motion to the right. For example, n=-3 means the frog jumps three units to the left each second, and n=5 means the frog jumps five units to the right each second.
Yasha2012-06-26 19:13:31 The order we have for guessing the values of n is 1, -1, 2, -2, 3, -3, and so forth.
Yasha2012-06-26 19:13:38 Of course, a different order can work just as well, such as -1, 1, -2, 2, -3, 3, and so forth.
Yasha2012-06-26 19:13:49 The important part is that every possibility for the frog's jumping strength and direction appears in this list, so, eventually, we will guess the correct value of n.
Yasha2012-06-26 19:13:59 OK, so assuming that the order in which we guess the values of n is 1, -1, 2, -2, 3, -3, etc., what should be the first integer we search?
Mathpather2012-06-26 19:14:12 1
dantx52012-06-26 19:14:12 1
MA7HL0V3R2012-06-26 19:14:12 1
yingted2012-06-26 19:14:12 1
viperstrike2012-06-26 19:14:12 1
spacebacon992012-06-26 19:14:12 1
frtennis12012-06-26 19:14:12 1
Yasha2012-06-26 19:14:20 If n=1, then after one second the frog will be at the integer 1, so we should search there. What about the second integer we search?
Mathpather2012-06-26 19:14:30 -2
Watermelon8762012-06-26 19:14:30 -2
yingted2012-06-26 19:14:30 -2
jared4292012-06-26 19:14:30 -2
dantx52012-06-26 19:14:30 -2
frtennis12012-06-26 19:14:30 -2
nemarci2012-06-26 19:14:30 -2
CDerwin12012-06-26 19:14:30 -2
numbertheorist172012-06-26 19:14:30 -2
MA7HL0V3R2012-06-26 19:14:30 -2
spacebacon992012-06-26 19:14:30 -2
esque2012-06-26 19:14:30 -2
Yasha2012-06-26 19:14:36 On our second guess, we decided to check the possibility that n=-1. If n=-1, then after two seconds the frog will be at the integer -2, so we search there. What about the third guess?
nemarci2012-06-26 19:15:00 6
Mathpather2012-06-26 19:15:00 6
viperstrike2012-06-26 19:15:00 then 6
dantx52012-06-26 19:15:00 6
MA7HL0V3R2012-06-26 19:15:00 6
Watermelon8762012-06-26 19:15:00 6
melikababadi2012-06-26 19:15:00 6
yingted2012-06-26 19:15:00 6
viperstrike2012-06-26 19:15:00 6
Yasha2012-06-26 19:15:08 On our third guess, we decided to assume that n=2. If n=2, then after three seconds the frog will be at the integer 6.
Yasha2012-06-26 19:15:17 Likewise, on our fourth guess, we assume that n=-2. Then, after four seconds, the frog will be at the integer -8.
Yasha2012-06-26 19:15:22 We now have a strategy for catching the frog: On our kth guess, we look at the kth term of the sequence 1, -1, 2, -2, 3, -3, .... This term gives us the value of n that we'd like to test. We multiply it by k (the number of seconds since the frog started) to find where the frog would be if our guess for n were correct, and we search at that integer.
jwp2012-06-26 19:16:14 but how do we know the pace of the frog?
Yasha2012-06-26 19:16:16 We don't, that's the beauty of it. We make a guess for the pace of the frog each time we search. Since we guess systematically, eventually we will be right.
Yasha2012-06-26 19:16:35 At some point, we will reach n in the sequence, and at that point we will search for the frog in the correct place and catch it.
marupiravi2012-06-26 19:16:59 how will we know we are right
Yasha2012-06-26 19:17:00 We'll know because there will be a frog at the integer we search, and then we stop.
Yasha2012-06-26 19:17:26 We have a strategy for catching the frog, so we've done everything the problem asked for, but let's go ahead and see if we can find an actual formula for the integer we should search on guess number k.
Yasha2012-06-26 19:17:38 How can we start?
Watermelon8762012-06-26 19:18:13 Write the sequence 1,-1,2,-2... in terms of t
Mathpather2012-06-26 19:18:13 n odd and n even are different
theGoodGuy2012-06-26 19:18:13 +- floor(k/2)*n
Yasha2012-06-26 19:18:34 We consider the cases where k is odd and even separately, because on odd guesses we guess positive values of n, and on even guesses we guess negative values of n.
Yasha2012-06-26 19:18:39 Let's first work with the case where k is odd. What is the value of n that we test on guess number k, in terms of k?
Yasha2012-06-26 19:19:05 The value of n that we'd like to test is the kth term of the sequence 1, -1, 2, -2, 3, -3, .... What is the kth term of this sequence, in terms of k?
Yasha2012-06-26 19:19:18 (We're working with odd k for now.)
yingted2012-06-26 19:19:31 (k+1)/2
Yasha2012-06-26 19:19:37
Yasha2012-06-26 19:19:42 You can check the first couple of values to make sure that you're not off by one: When k=1 our guess is n=1. When k=3 our guess is n=2, and so forth.
Yasha2012-06-26 19:19:58 At what integer should we search for the frog on our kth guess, where k is odd?
Watermelon8762012-06-26 19:20:24
Watermelon8762012-06-26 19:20:24
Mathpather2012-06-26 19:20:24 k(k+1)/2
esque2012-06-26 19:20:24 k(k+1)/2
Watermelon8762012-06-26 19:20:24
Yasha2012-06-26 19:20:29
Yasha2012-06-26 19:20:36 OK, let's move on to even values of k. What value of n do we test on our kth guess when k is even?
yingted2012-06-26 19:20:54 -k/2
MA7HL0V3R2012-06-26 19:20:54 -k/2
Yasha2012-06-26 19:21:00
Yasha2012-06-26 19:21:07 At what integer should we search for the frog?
Mathpather2012-06-26 19:21:24 -(k^2)/2
yingted2012-06-26 19:21:24 -k^2/2
nemarci2012-06-26 19:21:24 -k^2/2
Yasha2012-06-26 19:21:40
Yasha2012-06-26 19:21:51
frtennis12012-06-26 19:22:55 Around n*2 tries
Watermelon8762012-06-26 19:22:55 If n is positive, 2n-1. If n is neg. -2n
dantx52012-06-26 19:22:55 2n-1 odd, 2n even
Yasha2012-06-26 19:23:04
Yasha2012-06-26 19:23:22
Yasha2012-06-26 19:23:33 Any questions on this part?
Yasha2012-06-26 19:23:49
Yasha2012-06-26 19:24:07 Like in the previous part, let's let negative values of n denote moving to the left, and positive values of n denote moving to the right.
Yasha2012-06-26 19:24:24 What would we need to do in order to be able to use a strategy similar to the one we've used in the previous parts?
skycao2012-06-26 19:24:40 count all the lattice points
Maz9062012-06-26 19:24:52 check where the frog is for various (m,n)
frtennis12012-06-26 19:24:52 Find a way to systematically navigate all possible combinations of n and m
Watermelon8762012-06-26 19:24:52 We need a way to biject pairs of m and n to the natural numbers (time in seconds)
yingted2012-06-26 19:24:52 search the half-plane (m,n),n>0
Yasha2012-06-26 19:24:58 We'd need to go through all possible pairs (n,m), testing one of them each guess.
Yasha2012-06-26 19:25:04 Each guess, we'd search the integer where we'd expect the frog to be if our guess (n,m) of its jumping speed and initial position were correct. We want to eventually hit every possibility for (n,m). If we manage that, then at some point our guess (n,m) will be correct, and we'll find the frog.
Yasha2012-06-26 19:25:25 So, how can we go through all of the possibilities for (n,m)?
Yasha2012-06-26 19:25:55 Unfortunately, we can't let n=1 and go through all possibilities for m, and then move to n=-1, then n=2, and so forth: Since there are infinitely many values of m to try, we'll never actually get to n=-1!
Watermelon8762012-06-26 19:26:43 We can draw a spiral on the np-plane where p is the initial position
Mathpather2012-06-26 19:26:43 we spiral out from the origin of a cartesian plane
skycao2012-06-26 19:26:43 spiral
frtennis12012-06-26 19:26:43 Try a spiral in the coordinate plane of n and m
Watermelon8762012-06-26 19:26:43 You can do this by drawing a spiral starting from 0,0 to 0,1 to 1,1 to 0,1 to -1,1, to -1,0...
yingted2012-06-26 19:26:43 for all m: for all n from 0 to m: try (n,m-n)
Yasha2012-06-26 19:27:26 There are a lot of possible solutions, and all of them involve first taking those (n,m) where both n and m are small, and then gradually increasing them.
Yasha2012-06-26 19:27:56 A lot of these ways look like spiraling out from the origin.
Yasha2012-06-26 19:28:06 Here's one way to do it explicitly:
Yasha2012-06-26 19:28:11 First, guess all possibilities for n and m where both of their absolute values are less than or equal to one. There are finitely many of these:
Yasha2012-06-26 19:28:16
Yasha2012-06-26 19:28:31 Next, guess all possibilities for n and m where both of their absolute values are less than or equal to two, (ignoring the ones we have guessed already). There are also finitely many of these:
Yasha2012-06-26 19:28:40
Yasha2012-06-26 19:28:50 Next, we continue up to three:
Yasha2012-06-26 19:28:55
Yasha2012-06-26 19:29:00
Yasha2012-06-26 19:29:13 We can continue this way indefinitely, and we will eventually list every possible pair (n,m).
bobthesmartypants2012-06-26 19:29:21 like with concentric circles?
Yasha2012-06-26 19:29:29 Yup, sort of.
Yasha2012-06-26 19:29:34 If you want to visualize this process geometrically, consider the pairs (n,m) as locations in the plane. We first take all the pairs (n,m) in a small square around the origin. Then we increase the size of the square, and we take all of the new pairs (n,m) that are now inside of it. Then we increase the size of the square again, and so forth.
viperstrike2012-06-26 19:30:07 what is n and m
Yasha2012-06-26 19:30:09 n was the velocity of the frog, and m was its starting location.
marupiravi2012-06-26 19:30:39 all we have to do is make m zero and then move up on the value of n
Yasha2012-06-26 19:30:41 If you do that, you'll only test the possibilities where m=0. You'll never get to any other values of m.
frtennis12012-06-26 19:30:53 Can this be put into an explict formula though?
Yasha2012-06-26 19:31:11 You could, but it would be a very messy formula.
yingted2012-06-26 19:31:16 using sqrt and floor :(
marupiravi2012-06-26 19:31:39 you say n was so does that mean that the velocity of the frog has changed
Yasha2012-06-26 19:31:43 No, the velocity of the frog stays the same, but our guess for the velocity of the frog changes.
Yasha2012-06-26 19:32:58 Like I mentioned while we were doing part (b), the problem just asks for a strategy, which may or may not involve a formula.
Yasha2012-06-26 19:33:15 We came up with a formula in (b) because it was easy, but I won't do it for (c), though it is possible.
Yasha2012-06-26 19:33:38 In any case, we now have an infinite list of pairs (n,m).
Yasha2012-06-26 19:33:46
yingted2012-06-26 19:34:12 m_k+kn_k
nemarci2012-06-26 19:34:25 m_k+k*n_k
Watermelon8762012-06-26 19:34:25
Yasha2012-06-26 19:34:28
Yasha2012-06-26 19:34:39
Yasha2012-06-26 19:35:52 You wouldn't need an explicit formula to get full credit. However, writing an explicit formula is a good way to make sure that your solution is a complete strategy.
Yasha2012-06-26 19:36:11 Sorry, that was in response to the question:
Setalia2012-06-26 19:36:13 So, for parts (a) and (b), did we get more "credit" or were our solutions given more weight if we had an explicit formula for which spot to search and how long it would take?
spacebacon992012-06-26 19:36:55 so in the test, you write this whole explanation down?
Yasha2012-06-26 19:36:58 You'd want to write enough to have a complete argument. I've included more details and examples than is necessary for a complete solution.
viperstrike2012-06-26 19:37:15 how long will this take
Yasha2012-06-26 19:37:33 I'm aiming for two hours, but we'll see. If you get tired you don't have to stay of course.
Yasha2012-06-26 19:37:57 Or perhaps you were talking about catching the frog.
Yasha2012-06-26 19:38:13 You can work it out afterwards if you'd like.
marupiravi2012-06-26 19:38:16 yingted said m_k+kn_k what does mean (undrescore)
Yasha2012-06-26 19:38:21 The underscore means subscript.
Yasha2012-06-26 19:38:34
Yasha2012-06-26 19:38:40 We'll discuss all seven.
upleit2012-06-26 19:38:42 How many questions are we going to discuss?
Yasha2012-06-26 19:38:51 Anyways, let's move on.
Yasha2012-06-26 19:38:55 Problem 2: Coloring the number line.
Yasha2012-06-26 19:38:58 Each integer on the number line is colored with exactly one of three possible colors—red, green or blue—according to the following two rules: the negative of a red number must be colored blue, and the sum of two blue numbers (not necessarily distinct) must be colored red.
Yasha2012-06-26 19:39:03 (a) Show that the negative of a blue number must be colored red and the sum of two red numbers must be colored blue.
Yasha2012-06-26 19:39:20 Okay, so we have two rules that we know the coloring must satisfy. Let's give them names so that we can cite them in our argument.
Yasha2012-06-26 19:39:24 (I) The negative of a red number is blue. (II) The sum of two blue numbers is red.
Yasha2012-06-26 19:39:34 Let's start by showing that the negative of a blue number must be red. Where do we start?
Yasha2012-06-26 19:39:51
esque2012-06-26 19:40:16 -B_0 is red
AwesomeToad2012-06-26 19:40:16
jared4292012-06-26 19:40:16
Watermelon8762012-06-26 19:40:16 -b is red
Mathpather2012-06-26 19:40:16 -B_0 is red
Maz9062012-06-26 19:40:16 -B_0 is red
koel172012-06-26 19:40:16 -B_0 is red
spacebacon992012-06-26 19:40:16 -B_0 is red
Yasha2012-06-26 19:40:22
bobthesmartypants2012-06-26 19:40:39 by following the rules?
yingted2012-06-26 19:40:56 2b is red
Yasha2012-06-26 19:41:10
Yasha2012-06-26 19:41:14 Now what?
yingted2012-06-26 19:41:28 -2b is blue
Maz9062012-06-26 19:41:28 by (II) 2B_0 is red, by (I) -2B_0 is blue
Watermelon8762012-06-26 19:41:28 That means that -2b is blue
Yasha2012-06-26 19:41:32
Watermelon8762012-06-26 19:42:10 -b=-2b+b. We're done as those are both blue and sum to a red
esque2012-06-26 19:42:10 by rule 2, -2b+b = -b must be red and we win
Tonitruant2012-06-26 19:42:10 -2b+b=-b is red.
Yasha2012-06-26 19:42:18
Yasha2012-06-26 19:42:23
spacebacon992012-06-26 19:43:42 a red number is x. a blue number will be -x. if x is already negative, then the colors are switched...?
Yasha2012-06-26 19:43:46 Nope, so before proving this, we might have thought that 10 is blue and -10 is green. Now we know that that's impossible.
Yasha2012-06-26 19:44:11 Now let's show that the sum of two red numbers must be blue. How do we start?
Watermelon8762012-06-26 19:44:15
Yasha2012-06-26 19:44:23
Yasha2012-06-26 19:44:38
nemarci2012-06-26 19:44:56 -r_1-r_2 is red
viperstrike2012-06-26 19:45:02 thus -(n1+n2) is red and negating this sum n1+N2 IS BLUE
Maz9062012-06-26 19:45:04 then -R_1 + (-R_2) is the sum of two blue numbers, which must be red by (II)
Yasha2012-06-26 19:45:14
Yasha2012-06-26 19:45:20
spacebacon992012-06-26 19:46:07 what's the point of the green color?
Yasha2012-06-26 19:46:10 If it weren't for the green color, we could say that something that isn't red must be blue. But with the green color there, we don't know that.
Yasha2012-06-26 19:46:19 Okay, that wasn't too hard. But that part of the problem was just the warm-up!
Yasha2012-06-26 19:46:24 (b) Determine all possible colorings of the integers that satisfy these rules.
Yasha2012-06-26 19:46:41 First, let's restate our rules, now including the two new ones we've just proved.
Yasha2012-06-26 19:46:46 (I) The negative of a red number is blue, and the negative of a blue number is red. (II) The sum of two red numbers is blue, and the sum of two blue numbers is red.
Yasha2012-06-26 19:47:03 Just looking at these rules, do you notice anything important?
yingted2012-06-26 19:47:22 symmetry!
Maz9062012-06-26 19:47:22 no mention of green numbers
Yasha2012-06-26 19:47:31 Right. There is symmetry between red and blue numbers in the following sense: if you exchanged the words "red" and "blue" in the above rules, you would get the same set of rules.
Yasha2012-06-26 19:47:37 That implies that if we have any coloring that fits these rules, we can swap the red and blue numbers, and get another legal coloring! That kind of observation will definitely save us some work.
Yasha2012-06-26 19:47:52 Also, there is no mention of green in the rules.
Yasha2012-06-26 19:48:03 Can you think of any colorings that fit these rules?
viperstrike2012-06-26 19:48:20 all greens
skycao2012-06-26 19:48:20 all numbers are green
yingted2012-06-26 19:48:20 all green :(
CDerwin12012-06-26 19:48:20 everything green
Yasha2012-06-26 19:48:26 Of course. The rules don't place any restrictions on green numbers, so we could color all the integers green! Do you think that's the only possibility?
viperstrike2012-06-26 19:48:37 no
yingted2012-06-26 19:48:38 no
dantx52012-06-26 19:48:38 no
Yasha2012-06-26 19:48:43 Let's start with assuming there's at least one non-green integer, and see where we get. Suppose, say, 0 is blue.
bobthesmartypants2012-06-26 19:49:10 then that doesn't work
Tonitruant2012-06-26 19:49:10 -0 must be red?
Maz9062012-06-26 19:49:10 then it's also red, since -0 = 0 must be red
Watermelon8762012-06-26 19:49:10 then 0 is red because -0=0
nemarci2012-06-26 19:49:10 then 0 has to be red, because 0=-0
bobthesmartypants2012-06-26 19:49:10 cause then -0 = 0 must be red
Yasha2012-06-26 19:49:16 Uh-oh. 0 can't be blue, because then -0 is red.
Yasha2012-06-26 19:49:21 Obviously, 0 can't be red either, because then -0 would be blue. Therefore, 0 is green. Okay, one integer down.
Yasha2012-06-26 19:49:27 Let's start by assuming 1 is blue, then. Then 2 is red. What about 3?
Yasha2012-06-26 19:49:40 We don't have a rule for the sum of a red and blue number. Could 3 be blue?
yingted2012-06-26 19:49:57 -1+3=2
nemarci2012-06-26 19:50:01 no
Yasha2012-06-26 19:50:06 No, because then -3 is red, and -3 + 2 = -1 must be blue, which makes 1 red.
marupiravi2012-06-26 19:50:40 negative zero does not exict
Yasha2012-06-26 19:50:41 No, it's fine. -0 is equal to 0.
Yasha2012-06-26 19:50:51 Could 3 be red?
yingted2012-06-26 19:51:00 no
yingted2012-06-26 19:51:10 -1+3=2, -2+3=1
Yasha2012-06-26 19:51:18 No, because then -3 is blue, and -3 + 1 = -2 is red, which makes 2 blue. So 3 must be green.
dantx52012-06-26 19:51:35 -3 also green
melikababadi2012-06-26 19:51:39 -0 and 0 are equal therefore 0 cant be blue and red at the same time
Yasha2012-06-26 19:51:46 Exactly, that's why it has to be green.
Yasha2012-06-26 19:52:07 We can continue with this sort of logic, and we'll get a pattern:
Yasha2012-06-26 19:52:11
Yasha2012-06-26 19:52:16 What would have happened if we'd started with 1 being red?
yingted2012-06-26 19:52:25 flip
bobthesmartypants2012-06-26 19:52:32 red blue switcheroo
MA7HL0V3R2012-06-26 19:52:37 The same pattern, with blue and red interchanged.
Watermelon8762012-06-26 19:52:37 red and blue swapped, green stays the same
Yasha2012-06-26 19:52:42 Right, we'd just get the same pattern with red and blue exchanged. (You can also think of it as exchanging positive and negative numbers.)
Yasha2012-06-26 19:52:46
Yasha2012-06-26 19:52:52 Are these two the only possible solutions aside from the trivial all-green solution?
yingted2012-06-26 19:53:18 no
Watermelon8762012-06-26 19:53:18 No!!!
Yasha2012-06-26 19:53:25 Nope, we left out a case. What was it?
nemarci2012-06-26 19:53:38 1 is green
Yasha2012-06-26 19:53:48 We forgot to consider what would happen if 1 were green. If 1 is green, what do we know about 2?
bobthesmartypants2012-06-26 19:54:08 nothing?
Yasha2012-06-26 19:54:13 We don't know anything about 2, because none of our rules deal with green numbers. So 2 is free to be anything. Again, we'll assume it's blue.
Yasha2012-06-26 19:54:22 Applying the rules as before, we can easily prove that the even numbers fit the pattern we saw above:
yingted2012-06-26 19:54:26 double everything
Yasha2012-06-26 19:54:31
Yasha2012-06-26 19:54:36 But how can we fill in the odd numbers?
bobthesmartypants2012-06-26 19:54:47 all green
Tonitruant2012-06-26 19:54:47 Multiply everything by a constant, with the remaining numbers green.
yingted2012-06-26 19:54:47 green
MA7HL0V3R2012-06-26 19:54:51 They can all be green.
Yasha2012-06-26 19:55:14 We could certainly make them all green. That wouldn't cause any contradictions, since the even numbers can only affect other even numbers by rules (I) and (II).
Yasha2012-06-26 19:55:27 Are there any other ways to fill them in?
al872892012-06-26 19:55:41 no
Yasha2012-06-26 19:55:54 Probably not. If we play around with it, it doesn't seem like we can.
Yasha2012-06-26 19:55:59 Let's leave that issue aside for now, and come back to it later. So we've dealt with the case where 2 is our smallest positive non-green number. Are there any other cases?
Yasha2012-06-26 19:56:49 Sure. We could have 3 be our smallest positive non-green number, or 4, or 5, or any positive integer. And no matter which number it was, we'd get the same pattern for multiples of that number, with all three colors repeating.
Yasha2012-06-26 19:57:03 What if there's no smallest positive non-green number?
yingted2012-06-26 19:57:20 all green!
spacebacon992012-06-26 19:57:20 then all green
Tonitruant2012-06-26 19:57:20 All green.
bobthesmartypants2012-06-26 19:57:20 all greeen
Yasha2012-06-26 19:57:25 Then we're in the all-green case. So this should be all of them, if we can deal with the issue of filling in non-multiples. Let's now state our claim about the general form of a legal coloring under these rules:
Yasha2012-06-26 19:57:35
Yasha2012-06-26 19:57:40
Yasha2012-06-26 19:57:58 On our way, let's start with a lemma, which some of you mentioned already. What do you think is true of the sum of a red and blue number?
yingted2012-06-26 19:58:12 green
Watermelon8762012-06-26 19:58:13 green?
AwesomeToad2012-06-26 19:58:13 green
Yasha2012-06-26 19:58:21
Yasha2012-06-26 19:58:29
Yasha2012-06-26 19:58:40
Yasha2012-06-26 19:58:55 Therefore, R + B must be green!
bobthesmartypants2012-06-26 19:59:49 r+b=r+-r=0=green
Yasha2012-06-26 19:59:51 Not quite. You're assuming there that b=-r, but b could be any blue number.
Yasha2012-06-26 20:01:24 Unfortunately, I can't explain it that quickly. There are AoPS classes you could take to learn about it.
Yasha2012-06-26 20:01:34 (3(k + 1))n = (3k + 1)n + 2n, which is the sum of blue number and red number. So it must be green.
Yasha2012-06-26 20:01:43 (3(k + 1) + 1)n = (3k + 2)n + 2n, which is the sum of two red numbers. So it must be blue.
Yasha2012-06-26 20:01:50 Lastly, (3(k + 1) + 2)n = (3k + 1)n + 4n, which is the sum of two blue numbers. So it must be red.
Yasha2012-06-26 20:01:58 Now we've dealt with all the positive multiples of n. What about the ones that aren't multiples of n? What do we want to be true about them?
Watermelon8762012-06-26 20:02:14 They have to be green
yingted2012-06-26 20:02:18 green
bobthesmartypants2012-06-26 20:02:18 all green
Yasha2012-06-26 20:02:25 We want to show they're all green. Well, let's use proof by contradiction and suppose they're not. Let m be the least positive non-green integer that is not a multiple of n. What must be true of m?
Yasha2012-06-26 20:03:46 (Remember, we declared m to be the smallest positive non-green integer that is not a multiple of n.)
Yasha2012-06-26 20:03:55 Is there a convenient way we could write m, keeping in mind that it's not a multiple of n?
yingted2012-06-26 20:04:11 nk+d
Yasha2012-06-26 20:04:21 Let m = qn + r, where 0 < r < n is the remainder when m is divided by n. We know this remainder is nonzero, since m is not a multiple of n. We also know that q > 0, since m > n.
Yasha2012-06-26 20:04:36 Since m must be either blue or red, we should go through both cases and find a contradiction.
Yasha2012-06-26 20:05:27 OK, so let's suppose m is blue. Since m is blue, and -2n is blue, m - 2n = (q - 2)n + r is red. If q > 1, this gives us a smaller positive non-green integer that's not a multiple of n, which is a contradiction. But there's one minor problem here. What is it?
Watermelon8762012-06-26 20:05:44 q=1
Yasha2012-06-26 20:06:24 Right, we only get a contradiction if we find a _positive_ non-green integer smaller than n.
Yasha2012-06-26 20:06:36 We might fall into the negatives, if q = 1!
nemarci2012-06-26 20:06:53 if q=1, then (q-2)n+r=-n+r is red, then n-r is blue
Yasha2012-06-26 20:06:56 But if q = 1, and m - 2n = -n + r is red, that means that n - r is blue by (I), another contradiction. Whew.
Yasha2012-06-26 20:07:09 Now, we've shown that all the positive numbers which are multiples of n follow the mod 3 pattern, and all the ones which aren't multiples of n must be green. Do we need to show the same for negative numbers?
nemarci2012-06-26 20:07:42 no, because of symmetry
Watermelon8762012-06-26 20:07:42 no because each neg corresponds to a positive nongreen which is it's negative
Yasha2012-06-26 20:07:47 No! Rule (I) handles that all for us. So we're done.
Yasha2012-06-26 20:07:59 Questions before we go on?
Watermelon8762012-06-26 20:08:19 This was so cool. I never realized that R+B was green
Yasha2012-06-26 20:08:21 Yup, it's a pretty cool problem.
Yasha2012-06-26 20:08:27 Problem 3: Campers in a circle with cookies.
Yasha2012-06-26 20:08:38
Yasha2012-06-26 20:09:03 Ah, one sec, a question on the previous problem:
viperstrike2012-06-26 20:09:06 there are provable rules which apply to the green numbers which limit the possible solutions to all green and the alternating green red blue arrangements can you write a full solutions in sumary please
Yasha2012-06-26 20:09:45 So the solutions are either all green, or there is some n where multiples of n cycle blue, red, green, and all non-multiples of n are green.
Yasha2012-06-26 20:10:06 Of course, to get full credit, you need a proof, which is what we came up with earlier.
Yasha2012-06-26 20:10:30 OK, anyways, on to problem 3.
Yasha2012-06-26 20:10:46 Before we go any further, let's try some small cases so we get an idea of what's going on. The smallest case to try is p=3. What are the labels of the campers who get cookies, in order?
Watermelon8762012-06-26 20:12:00 1,2,1,1,2,1,...
nemarci2012-06-26 20:12:00 1;2;1;1;2;1;1;2;1...
xsad13002012-06-26 20:12:00 12112112112112112
AwesomeToad2012-06-26 20:12:00 1, 2, 1, 1, 2, 1, ...
Yasha2012-06-26 20:12:11 First camper 1 says 1, then camper 2, says 2. We skip over camper 3 and go back to camper 1, who says 3. Three spaces around the circle gets us back around to camper 1 again, who says 4. Continuing, we see that the sequence of campers who get cookies is 1, 2, 1, 1, 2, 1, 1, 2, 1, and so forth, repeating every three.
marupiravi2012-06-26 20:12:23 the smallest case we can try is case p=1
Yasha2012-06-26 20:12:24 Ah, nope, 1 is not considered to be a prime.
Yasha2012-06-26 20:12:32 What about if p=5? What's the sequence of camper labels then?
viperstrike2012-06-26 20:12:55 1,2,4,2,1
nemarci2012-06-26 20:12:55 1;2;4;2;1;1;2;4;2;1...
Yasha2012-06-26 20:13:01 With p=5, we can compute that the sequence of campers who get cookies is 1, 2, 4, 2, 1, 1, 2, 4, 2, 1, and so forth, repeating every five. What about if p=7?
Yasha2012-06-26 20:14:55 (a) Show that there is a camper who never gets a cookie.
Yasha2012-06-26 20:15:03 Looking at our examples, what can we do?
spacebacon992012-06-26 20:15:33 camper 3 never gets cookie
Yasha2012-06-26 20:15:38 It looks like camper 3 never gets a cookie, but it turns out that if we were to try more examples, we'd find a value for p where camper 3 does get a cookie. For example, if p=13, then the sequence of campers is 1, 2, 4, 7, 11, 3, 9, 3, 11, 7, 4, 2, 1, and then it repeats.
Watermelon8762012-06-26 20:16:11 we can show that camper 1 recieves at least two cookies. Or more generally that it's symmetric
yingted2012-06-26 20:16:11 show one camper always gets more than one
Yasha2012-06-26 20:16:18 Based on our examples, it looks like the sequence of campers getting cookies repeats every p cookies, and it looks like camper 1 gets the first and last cookie in each repeating block. Since there are p cookies given out each block and there are p campers, if camper 1 gets two cookies in a block, then that means some other camper has to have gotten no cookies.
nemarci2012-06-26 20:16:28 in every period, 1 gets the first and last cookie
yingted2012-06-26 20:16:32 since camper 1 always gets at least 2, some camper can't get any :(
vlchen8882012-06-26 20:16:53 how can u prove the pattern though?
Yasha2012-06-26 20:16:58 Don't worry, we'll get there.
upleit2012-06-26 20:17:23 So, it circulates, and everybody ends up getting a cookie, right?
Yasha2012-06-26 20:17:26 Nope, it looks like not everyone gets a cookie, and some people get lots of them.
Yasha2012-06-26 20:17:36 Let's see if we can prove our hunches. First, can anyone find a way to determine the label of the camper who says the number n, in terms of n?
nemarci2012-06-26 20:18:09 1+n(n-1)/2
Maz9062012-06-26 20:18:23 a_n = 1 + T_(n-1), where T_n is the nth triangular number, and a_n is the label of the camper that shouts n
MA7HL0V3R2012-06-26 20:18:27 Camper = n(n-1)/2+1 (mod p)
viperstrike2012-06-26 20:18:33 n(n-1)/2 +1
Mr.1172012-06-26 20:18:33 1+(n-1)(n-2)/2
viperstrike2012-06-26 20:18:33 mode p
Yasha2012-06-26 20:18:36
Yasha2012-06-26 20:18:47 We'd like to show that the sequence of campers who speak is periodic with period p. That is, if a camper says n, then that camper will also say n+p.
Yasha2012-06-26 20:19:07 How can we show that this statement is true?
bobthesmartypants2012-06-26 20:19:26 mod
Yasha2012-06-26 20:19:35 We could use some facts from modular arithmetic.
nemarci2012-06-26 20:19:38 p(p-1)/2 is evenly divisible by p, because p-1 is even
Yasha2012-06-26 20:19:48 We could also do it directly.
Yasha2012-06-26 20:19:55
Yasha2012-06-26 20:20:04
Yasha2012-06-26 20:20:16
Yasha2012-06-26 20:20:29 We conclude that the camper who says n also says n+p, so the sequence of campers who speak is periodic with period p.
Yasha2012-06-26 20:20:37 Now, all we have left to show is that camper 1 gets two cookies in each repeating block, and from there we'll be able to conclude that some other camper must get no cookies.
Yasha2012-06-26 20:20:51 How can we show that camper 1 gets two cookies in each round of p cookies?
Yasha2012-06-26 20:22:06 Yeah, it's easy to see for p=3, but we need to prove it for all values of p.
yingted2012-06-26 20:22:34 he also gets the pth
yingted2012-06-26 20:22:34 /she
vlchen8882012-06-26 20:22:34 plug p into the equation
yingted2012-06-26 20:22:34 if someone gets the pth cookie, they also get the (p+1)th cookie
Yasha2012-06-26 20:23:06 So, we can be clever and notice that whoever says p also says the next number, since going p spaces around the circle gets you back to where you started.
Yasha2012-06-26 20:23:20 We proved periodicity, so we know that camper 1 says p+1, so therefore camper 1 must say p.
Yasha2012-06-26 20:23:44 However, even without a clever argument, you can still just plug p into the formula.
Yasha2012-06-26 20:23:55
upleit2012-06-26 20:24:17 Can other campers stil get more than 1 cookie?
Yasha2012-06-26 20:24:28 Yes, in fact it seems like they do, from our examples.
bobthesmartypants2012-06-26 20:24:43 wouldn't (p-1)/2 people get at least 2 cookies?
Yasha2012-06-26 20:24:51 It sure looks that way, but we haven't proved it yet.
Yasha2012-06-26 20:25:03 We showed that camper 1 gets two cookies in the first p turns. In the first p turns, there are p cookies to be distributed among p campers. If camper 1 gets two cookies, that means that some other camper must have gotten no cookies in the first p turns.
Yasha2012-06-26 20:25:13 But we showed that the campers who speak are periodic with period p. Therefore, if a camper doesn't get a cookie in the first p turns, that camper will never get a cookie.
Yasha2012-06-26 20:25:16 :(
marupiravi2012-06-26 20:25:44 we can prove it with the example in the begginging
marupiravi2012-06-26 20:25:44 of p=13
Yasha2012-06-26 20:26:00 Again, that will just prove it for p=13, but we wouldn't know if it is true for p=17.
Yasha2012-06-26 20:26:53 We were trying to show that some camper gets no cookies.
Yasha2012-06-26 20:27:05 We're done with that now, so we're about to move to the next part.
upleit2012-06-26 20:27:21 Is it possible to get an example that takes care of all cases?
Yasha2012-06-26 20:27:23 Sometimes that's possible, but not in this problem.
jared4292012-06-26 20:27:45 is there anything special about it being prime, or would it work if p were just some odd number?
Yasha2012-06-26 20:27:47 Yes, there is something special about p being prime, though I don't think we've used it yet.
Yasha2012-06-26 20:28:16 (b) Of the campers who do get cookies, is there one who at some point has at least ten more cookies than the others?
Yasha2012-06-26 20:28:31 Based on our examples, what do we think the answer is?
Watermelon8762012-06-26 20:28:55 No except if p=3
bobthesmartypants2012-06-26 20:28:56 no
nemarci2012-06-26 20:28:56 no
Yasha2012-06-26 20:29:01 It looks like, in each block of p cookies, some campers get no cookies, some campers get two cookies, and one camper gets one cookie. For p>3, it looks like there are at least two campers who get two cookies each block, so none of them can get far ahead of all of the others. The answer seems to be no.
Yasha2012-06-26 20:29:35 So just to be clear about the question: It is true that some campers get ahead of some other campers, but no camper gets ahead of all of the other campers.
Yasha2012-06-26 20:29:53 When p=3, the sequence of campers who get cookies is 1, 2, 1, 1, 2, 1, and so forth. Camper 1 gets two cookies each block, camper 2 gets one cookie each block, and camper 3 gets no cookies. Therefore, after thirty turns, camper 1 will have twenty cookies, and camper 2 will have ten cookies. For p=3, the answer is yes.
Yasha2012-06-26 20:30:17 OK, so now we still have to prove our theory that some campers get two cookies, some campers get no cookies, and one camper gets one cookie.
upleit2012-06-26 20:30:43 So, it's only if p is smaller than 3?
Yasha2012-06-26 20:31:08 There's only one odd prime less than or equal to 3, namely 3 itself. For p=3, the answer is yes. Otherwise, it is no.
Yasha2012-06-26 20:31:44 Let's look at one of our examples again, with p=13: The sequence of campers who speak is 1, 2, 4, 7, 11, 3, 9, 3, 11, 7, 4, 2, 1, and then it repeats.
Yasha2012-06-26 20:31:53 Lots of people have noticed that there is symmetry.
vlchen8882012-06-26 20:31:57 the middle guy gets 1 cookie
Yasha2012-06-26 20:32:19 Also, you can notice that the person who says the "middle" number is the one who only gets one cookie per block of p cookies.
marupiravi2012-06-26 20:32:46 once angaing can you prove it with other numbers
Yasha2012-06-26 20:33:02 We'll get there. We're looking at an example to get an idea for how to prove it, and then we'll prove it for all odd primes.
Yasha2012-06-26 20:33:49 Anyways, the sequence seems to be a palindrome, which means that it is the same when read forwards and backwards. Can we prove that the sequence is always a palindrome, no matter what p is?
Yasha2012-06-26 20:34:46 To show that the sequence is a palindrome, we have to show that the camper who says n also says p+1-n. (Check a few values of n in our example to convince yourself that p+1-n is the right formula for the "mirror image" of n in our palindrome.)
Yasha2012-06-26 20:35:05 To do this, we can use our formula.
Yasha2012-06-26 20:35:18
Yasha2012-06-26 20:35:25
Yasha2012-06-26 20:35:40
Yasha2012-06-26 20:35:50 OK, so we know that the sequence of campers who say the first p numbers is a palindrome. Therefore, a camper who speaks in the first block of p numbers will speak a second time in that block, except for the camper who speaks in the exact middle of the block. (In our example with p=13, that camper was labeled 9.)
Yasha2012-06-26 20:36:01 Are we done? Have we successfully shown that some campers speak exactly twice, and one camper speaks exactly once?
Mr.1172012-06-26 20:36:14 have we proved the numbers in the palindrome only appear once/twice and not more?
Yasha2012-06-26 20:36:41 We haven't ruled out the case where a camper speaks three times, or four times. What if the sequence instead looked like 1, 2, 4, 9, 11, 3, 9, 3, 11, 9, 4, 2, 1? Or 1, 2, 3, 7, 11, 3, 9, 3, 11, 7, 3, 2, 1? It's still a palindrome, but there is one camper that speaks more than twice and will eventually have ten more cookies than anyone else. We need to rule out this possibility.
yingted2012-06-26 20:36:58 consider 15
Yasha2012-06-26 20:37:20 Apparently, this happens if you try p=15, according to yingted.
nemarci2012-06-26 20:37:28 15 is not prime
bobthesmartypants2012-06-26 20:37:28 not a prime
Yasha2012-06-26 20:37:30 Fortunately for us, 15 is not a prime.
Yasha2012-06-26 20:37:47 This answers the earlier question of whether or not it matters that p is prime.
Yasha2012-06-26 20:37:56
Yasha2012-06-26 20:38:05
Yasha2012-06-26 20:38:12 A value of n satisfies this equation if and only camper c says the number n.
Yasha2012-06-26 20:38:16 What does this equation look like?
Watermelon8762012-06-26 20:38:28 A quadratic equation!
Yasha2012-06-26 20:38:33 It looks like a quadratic equation in n. What do we know about quadratic equations, at least when dealing with real numbers?
yingted2012-06-26 20:39:05 <=2 solutions
MA7HL0V3R2012-06-26 20:39:05 They have at most two roots.
nemarci2012-06-26 20:39:05 they don't have more than 2 solutions
Yasha2012-06-26 20:39:11 Quadratic equations over the real numbers have at most two solutions. This fact is also true over the integers mod p, if p is a prime. (In fact, quadratic equations have at most two solutions over any field, but don't worry if you haven't heard of fields before.)
Yasha2012-06-26 20:39:42 There's also a way to do this problem without using this fact, but I'm going to use it to save time.
Yasha2012-06-26 20:39:57
bobthesmartypants2012-06-26 20:40:21 there can be 2 solutions at most
bobthesmartypants2012-06-26 20:40:24 not over 2
Yasha2012-06-26 20:40:31 The camper c says the number n if and only if that equation is satisfied. Therefore, in the first p turns, camper c will speak at most twice.
Yasha2012-06-26 20:40:36 We showed that the sequence of campers who say the first p numbers is a palindrome. How can we use the fact that no camper speaks more than twice to finish the problem?
Yasha2012-06-26 20:41:20
Yasha2012-06-26 20:41:26 Since no camper can speak more than twice in a block, the campers who get the most cookies are the ones who do speak twice.
nemarci2012-06-26 20:41:37 if p>=5, then there are more than 1 campers who get 2 cookies, but no one gets more
Yasha2012-06-26 20:41:48 Exactly. For p>3, there are at least two numbers in the first half of the palindrome, and the campers who say those numbers have to different. (If a camper were to say two numbers in the first half of the palindrome, they would have to also say two numbers in the second half of the palindrome, for a total of four times.)
Tonitruant2012-06-26 20:41:55 And there are (p-1)/2 of them.
Yasha2012-06-26 20:42:09 Yes.
bobthesmartypants2012-06-26 20:42:16 and there are more than one 2-cookie camper in p<=5
Yasha2012-06-26 20:42:23 Ah, nope, not for p=3, as we've seen.
bobthesmartypants2012-06-26 20:42:39 sorry, typo
bobthesmartypants2012-06-26 20:42:44 p>=5
Yasha2012-06-26 20:42:47 ah ok.
Yasha2012-06-26 20:43:00 There are at least two campers who speak twice, and so they can't get far ahead of all the others in terms of number of cookies.
Yasha2012-06-26 20:43:26 Any questions on this part?
Yasha2012-06-26 20:43:36 (c) Of the campers who do get cookies, is there one who at some point has at least ten fewer cookies than the others?
Yasha2012-06-26 20:43:45 Based on our work, what is the answer?
bobthesmartypants2012-06-26 20:43:59 yes
Watermelon8762012-06-26 20:43:59 yes
nemarci2012-06-26 20:43:59 yes
bobthesmartypants2012-06-26 20:43:59 the one who gets one cookie
yingted2012-06-26 20:43:59 yes
Yasha2012-06-26 20:44:04
Yasha2012-06-26 20:44:12 All of the other campers who speak get two cookies in the first block of p numbers, since a camper who says a number in the first half of the palindrome also says a number in the second half, and vice versa.
Yasha2012-06-26 20:44:21 Therefore, after ten blocks of p numbers, the camper who said (p+1)/2 will have ten cookies, whereas all of the other campers who got cookies will have twenty of them.
Watermelon8762012-06-26 20:45:36 In the interest of saving time, maybe we can take a poll and find out which problems/sections people want to see. I only need to see 4 and 6 (maybe 5f). Anything else is just extra. Similarly I'm sure some people only need to see 5de or something...
Yasha2012-06-26 20:45:38 Yes, as you can tell, we're going to run a bit over, which is fine. However, since some people might need to leave, I'd be up for switching up the order. How about we take a vote on which problem to see next. If there's a clear winner, I'll go with that, and if not I'll just move on to 4.
bobthesmartypants2012-06-26 20:46:32 what is this for?
bobthesmartypants2012-06-26 20:46:32 I just randomly joined
Yasha2012-06-26 20:46:44 Ah, these are the Mathcamp qualifying quiz problems.
Yasha2012-06-26 20:47:28 There's no clear consensus, so I'm going to move on to 4.
Yasha2012-06-26 20:47:41 There will be a transcript at the end, so if you have to leave you can look back at it tomorrow.
Yasha2012-06-26 20:47:50 Problem 4: Lollipops.
Yasha2012-06-26 20:47:56 Let a be a rational number with 0 < a < 1. A lollipop in the xy-plane with base (a,0) consists of a line segment from (a,0) to some point (a,b) with b > 0, together with a filled in disc of radius less than b, centered at (a,b). Determine whether or not it is possible to have a set of lollipops in the xy-plane satisfying both of the following conditions: * for every rational number a with 0 < a < 1, there is a lollipop whose base is the point (a,0), * no two lollipops touch or overlap each other. If such a set of lollipops exists, explain how to construct it. If not, justify why not.
Yasha2012-06-26 20:48:23 First, are there any questions about what a lollipop is?
Watermelon8762012-06-26 20:49:10 You could just troll and say that the radius is 0 I guess
Yasha2012-06-26 20:49:13 True, so we assume that the radius has to be positive.
Yasha2012-06-26 20:49:21 So, our task to construct a non-overlapping set of lollipops with bases at every rational number between 0 and 1. Or prove that it's impossible. How many lollipops is that?
jared4292012-06-26 20:49:42 infinite
robinpark2012-06-26 20:49:43 infinite
Watermelon8762012-06-26 20:49:43 aleph-null
bobthesmartypants2012-06-26 20:49:43 rational number? there are infinite of those
MA7HL0V3R2012-06-26 20:49:46 Infinite number.
Global2012-06-26 20:49:49 infinitely many
Yasha2012-06-26 20:49:54 Yes, there are infinitely many. (In fact, the rationals between 0 and 1 form a countably infinite set.) Hmm, that's going to make things tough. We need to find some way to deal with this huge set of numbers.
Yasha2012-06-26 20:50:02
Yasha2012-06-26 20:50:06 It would be really convenient if we knew where to start. Is there a natural order to consider these numbers in? (That is, does S have a least element?)
bobthesmartypants2012-06-26 20:50:32 0
Yasha2012-06-26 20:50:33 Nope, it says 0<r.
Maz9062012-06-26 20:50:40 ...no?
robinpark2012-06-26 20:50:40 S doesn't have a least element.
Yasha2012-06-26 20:50:51 Unfortunately, the standard order of numbers is not going to be helpful. What's the first positive rational number?
Mr.1172012-06-26 20:51:05 infinitely small...
Yasha2012-06-26 20:51:09
Yasha2012-06-26 20:51:20 Is there a better way to put S in order?
yingted2012-06-26 20:51:37 by denominato
yingted2012-06-26 20:51:37 you need to minimize the denominator, then the numerator
nemarci2012-06-26 20:51:37 yes, we can put the elements of S into an order, similarly to 1c
Tonitruant2012-06-26 20:51:37 Use the spiral method from the frog problem.
yingted2012-06-26 20:51:39 by denominator
Yasha2012-06-26 20:51:44 Right, we can sort them by denominator. First 1/2, then 1/3 and 2/3, then 1/4 and 3/4, then 1/5, 2/5, 3/5, and 4/5...
Yasha2012-06-26 20:51:54 To make this more precise, each element of S can be written in lowest terms as p/q, where p and q have no common factors.
Yasha2012-06-26 20:51:58
Yasha2012-06-26 20:52:02
Yasha2012-06-26 20:52:07
Yasha2012-06-26 20:52:11
Yasha2012-06-26 20:52:15
Yasha2012-06-26 20:52:19 And so on.
Yasha2012-06-26 20:52:22
Yasha2012-06-26 20:52:28
Yasha2012-06-26 20:52:35 Can our set of lollipops all be the same height?
yingted2012-06-26 20:52:45 no
Watermelon8762012-06-26 20:52:45 no
Global2012-06-26 20:52:49 no
Mr.1172012-06-26 20:52:49 no
bobthesmartypants2012-06-26 20:52:49 no
Yasha2012-06-26 20:52:56 No, that definitely won't work. If they're all the same height, then the discs will certainly overlap.
Yasha2012-06-26 20:53:05 So what should we do with the heights as we go along? We could have them increase or decrease. (Or something more complicated, of course, but let's start with those.)
Yasha2012-06-26 20:54:01 Increasing heights won't help us. Once we have a lollipop of radius c > 0 based at 1/2, then any taller lollipop based at a number within c of 1/2 is going to overlap.
Watermelon8762012-06-26 20:54:08 If we decrease, we don't have a chance of the stalk of the lollipop colliding with a disc
Yasha2012-06-26 20:54:21 So we want the lollipops to be shorter and shorter as the denominator increases.
Yasha2012-06-26 20:54:26
yingted2012-06-26 20:54:50 make the top below the bottom
Yasha2012-06-26 20:54:55
bobthesmartypants2012-06-26 20:55:10 make them just below touching
Yasha2012-06-26 20:55:13
Yasha2012-06-26 20:55:24
Yasha2012-06-26 20:55:42 yingted has provided us with a pretty awesome illustration:
yingted2012-06-26 20:55:44
Yasha2012-06-26 20:56:19 Again, let's make this precise. What a good way to divide the positive y-axis into infinitely many disjoint intervals that go to zero?
yingted2012-06-26 20:57:11 1/denominator
Watermelon8762012-06-26 20:57:11 geometric series?
Mr.1172012-06-26 20:57:11 by denominators.
nemarci2012-06-26 20:57:11 1/2; 1/4; 1/8; 1/16 ...
Yasha2012-06-26 20:57:17
Yasha2012-06-26 20:57:21
Yasha2012-06-26 20:57:25
Yasha2012-06-26 20:57:30
Yasha2012-06-26 20:57:33 And so on.
Yasha2012-06-26 20:57:41
Yasha2012-06-26 20:57:48
Yasha2012-06-26 20:58:03 For example, the lollipop on 1/2 is centered at (1/2, 3/4) and has radius 1/4. So the y-values of the disc are all between 1/2 and 1.
Yasha2012-06-26 20:58:07 The lollipops on 1/3 and 2/3 are centered at (1/3, 5/12) and (2/3, 5/12) and have radius 1/12. So the y-values of the discs are all between 1/3 and 1/2.
Yasha2012-06-26 20:58:12 Now we're guaranteed that lollipops in different layers won't have their discs overlap.
Yasha2012-06-26 20:58:16 What about lollipops within the same layer?
marupiravi2012-06-26 20:58:36 they will not touch
Yasha2012-06-26 20:58:46
bobthesmartypants2012-06-26 20:58:50 their distance is greater then their combined diameters
yingted2012-06-26 20:58:54 they are too small to touch
Watermelon8762012-06-26 20:58:54 too far apart. The closest they could be is within 1/(q-1) of each other and the radius is less
Yasha2012-06-26 20:59:05 (Unless, of course, q = 2, but in that case there's only one lollipop in the layer!)
Yasha2012-06-26 20:59:11 So lollipops in the same layer can't have overlapping discs either.
Yasha2012-06-26 20:59:16 Are we done?
Yasha2012-06-26 20:59:53 No! There's one other issue to deal with: the possible overlap between discs and the vertical line segments. We haven't dealt with that possibility at all.
nemarci2012-06-26 21:00:02 no, because circles can cross line segments parallel with y-axis
Yasha2012-06-26 21:00:06 Suppose we have a lollipop whose disc intersects some vertical segment belonging to a second lollipop. Which one is taller, the first or the second lollipop?
Watermelon8762012-06-26 21:00:21 the second
MA7HL0V3R2012-06-26 21:00:26 second
bobthesmartypants2012-06-26 21:00:33 2nd
Yasha2012-06-26 21:00:35
yingted2012-06-26 21:00:58 anything q'<q
Yasha2012-06-26 21:01:01
Yasha2012-06-26 21:01:06 Now, since there are only finitely many lollipops taller than any given lollipop, we could simply shrink each radius so that it doesn't touch any of their stems.
Yasha2012-06-26 21:01:13
Yasha2012-06-26 21:01:18 Let's prove it.
Yasha2012-06-26 21:01:24
Yasha2012-06-26 21:01:31 How can we find out whether this happens?
yingted2012-06-26 21:02:31 subtraction
Yasha2012-06-26 21:02:36
bobthesmartypants2012-06-26 21:03:12 They do not overlap!
Yasha2012-06-26 21:03:15
Yasha2012-06-26 21:03:23 The first inequality is true because the distance can't be zero, so the numerator has absolute value at least one. And the following one is true because n < q, so n is at most q - 1.
Yasha2012-06-26 21:03:33
Yasha2012-06-26 21:03:41 We proved earlier that there were no overlaps between discs and discs, and obviously two stems will never overlap, so our construction works and we're done!
Yasha2012-06-26 21:03:43 Yay!
Yasha2012-06-26 21:03:52 Any questions on this problem?
bobthesmartypants2012-06-26 21:04:27 I have a question: What grade is this? I'm in 6th
Yasha2012-06-26 21:04:35 Mathcamp is a summer program for people 13 and above.
Watermelon8762012-06-26 21:04:58 I propose we move on to 6 and skip 5
Yasha2012-06-26 21:05:14 Some people wanted to see 5 though, so I think we'll go in order.
Technik2012-06-26 21:05:18 What is this MathJam?
Yasha2012-06-26 21:05:25 This is the Mathcamp qualifying quiz Mathjam.
Yasha2012-06-26 21:05:33 Problem 5: P-distances.
Yasha2012-06-26 21:05:38 A convex body in the plane is a region with positive area such that for any two points in this region, the entire line segment between them also lies within the region. Let P be the perimeter (i.e., boundary) of a convex body in the plane. We will assume throughout this problem that P is centrally symmetric: that is, if (a,b) is a point on P, then so is (-a,-b).
Yasha2012-06-26 21:05:42 For any nonnegative real number k, we define kP to be the subset of the plane obtained by multiplying all the points of P by k in each coordinate. In other words, for each point (a,b) of P, the point (ka,kb) is in kP.
Yasha2012-06-26 21:05:46 If (x_1,y_1), (x_2,y_2) are two points in the plane, we define the P-distance between them to be the smallest nonnegative real number k such that when the set kP is translated by (x_1,y_1) (that is, by x_1 units horizontally and by y_1 units vertically), the point (x_2,y_2) lies on it. For example, suppose that P is the square with vertices (0,1), (1,0), (0,-1), (-1,0); then the P-distance between (3,5) and (4,10) is 6.
Yasha2012-06-26 21:06:24 Before we start trying to prove part (a), let's make some observations about the notion of P-distance.
Yasha2012-06-26 21:06:30
yingted2012-06-26 21:06:53 yes
yingted2012-06-26 21:06:57 f(x_1-x_2,y_1-y_2)
Watermelon8762012-06-26 21:07:03 P-distance is in terms of a shape
nemarci2012-06-26 21:07:06 the P-distance of (0;0) and (x_2-x_1;y_2-y_1)
Yasha2012-06-26 21:07:12
Yasha2012-06-26 21:07:20 So right off the bat, we can just think about P-distances from the origin. That means no translations are necessary; the P-distance between (0,0) and (x,y) is just the smallest factor of k such that kP contains (x,y).
Yasha2012-06-26 21:07:30 How can we find this k?
yingted2012-06-26 21:07:55 magnitude divided by projection
Global2012-06-26 21:07:55 write P as a polar curve r=r(\theta)
Yasha2012-06-26 21:08:12 We could write P as a polar curve, but you can do it without that.
Yasha2012-06-26 21:08:17 We can draw a ray from the origin to (x,y), and see where that ray intersects P.
Yasha2012-06-26 21:08:21 Suppose the ray from the origin to (x,y) intersects P in a point p. Then what is the P-distance between the origin and (x,y)?
yingted2012-06-26 21:08:51 or for an intuitive approximation, draw several shapes for each integer k
Global2012-06-26 21:08:55 yes; describe P as a polar curve, and use the radius
Global2012-06-26 21:08:55 then, if tan \theta = y/x, the P-distance between origin and (x,y) is \sqrt(x^2+y^2)/r(\theta)
Yasha2012-06-26 21:09:22 There's one subtlety here: how do we know that the ray intersects the perimeter P in only one point?
yingted2012-06-26 21:09:37 convex
MA7HL0V3R2012-06-26 21:09:41 P is convex.
Yasha2012-06-26 21:09:46 We have to use convexity. It's not hard, so I'll leave the details to you.
Yasha2012-06-26 21:09:50 Okay, now let's tackle part (a).
Yasha2012-06-26 21:09:58 (a) Let P be the perimeter of a disc of radius 1 centered at the origin. Find a formula for the P-distance between any two points (a,b) and (c,d) in the plane.
Yasha2012-06-26 21:10:15 By our previous work, we just need the P-distance between the origin and (a - c, b - d).
Yasha2012-06-26 21:10:19 What is that distance?
nemarci2012-06-26 21:10:38 it equals the normal distance
Watermelon8762012-06-26 21:10:38 d(a - c, b - d)
yingted2012-06-26 21:10:38 euclidean
Yasha2012-06-26 21:10:42 It doesn't matter where the ray intersects the disc, since for all points on the perimeter of the disc, d((0,0), p) = 1. That's the definition of a disc!
Yasha2012-06-26 21:10:49
Yasha2012-06-26 21:10:59 (b) Let P be the perimeter of a rhombus with vertices (2,0), (-2,0), (0,3), (0,-3). Find a formula for the P-distance between any two points (a,b) and (c,d) in the plane.
Yasha2012-06-26 21:11:09 Can we save ourselves some trouble here by narrowing our cases?
nemarci2012-06-26 21:11:36 study just one quarter of the plane
Watermelon8762012-06-26 21:11:42 Yes, consider only first quadrant for now
Yasha2012-06-26 21:11:46 By symmetry of the rhombus, we can get most of our information from the case where (a - c, b - d) is in the first quadrant. What's the equation for the line which is the portion of P in the first quadrant?
bobthesmartypants2012-06-26 21:12:11 assuming that the center of the rhombus is at 0,0?
Yasha2012-06-26 21:12:25 Yup, when we compute the distance from the origin, we don' tneed to translate.
MA7HL0V3R2012-06-26 21:12:42 y=-3/2x+3
nemarci2012-06-26 21:12:42 3x+2y=6
Yasha2012-06-26 21:12:47 It's just 3x + 2y = 6.
Yasha2012-06-26 21:12:51 When we scale this by a factor of k, what do we get?
nemarci2012-06-26 21:13:12 3x+2y=6k
Yasha2012-06-26 21:13:18 Right, 3x + 2y = 6k.
Yasha2012-06-26 21:13:31 And if we want (a - c, b - d) to lie on this line, what should we do?
Maz9062012-06-26 21:14:07 substitute that in and solve for k
bobthesmartypants2012-06-26 21:14:07 replace x and y with the variables?
Global2012-06-26 21:14:07 3(a-c)+2(b-d)=6k
adi122012-06-26 21:14:21 3(a-c)+2(b-d)=6k
Yasha2012-06-26 21:14:24 We just plug them in to get 3(a - c) + 2(b - d) = 6k.
Yasha2012-06-26 21:14:29
Yasha2012-06-26 21:14:34 What if a - c is negative?
nemarci2012-06-26 21:15:00 then use c-a
Global2012-06-26 21:15:00 use symmetry of rhombus for other cases
MA7HL0V3R2012-06-26 21:15:00 Use absolute values?
Watermelon8762012-06-26 21:15:00 turns out if we drop absolute value bars around both terms, stuff works out
Yasha2012-06-26 21:15:11 Then the x-coefficient in the equation for the line would also be negative, so it would still have a positive contribution to the distance.
yingted2012-06-26 21:15:15 abs
Yasha2012-06-26 21:15:22
Yasha2012-06-26 21:15:29 (c) In part (a), we took it for granted that a filled-in disc of radius 1 is a convex body. Prove this rigorously, using the definition of convexity given above.
Yasha2012-06-26 21:15:58 So we want to prove that, for any two points p and q in a disc of radius 1, every point r on the line segment connecting p to q is still within the disc.
Yasha2012-06-26 21:16:05 Can we make any sort of assumptions about p and q to cut down the possibilities?
bobthesmartypants2012-06-26 21:16:39 the circle doesn't have any part where it caves in
Yasha2012-06-26 21:16:40 Yup, but we want to prove it rigorously.
Yasha2012-06-26 21:16:58 If we extend the the line pq, it'll hit the boundary circle on both sides. So without loss of generality, it suffices to prove that r is in the disc whenever p and q are on the boundary circle.
Yasha2012-06-26 21:17:11 Can we narrow things down even more? Is there a particular place we could put p without losing any generality?
Maz9062012-06-26 21:17:44 (1,0)?
Tonitruant2012-06-26 21:17:44 (0,1)?
Yasha2012-06-26 21:17:49 Sure! The disc has rotational symmetry, so we can rotate everything until p = (0,1).
nemarci2012-06-26 21:17:53 draw a segment between two points of the circle, and search the furthest point of the origin on the segment. It's going to be at the end of the segment
Yasha2012-06-26 21:17:59 Yeah, we're going to do something like that.
Yasha2012-06-26 21:18:06
yingted2012-06-26 21:18:29 r=zp+(1-z)q
Yasha2012-06-26 21:18:35
Global2012-06-26 21:18:46 (0(1-t)+xt, 1(1-t)+yt), 0<=t<=1
Yasha2012-06-26 21:18:54 How can we tell if such a point is within the disc of radius 1 centered at the origin?
CDerwin12012-06-26 21:19:35 distance from origin
Maz9062012-06-26 21:19:35 if the distance between r and the origin is less than 1
Global2012-06-26 21:19:35 show that (1-t+tx)^2 + (ty)^2 <=1
Yasha2012-06-26 21:19:47 We can just take the distance squared using coordinates.
Yasha2012-06-26 21:19:51
Yasha2012-06-26 21:19:57
Yasha2012-06-26 21:20:08 And we're done.
Yasha2012-06-26 21:20:17 (d) Suppose P is a convex quadrilateral. What are the possible P-distances between vertices of P? What about when P is a convex hexagon? (Remember: P must still be centrally symmetric!)
Yasha2012-06-26 21:20:36 Is there anything special about a centrally symmetric convex quadrilateral?
bobthesmartypants2012-06-26 21:20:54 parallellogram
AwesomeToad2012-06-26 21:20:54 it's a parallelogram
yingted2012-06-26 21:20:58 parallelogram
bobthesmartypants2012-06-26 21:20:58 paralellogram*
Yasha2012-06-26 21:21:06 It must be a parallelogram! Given any side, multiplying it by -1 takes it to another side by central symmetry, and that new side is parallel to the original.
Yasha2012-06-26 21:21:11 That should make things easier. Let's suppose our parallelogram has two adjacent corners p and q, so that its other corners are -p and -q.
Yasha2012-06-26 21:21:17 What is the P-distance between any corner and its opposite?
Watermelon8762012-06-26 21:21:26 2
yingted2012-06-26 21:21:29 2
Yasha2012-06-26 21:21:34 The p-distance between p and -p or q and -q is always 2, since the length of the diagonal is twice the distance from the origin to the corner.
Yasha2012-06-26 21:21:40 What is the P-distance between p and q?
Watermelon8762012-06-26 21:21:53 It turns out to be 2!
yingted2012-06-26 21:22:02 2
Yasha2012-06-26 21:22:04 This time, p - q corresponds to the length of the side connecting them, which is twice the length of the segment parallel to that side connecting the center and boundary. So the distance is 2 again.
Yasha2012-06-26 21:22:11 And of course the P-distance between any vertex and itself is 0. So the possibilities are 0 and 2.
iamawesome1232012-06-26 21:22:18 2
Yasha2012-06-26 21:22:19 What happens in the case of a hexagon?
yingted2012-06-26 21:22:48 2
Watermelon8762012-06-26 21:22:48 This logic doesn't work as well
bobthesmartypants2012-06-26 21:22:48 1?
Yasha2012-06-26 21:22:55 Here's one example of how to do it: take the hexagon with vertices at (0,2), (1,d), (1,-d), (0,-2), (-1,-d), and (-1,d), where 0 < d < 2.
Yasha2012-06-26 21:23:06 It's easy to show that the P-distance from (1,d) to (1, -d) is d. So we can get anything between 0 and 2 using adjacent vertices.
Yasha2012-06-26 21:23:13 But we can also get 2 using opposite vertices, as before, and 0 using the distance from a vertex to itself.
Yasha2012-06-26 21:23:20 However, we haven't shown that it's impossible to get anything higher than 2. Maybe there's some weird hexagon where the P-distance between adjacent vertices is 3. We'll need to use the proof of the next part to show that [0,2] is indeed all that's possible.
Yasha2012-06-26 21:23:37 (e) Let P be the perimeter of some centrally symmetric convex body, and let (a,b) be a point on P. What is the largest possible P-distance from (a,b) to another point on P? Will (a,b) be at this P-distance from just one other point on P or from multiple other points? (If any of your answers depend on the geometry of P and/or on the choice of (a,b), explain how.)
Yasha2012-06-26 21:23:58 Given what we've tried so far, we might suspect the answer is 2.
Yasha2012-06-26 21:24:04 Is there definitely another point q on P such that the P-distance between p = (a,b) and q is 2?
Watermelon8762012-06-26 21:24:59 It can be as large as we want it do if we construct a hexagon with vertices (-1,1) (0,2a) (a,a) and the reflections
Yasha2012-06-26 21:25:01 That might not be convex for all a.
nemarci2012-06-26 21:25:12 mirrored to the center of P
yingted2012-06-26 21:25:12 there is one, but not guaranteed to be many
Watermelon8762012-06-26 21:25:12 yes the opposite vvertex
yingted2012-06-26 21:25:21 antipodal point?
Yasha2012-06-26 21:25:26 Sure. We'll take q = (-a,-b), which must be on P by central symmetry. That gives us a P-distance of 2 between p and q.
Yasha2012-06-26 21:25:32 Now, we want to prove that no greater distance is possible. Let's take an arbitrary p and q. We want to show the P-distance is less than or equal to 2. That means that the ray in the direction p - q intersects P at a point whose distance from the origin is at least 1/2 the length of p - q.
Yasha2012-06-26 21:25:46 In other words, we want to show that the point 1/2 (p - q) is on or inside P. How can we do that?
yingted2012-06-26 21:26:38 go to origin and back
Yasha2012-06-26 21:26:41 You might be assuming the triangle inequality, which we haven't actually proved for P-distances!
Yasha2012-06-26 21:27:10 But we can show that 1/2(p-q) is on or inside P from more basic principles.
Yasha2012-06-26 21:27:23 Ideas?
Yasha2012-06-26 21:27:46 We'll use symmetry first. Since q is on P, -q is on P. Then what?
Yasha2012-06-26 21:28:34 So, p is on P, and -q is on P. What can we say about (p-q)/2?
Yasha2012-06-26 21:29:05 Geometrically, how does the point (p-q)/2 relate to p and -q?
yingted2012-06-26 21:29:17 midpoint!
yingted2012-06-26 21:29:22 which is insode
Yasha2012-06-26 21:29:27 1/2 (p - q) is the midpoint of the segment connecting p and -q. So 1/2 (p - q) is on or inside P, by convexity. Therefore, the ray in the direction p - q intersects P at a point whose distance from the origin is at least as far as 1/2 (p - q), which means the P-distance is at most 2.
Yasha2012-06-26 21:29:49 Now let's answer the other parts: will (a,b) be P-distance 2 from just one other point, or multiples?
yingted2012-06-26 21:30:09 can be multiple
Yasha2012-06-26 21:30:19 Sure, the parallelogram gives us that example.
Yasha2012-06-26 21:30:32 Could it be just one point?
yingted2012-06-26 21:30:42 circle
Yasha2012-06-26 21:30:55 Yeah, on the circle the opposite point is the only point distance two away.
Yasha2012-06-26 21:31:05 The answer will depend on the geometry of P.
yingted2012-06-26 21:31:27 you have to project it
yingted2012-06-26 21:31:27 and find the tangent
yingted2012-06-26 21:31:27 to get the solution set
Yasha2012-06-26 21:31:37 For a convex quadrilateral, it's achieved for all other corners. In fact, it's not hard to show, using convexity, that the P-distance remains 2 on any line segment connecting two corners which are both P-distance 2 from the original point.
Yasha2012-06-26 21:32:03 For general shapes, that is.
Yasha2012-06-26 21:32:09 (f) In principle, we could define P-distance even when P doesn't come from a convex body and/or is not centrally symmetric. But it turns out that in both of these cases, the definition is problematic: the resulting quantity doesn't behave in the ways we expect a "distance" to behave. Can you determine what problematic issues arise?
Yasha2012-06-26 21:32:20 First, what happens if we lose central symmetry?
yingted2012-06-26 21:32:33 you lose actual symmetry
Yasha2012-06-26 21:32:37 Then the P-distance from x to y is not necessarily the same as the P-distance from y to x, because we're taking the point in the opposite direction! This is bad; we want distance not to depend on order.
Watermelon8762012-06-26 21:32:50 if it's not centrally symmetric, the distance between two points one way is different than the other way
Yasha2012-06-26 21:32:52 What happens if we throw out convexity?
yingted2012-06-26 21:33:02 triangle
Yasha2012-06-26 21:33:07
Yasha2012-06-26 21:34:35
bobthesmartypants2012-06-26 21:34:59 multiple points?
Yasha2012-06-26 21:35:12 It could be multiple points, but there is one point where we know it must hit P.
yingted2012-06-26 21:35:25 it is at p
yingted2012-06-26 21:35:32 but the magnitude is only 1-lambda
ehhhhe2012-06-26 21:35:35 p
Yasha2012-06-26 21:35:55
yingted2012-06-26 21:36:07 lambda
Yasha2012-06-26 21:36:11
Yasha2012-06-26 21:36:17
yingted2012-06-26 21:36:39 more than 1!?!
Yasha2012-06-26 21:36:46 We don't know, but it must be more than 1, because that point is outside P.
Yasha2012-06-26 21:36:51 So we've violated the triangle inequality! We can get from the origin to that point with total P-distance equal to 1, if we take an indirect route, but if we go straight there, the distance is greater than 1.
yingted2012-06-26 21:37:23 the worst, though, is the loss of the "function" part
Yasha2012-06-26 21:37:40 Well, so if you go by the original definition, I think it's still well-defined.
Yasha2012-06-26 21:37:56 The original definition asks for the minimum k that puts the second point inside the region, which is still well-defined.
Yasha2012-06-26 21:38:39 For a non-convex body, as you grow your region, you might eat up the point and then spit it out again and then eat it a third time, but the minimum value of k is still well-defined.
Yasha2012-06-26 21:38:53 Anyways, onwards to problem 6:
Yasha2012-06-26 21:38:57 Problem 6: Islands and Bridges.
Yasha2012-06-26 21:39:01 An ocean has infinitely many islands. Every island is labeled by one of the integers ...,-3,-2,-1,0,1,2,3,..., with no two islands having the same label and every integer being the label of some island. Two islands are connected by a bridge if their labels differ by a power of two. For instance, there is a bridge connecting island 7 and island -25.
Yasha2012-06-26 21:39:05 We define the distance between two islands k_1 and k_2 to be the minimum number of bridges needed to get from k_1 to k_2. For instance, the distance between the islands 0 and 7 is 2. (You can move from island 0 to island 8, then to island 7; this is the minimum, since you can't go from 0 to 7 using just one bridge.)
Yasha2012-06-26 21:39:19 First, let's think a bit about this notion of distance. Does it have any nice properties?
Watermelon8762012-06-26 21:39:49 the distance from x to y is the same as the distance from y to x
yingted2012-06-26 21:39:49 symmetry?
Yasha2012-06-26 21:40:05 Sure, since you can go both directions along bridges, there is symmetry.
bobthesmartypants2012-06-26 21:40:09 you can get from any island to any other one
hatchguy2012-06-26 21:40:59 bridges only join islands with same parity
Yasha2012-06-26 21:41:01 Not quite, since 1 counts as a power of 2.
Watermelon8762012-06-26 21:41:09 translation independent
Watermelon8762012-06-26 21:41:09 the distance between x+n and y+n is the same as the distance between x and y
Yasha2012-06-26 21:41:17 For one thing, distance is translation invariant. That is, d(x,y) = d(x + c, y + c) for any number c. Any sequence of adding or subtracting powers of two that will take us from x to y will also take us from x + c to y + c.
Yasha2012-06-26 21:41:25 This means that if we want to find two islands that are distance r from each other, what do we actually need to find?
yingted2012-06-26 21:41:51 you can erase a variable
Watermelon8762012-06-26 21:41:51 an island a distance r from 0
Yasha2012-06-26 21:41:58 It is necessary and sufficient to find an island x such that d(0,x) = r.
MA7HL0V3R2012-06-26 21:42:10 An island r distance from island 0?
Yasha2012-06-26 21:42:23 Can anyone think of such an island?
yingted2012-06-26 21:42:33 basically space out powers
Tonitruant2012-06-26 21:42:40 Binary number 101010...
Yasha2012-06-26 21:42:46
Yasha2012-06-26 21:42:52 (We decided to write all our numbers in binary because the central definition is about powers of two.)
Yasha2012-06-26 21:43:02 It's certainly obvious that d(0,x) is no more than r. But is it obvious that it's actually equal to r?
Yasha2012-06-26 21:43:45 The distance is no more than r because the binary expansion gives us the r powers of 2 we need to add to get to x.
Yasha2012-06-26 21:43:54 It's not obvious that there's no shorter path to get to x from 0, though.
yingted2012-06-26 21:43:57 use induction
Yasha2012-06-26 21:44:05 Perhaps surprisingly, proving this one simple fact is the heart of this whole problem! We're going to state it as a stronger lemma, which will make the future parts of the problem easier.
Yasha2012-06-26 21:44:17
Yasha2012-06-26 21:44:29
Yasha2012-06-26 21:44:33 Does anyone see one part of this equality which is straightforward?
Yasha2012-06-26 21:45:07 It's not hard to see that the distance from 0 to any such x or y must be less than or equal to r, since each of these numbers only has r nonzero digits in binary, which provide a sequence of powers of two to add! So we really only need to show that it's at least r.
Yasha2012-06-26 21:45:23 As yingted suggests, we can use induction.
Yasha2012-06-26 21:45:31 The base cases for r = 1 and 2 are easy to check.
Yasha2012-06-26 21:45:35 Now, let's assume it's true for all values up to r, and try to prove it for r+1.
Yasha2012-06-26 21:45:42 This next part gets quite complicated, so I encourage you to try to go through it on your own afterwards; it's entirely possible you won't fully understand everything the first time.
Yasha2012-06-26 21:46:00 In the interests of time, I'll just go through it quickly.
Yasha2012-06-26 21:46:08
Yasha2012-06-26 21:46:22
Yasha2012-06-26 21:46:35
Yasha2012-06-26 21:46:46 Therefore, d(0,y) = r+1.
Yasha2012-06-26 21:46:53
Yasha2012-06-26 21:46:58 But in either case, that gives us that d(0,y) = r + 1 as before.
Yasha2012-06-26 21:47:03
Yasha2012-06-26 21:47:11
Yasha2012-06-26 21:47:16
Yasha2012-06-26 21:47:22
Yasha2012-06-26 21:47:27
Yasha2012-06-26 21:47:34
Yasha2012-06-26 21:47:38
Yasha2012-06-26 21:47:43
Yasha2012-06-26 21:47:48 Again, this was a rather complicated induction proof, so don't panic if you didn't catch all the details. The big picture is what counts: numbers with isolated by 1's, separated by zeroes in their binary expansion, have distance to 0 equal to the number of 1's in the expansion.
Yasha2012-06-26 21:48:26 I'll pause for a bit to make sure you get this point, and then we'll move forward.
Yasha2012-06-26 21:48:42
Yasha2012-06-26 21:48:47 (b) An infinite path in our ocean consists of an infinite set of islands I and an infinite set of bridges B, with each bridge in B joining two islands in the set I, such that: * every island in I is connected to exactly two bridges in B, * for any two islands in I, you can get from one to the other using only bridges in B. Here is one example of an infinite path: let I be the set of all odd-numbered islands and let B be the set of bridges between islands in I whose labels differ by 2. It is easy to see that I and B satisfy both conditions for an infinite path: * Each island k in I is connected to exactly two bridges in B – namely, the bridges leading to islands k-2 and k+2; * To get from any odd-numbered island to any other using bridges in B, you start at the smaller number and keep adding 2. As a warm-up, can you create an infinite path with the same I as in the example above, but with a different B?
Yasha2012-06-26 21:49:15 Does anyone see a simple way to modify the above example to make a different infinite path with the same set of islands?
Watermelon8762012-06-26 21:50:11 0 to 4,4 to 2, etc.
Yasha2012-06-26 21:50:34 Er, odd-numbered islands, but apart from that it's the right strategy.
Yasha2012-06-26 21:50:43 We can just make one small local change. Here's an example:
Yasha2012-06-26 21:50:48
Yasha2012-06-26 21:50:52
Yasha2012-06-26 21:51:07 Of course, there are lots of other possible examples.
Yasha2012-06-26 21:51:12 (c) Is it possible to construct an infinite path in our ocean such that, for any two islands k_1, k_2 in I, the minimum number of bridges in B needed to get from k_1 to k_2 is exactly the distance between k_1 and k_2? For instance, the infinite path in our example does not have this property: it takes two bridges in B to go from island 1 to island 5 (you have to go via island 3), even though the distance between these two islands is 1 (there is a single bridge not in B that connects them to each other). If your answer is yes, give an example of sets I and B that work. If your answer is no, prove that it can't be done.
Yasha2012-06-26 21:51:53 We'll make heavy use of the lemma we proved earlier.
Yasha2012-06-26 21:52:02
Yasha2012-06-26 21:52:13 If we subtract any two of these, we get a number with no consecutive ones, so it's easy to see that the minimum number of bridges is in fact the distance.
Yasha2012-06-26 21:52:21 How can we extend this chain to be infinite in both directions?
nemarci2012-06-26 21:53:05 -100, -1000100, ...
MA7HL0V3R2012-06-26 21:53:08 0, 100, 1000100, 10001000100, etc.
Yasha2012-06-26 21:53:14 We can just go into the negatives, but placing the ones slightly differently.
Yasha2012-06-26 21:53:20
Yasha2012-06-26 21:53:25 A few applications of the lemma will show that this infinite path has the desired properties. (I'll skip them for now in the interest of time.)
Yasha2012-06-26 21:53:31 (d) Does there exist a set S of 9 islands such that: * the configuration of bridges connecting pairs of islands in S is exactly as in the picture below (with no additional bridges between any of the islands), and * the distance between any two islands in S equals the minimum number of bridges needed to get from one island to the other via islands in S?
Yasha2012-06-26 21:54:00 What if, instead of a 3x3 grid, we had an nxn grid: for which n is such a configuration possible?
Yasha2012-06-26 21:54:09 We'll dive straight into the nxn version.
yingted2012-06-26 21:54:13 65,1,5,64,0,4,80,16,20
Yasha2012-06-26 21:54:28 That looks like it might work for the 3 by 3 case, though I haven't checked it.
Yasha2012-06-26 21:54:34
Yasha2012-06-26 21:54:42 The sequences we used in part (c) will help us out here:
yingted2012-06-26 21:54:59 you must split each row or column
yingted2012-06-26 21:54:59 in base 4, assign a 1 to one side and 0 to the other side
Yasha2012-06-26 21:55:11 That sounds somewhat similar to the solution I have,
Yasha2012-06-26 21:55:15
Yasha2012-06-26 21:55:19
Yasha2012-06-26 21:55:23
Yasha2012-06-26 21:55:28 Checking that the distances are correct uses similar logic to part (c), including the lemma from part (a).
Yasha2012-06-26 21:55:39
Yasha2012-06-26 21:55:44
Yasha2012-06-26 21:55:49 Again, we'll skip the the other cases in the interest of time.
Yasha2012-06-26 21:56:00 (e) Suppose, in a sea far away, we have islands labeled in the same way, with two islands connected by a bridge if their labels differ by a power of 3. Is there a one-to-one correspondence between islands in the ocean and islands in the far-away sea, such that two islands in the ocean are connected by a bridge precisely when the corresponding two islands in the sea are connected by a bridge?
yingted2012-06-26 21:56:19 no
yingted2012-06-26 21:56:19 0,1,2
yingted2012-06-26 21:56:19 use parity
Yasha2012-06-26 21:56:27 If you start by drawing the appropriate graph of bridges for small integers, you may suspect that there is no such correspondence.
Yasha2012-06-26 21:56:30 Perhaps surprisingly, there's a way to prove this without any use of the preceding work.
Yasha2012-06-26 21:56:59 We look for features that our set of islands in the Binary Sea has that the islands in the Ternary Sea might not have.
Yasha2012-06-26 21:57:11 As yingted mentions, we have a triangle in the Binary sea: 0, 1, 2.
Yasha2012-06-26 21:57:16 Is it possible for there to be three islands in the Ternary Sea with bridges connecting them?
MA7HL0V3R2012-06-26 21:57:26 No
Yasha2012-06-26 21:57:31 No, it isn't! Why not?
yingted2012-06-26 21:58:03 all powers of 3 are odd
Yasha2012-06-26 21:58:08 Powers of three are all odd. There's no way to add or subtract three of them and get back to zero, because you'll always get an odd number.
Yasha2012-06-26 21:58:15 So the answer is no: there is no such correspondence.
Watermelon8762012-06-26 21:58:37 that's a nice proof
Yasha2012-06-26 21:58:49 Indeed, it's a neat trick for (e).
nemarci2012-06-26 21:59:01 there are no a+b=c numbers, where a, b and c are power of 3
Yasha2012-06-26 21:59:09 Yeah, that's another way of putting it.
Yasha2012-06-26 21:59:13 OK, onwards to the last problem!
Yasha2012-06-26 21:59:40 This one has some easier parts at the beginning, which will be refreshing after these tricky problems.
Yasha2012-06-26 21:59:47 Problem 7: Mentor Pyramids.
Yasha2012-06-26 21:59:53 Last summer, the graduate students teaching at Mathcamp (we call them "mentors") arranged themselves into a pyramid with four layers:
Yasha2012-06-26 21:59:58
Watermelon8762012-06-26 22:00:13 This was my favorite question
Yasha2012-06-26 22:00:15 Awesome!
Yasha2012-06-26 22:00:21
Yasha2012-06-26 22:00:31
Yasha2012-06-26 22:00:42 (a) Find the weight supported by the mentor at the bottom left corner of a pyramid with n layers.
Yasha2012-06-26 22:00:53 The weight that someone supports does not depend on what happens below them. Therefore, the first few values for the weight supported by the mentor at the bottom left corner of a pyramid with n layers can be read off of the left side of the example above: 1, 3/2, 7/4, 15/8, etc.
Yasha2012-06-26 22:01:01 Based on the first few values, what do we think the answer is?
jared4292012-06-26 22:01:30
yingted2012-06-26 22:01:30 2-2^(1-n)
nemarci2012-06-26 22:01:30 2-(1/2)^(n-1)
Yasha2012-06-26 22:01:36
JonathanLi2012-06-26 22:01:41 Will the transcript be posted online? if so, where?
Yasha2012-06-26 22:02:16 Yeah, if you go to the MathJams page, there's a link for archived MathJams. You can see transcripts of old MathJams there, and this one will be there in the next couple days.
Yasha2012-06-26 22:02:26 Probably by tomorrow.
Yasha2012-06-26 22:02:43 OK, so we have a guess for the formula for the weight supported by the bottom left mentor.
Yasha2012-06-26 22:02:49 What is a technique we can use to prove this formula?
yingted2012-06-26 22:02:58 induction ...
Yasha2012-06-26 22:03:02 We can use induction. What is our base case?
yingted2012-06-26 22:03:18 top
Watermelon8762012-06-26 22:03:18 the top person who supports 1
Yasha2012-06-26 22:03:22
Yasha2012-06-26 22:03:28
Yasha2012-06-26 22:03:43 Let's consider a pyramid with k+1 layers, and let's call the mentor at the bottom left corner Ian. Let's call the mentor right above him Paddy. Here's our picture again:
Yasha2012-06-26 22:03:49
Yasha2012-06-26 22:03:56 What is the weight supported by Paddy, in terms of k?
nemarci2012-06-26 22:04:39 2-1/2^(k-1)
Yasha2012-06-26 22:04:45
Yasha2012-06-26 22:04:52 How can we find the weight that Ian supports?
Watermelon8762012-06-26 22:05:17 1+ 1/2 of the weight Paddy supports
nemarci2012-06-26 22:05:17 1+1/2*(2-1/2^(k-1))
Yasha2012-06-26 22:05:22 Ian supports his weight of 1, plus half of the weight supported by the mentors leaning on him, which in this case is just the weight supported by Paddy.
Yasha2012-06-26 22:05:27
Yasha2012-06-26 22:05:31 This statement is exactly what we wanted to show, so it completes the induction.
Yasha2012-06-26 22:05:34
Yasha2012-06-26 22:05:38 Any questions on this part?
Watermelon8762012-06-26 22:06:12 I thought n was different so my answers have n instead of n-1
Yasha2012-06-26 22:06:40 I of course don't remember your particular case, but generally we don't take off points for typos like that.
Yasha2012-06-26 22:06:58 as long as the math is correct
Yasha2012-06-26 22:07:06
Yasha2012-06-26 22:07:20 Let's call the (k+1)-th mentor from the left in the (m+1)-th layer Mo, and let's call the mentor directly above and to the left Susan, and the mentor above and to the right Mathieu. Here's the picture again:
Yasha2012-06-26 22:07:26
Yasha2012-06-26 22:07:34 What layer is Susan in, and how far is she from the left?
nemarci2012-06-26 22:08:25 mth layer, kth from left
Yasha2012-06-26 22:08:31 Susan is in the mth layer. Counting carefully, we see that she is the kth mentor from the left in that layer. What about Mathieu?
Watermelon8762012-06-26 22:08:57 mth layer k+1 th from left
nemarci2012-06-26 22:08:57 mth layer, (k+1)-th from left
Yasha2012-06-26 22:09:03 Mathieu is in the mth layer and is the (k+1)-th mentor from the left.
Yasha2012-06-26 22:09:07 In terms of W, what is the weight supported by Susan?
Watermelon8762012-06-26 22:09:30 W(k-1,m-1)
Yasha2012-06-26 22:09:35 By definition, the weight supported by Susan is W(k-1,m-1). What about the weight supported by Mathieu?
Watermelon8762012-06-26 22:09:41 W(k,m-1)
Yasha2012-06-26 22:09:47 By definition, the weight supported by Mathieu is W(k,m-1).
Yasha2012-06-26 22:09:50 What is the weight supported by Mo?
yingted2012-06-26 22:09:57 1+(left+right)/2
nemarci2012-06-26 22:10:03 W(k,m)=1+1/2(W(k-1,m-1)+W(k,m-1))
Yasha2012-06-26 22:10:10 By the definition of W, it is W(k,m). But, by the problem statement, the weight supported by Mo is equal to his weight plus half of the weight supported by Susan and half of the weight supported by Mathieu.
Watermelon8762012-06-26 22:10:41 But wait! What if susan doesn't exist because mathew's the leftmost?
yingted2012-06-26 22:10:41 base cases
nemarci2012-06-26 22:10:41 k=0
MA7HL0V3R2012-06-26 22:10:41 The camper could be on the edge
jared4292012-06-26 22:10:48 not defined for W(0,0)
ychen4282012-06-26 22:10:55 First case
Yasha2012-06-26 22:11:01 There is a problem with the mentors on the edges of the pyramid. They only have one mentor above them, so the formula doesn't quite make sense.
Yasha2012-06-26 22:11:06
Yasha2012-06-26 22:11:25 W(-1,m-1) represents the weight supported by the 0th mentor from the left in the mth row, which doesn't make sense. W(m,m-1) represents the weight supported by the (m+1)-th mentor in the mth row, which also doesn't make sense.
Yasha2012-06-26 22:11:30 What should we do?
Watermelon8762012-06-26 22:11:43 but wait! we have a formula for mentors on the edge!
Yasha2012-06-26 22:11:46 We can do that, but there's something simpler we can do.
yingted2012-06-26 22:11:58 basically, let everything else =0
nemarci2012-06-26 22:11:58 their weights should be 0
yingted2012-06-26 22:11:58 let the air =0
Yasha2012-06-26 22:12:03 We can set W(k,m) to zero whenever it is out of the range of the pyramid. Doing this will make our recursive formula for W still correct even when we look at W(0,m) and W(m,m).
Yasha2012-06-26 22:12:23 What values of k and m correspond to actual mentors in the pyramid?
yingted2012-06-26 22:12:51 k is between 0 and m
nemarci2012-06-26 22:12:57 0<=k<=m
Yasha2012-06-26 22:13:00
Yasha2012-06-26 22:13:08
Yasha2012-06-26 22:13:15
Yasha2012-06-26 22:13:23 Any questions on this part?
jared4292012-06-26 22:13:53 what exactly is the final solution?
Yasha2012-06-26 22:14:22
Yasha2012-06-26 22:14:31 Otherwise, W(k,m)=0.
Yasha2012-06-26 22:15:11 Onwards!
Yasha2012-06-26 22:15:16
Yasha2012-06-26 22:15:34 What's a technique that might be helpful for this problem?
Maz9062012-06-26 22:15:51 generating functions?
Yasha2012-06-26 22:16:01 There is a solution with generating functions, but I won't go through that one here.
Watermelon8762012-06-26 22:16:17 Solving part d first and then looking at part c
Yasha2012-06-26 22:16:34 It's unclear if there's a closed form for general W(k,m) actually. There is one in this special case, though.
nemarci2012-06-26 22:16:50 induction
Yasha2012-06-26 22:16:57 We could try induction. That is, we could try to find W(n,2n) in terms of W(n-1,2(n-1)).
Yasha2012-06-26 22:17:21 It's not immediately clear how to get induction to work, but here's a picture that hints at one possible method:
Yasha2012-06-26 22:17:27
BarbieRocks2012-06-26 22:18:24 i experimented and found a summation that worked for (d), proved it by induction, then proved a hard identity to make (c) simpler
Yasha2012-06-26 22:18:27 Yes, you can certainly find a summation formula for (d) and then try to sum it in part (c). We won't go that route though.
Yasha2012-06-26 22:18:43 Anyways, the picture describes trying to compute W(3,6), which is the weight supported by the circled mentor.
Yasha2012-06-26 22:18:50 Let's call the circled mentor Mo.
Yasha2012-06-26 22:18:55 We're going to work with the special case n=3 for a while, and then we will see how we can generalize our method.
Yasha2012-06-26 22:19:00 One way to compute the weight supported by Mo is to add up the contributions to that weight from all of the mentors in the pyramid (including Mo himself).
Yasha2012-06-26 22:19:13 What can you say about the weight contributed by the mentors in the gray regions to the weight that Mo supports?
nemarci2012-06-26 22:19:40 0
Maz9062012-06-26 22:19:40 no contribution
yingted2012-06-26 22:19:40 none?
Yasha2012-06-26 22:19:46 There is no downward path from the mentors in the gray regions to Mo, so Mo does not support any of their weight. The only mentors whose weights matter are the ones in the diamond-shaped non-gray region.
Yasha2012-06-26 22:20:23 Er, as in you can find a formula that involves a sum, but as far as I know you can't evaluate the sum to find a closed form solution.
Yasha2012-06-26 22:20:28 For part (d), that is.
Yasha2012-06-26 22:20:49 For part (c), we will find a closed form solution, as the problem asks.
Yasha2012-06-26 22:21:01 Anyways, I think the picture might have floated all the way up, so I will repost it.
Yasha2012-06-26 22:21:05
Yasha2012-06-26 22:21:20 As we saw, we can ignore the gray vertices.
Yasha2012-06-26 22:21:26 What can we say about the total weight contributed by the mentors in the black region?
Watermelon8762012-06-26 22:21:56 equal to W(k-1, 2k-2)
yingted2012-06-26 22:22:03 we can use induction
Maz9062012-06-26 22:22:03 W(n-1, 2(n-1)
Yasha2012-06-26 22:22:11 The mentors in the black region form a smaller diamond, which can be viewed as part of a smaller pyramid where Mo is in the middle of the fourth row.
Yasha2012-06-26 22:22:17 Therefore, the weight contributed by the mentors in the black region is W(2,4).
Yasha2012-06-26 22:22:41 When we do the general case, it will be W(n-1,2(n-1)), as people suggest.
Yasha2012-06-26 22:22:47 OK, so now all we have to do is figure out the weights contributed by the mentors in the blue and red regions. (We will think of the top mentor as being both blue and red, and hence colored purple.)
Yasha2012-06-26 22:22:56 As an example, let's pick one of the blue mentors and call her Quirk, and figure out how much weight Quirk contributes to the weight that Mo supports.
Yasha2012-06-26 22:23:22 How many such downward paths are there from Quirk to Mo?
yingted2012-06-26 22:23:42 5C2=5C3?
nemarci2012-06-26 22:23:42 10
Yasha2012-06-26 22:23:52
Yasha2012-06-26 22:24:07 Given a particular path, how much of Quirk's weight "flows" along that path to Mo?
yingted2012-06-26 22:24:25 2^-5
Maz9062012-06-26 22:24:25 1/32=1/(2^5)
Yasha2012-06-26 22:24:31
Yasha2012-06-26 22:24:35 What then is the total weight that Quirk contributes to the weight supported by Mo?
Maz9062012-06-26 22:24:54 ncr(5, 2)*1/(2^5)
nemarci2012-06-26 22:24:54 10/32
Maz9062012-06-26 22:24:54 10/32 = 5/16
yingted2012-06-26 22:24:54 multiply them
Yasha2012-06-26 22:24:59
Yasha2012-06-26 22:25:04 What about the top mentor? How much weight does the top mentor contribute to Mo?
Watermelon8762012-06-26 22:25:43 6 choose 3 times 2^{-6}
yingted2012-06-26 22:25:48 6C3 2^-6
nemarci2012-06-26 22:25:49 6C3/2^6=20/64=5/16
Yasha2012-06-26 22:25:54
Yasha2012-06-26 22:25:58 What about the blue mentor in the third row?
Yasha2012-06-26 22:26:24 (Third from the top, that is, so in position (0,2).)
yingted2012-06-26 22:26:55 4C1 2^-4
Maz9062012-06-26 22:26:55 4/16=1/4
nemarci2012-06-26 22:26:55 4C3/2^4=1/4
Yasha2012-06-26 22:27:00
Yasha2012-06-26 22:27:07 And the blue mentor in the fourth row?
yingted2012-06-26 22:27:25 3C0 2^-3
Yasha2012-06-26 22:27:31
nemarci2012-06-26 22:27:41 1/8
Yasha2012-06-26 22:27:48 What is the total contribution of the mentors in the blue region to the weight supported by Mo? (Remember that the top purple mentor counts as both blue and red.)
nemarci2012-06-26 22:28:13 1
Yasha2012-06-26 22:28:18
Yasha2012-06-26 22:28:25 Therefore, the total weight contributed by the mentors in the red region is also 1.
Yasha2012-06-26 22:28:34 What, then, is a recursive formula for W(3,6) in terms of W(2,4)?
Yasha2012-06-26 22:29:35 It's tempting to add up the contributions from the black, blue, red regions. However, W(3,6) is not equal to W(2,4)+2. What is the problem?
nemarci2012-06-26 22:29:55 we counted the top person twice
yingted2012-06-26 22:30:05 overcounting?
Maz9062012-06-26 22:30:05 overcounted the top most mentor - two times
Yasha2012-06-26 22:30:07 We've double-counted the weight contributed by the purple mentor, since the purple mentor is both in the blue region and in the red region. What then is the correct formula for W(3,6) in terms of W(2,4)?
Maz9062012-06-26 22:30:56 W(3,6)=W(2,4) + 27/16
nemarci2012-06-26 22:30:56 W(3,6)=W(2,4)+2-6C3/2^6
Yasha2012-06-26 22:31:01
Yasha2012-06-26 22:31:06 In the case where n=3, the sum of the contributions from the blue region turned out to be exactly one. Assuming that this wasn't just a coincidence, what is the correct generalization of the above formula to a general n?
bluepie2012-06-26 22:32:24 what is this class about?
Yasha2012-06-26 22:32:26 We're going over the solutions of the Mathcamp qualifying quiz, available at
nemarci2012-06-26 22:32:33 W(n,2n)=W(n-1,2(n-1))+2-2nCn/2^(2n)
Yasha2012-06-26 22:32:45 For a general n, we can split up the pyramid into gray, black, blue, and red regions analogously.
Yasha2012-06-26 22:32:49 As before, the contribution of the gray region is zero and the contribution of the black region is W(n-1,2(n-1)).
Yasha2012-06-26 22:32:52 We're assuming for now that the blue and red regions each contribute a weight of 1. (I'm leaving the proof of this fact for later.)
Yasha2012-06-26 22:32:56
Yasha2012-06-26 22:33:06
Yasha2012-06-26 22:33:13
Yasha2012-06-26 22:33:23 It's not immediately obvious how to compute this sum.
Yasha2012-06-26 22:33:38 There are a couple approaches to tackling it. A natural thing to try first is to multiply through by 2^(2n) to get rid of the fractions, and then compute some of the first few values of the series.
Yasha2012-06-26 22:33:43 The result is 1, 6, 30, 140, 630, 2772, 12012, 51480, and so forth.
Yasha2012-06-26 22:33:52 Since you're allowed to use the internet, one thing to do after that is to try to find the sequence in the online encyclopedia of integer sequences at
Yasha2012-06-26 22:34:26 You'll find it there, but of course you still have to prove it.
Yasha2012-06-26 22:34:32 Without the internet, you could try to find a pattern. For example, if you try factoring, you might notice that this sequence is evenly divisible by the sequence 1, 3, 5, 7, 9, 11, 13, 15, etc. Dividing through, we get the sequence 1, 2, 6, 20, 70, 252, 924, 3432. If you look at Pascal's triangle, you'll notice that these numbers appear in the middle of the even-numbered rows.
yingted2012-06-26 22:34:37 lol
Yasha2012-06-26 22:34:49 You might also instead try dividing by the sequence 1, 2, 3, 4, 5, 6, 7, 8, etc. It turns out that you'd also find yourself with entries of Pascal's triangle.
Yasha2012-06-26 22:35:01
ychen4282012-06-26 22:35:19 WE COULD USE OEIS????
Yasha2012-06-26 22:35:21 The quiz says that you're allowed to use the internet :)
Yasha2012-06-26 22:35:29 Of course, finding it in OEIS is not enough.
Yasha2012-06-26 22:35:47 OEIS will give you a formula that matches the first few terms, but you still have to prove that that formula works for all n.
ychen4282012-06-26 22:35:59 oh okay :D
Yasha2012-06-26 22:36:22 OK, so as I said, a guess for the formula is not good enough.
Yasha2012-06-26 22:36:29 We need proof.
Yasha2012-06-26 22:36:30
Yasha2012-06-26 22:36:50 How can we prove this?
melikababadi2012-06-26 22:37:08 induction
AwesomeToad2012-06-26 22:37:08 induction?
yingted2012-06-26 22:37:08 combinatorics or induction
Yasha2012-06-26 22:37:16 Yes, induction seems like a good bet.
Yasha2012-06-26 22:37:28 There might be a combinatorial way, but it would be quite tricky to come up with it.
Yasha2012-06-26 22:37:33
Yasha2012-06-26 22:37:40 Now we assume that the statement is true for n=m-1, and we'd like to prove it for n=m.
Yasha2012-06-26 22:37:44
Yasha2012-06-26 22:37:55
Yasha2012-06-26 22:37:58 What now?
Yasha2012-06-26 22:38:37 What can we do to the right hand side to start simplifying?
Yasha2012-06-26 22:39:18 So, we have the inductive hypothesis to work with.
Yasha2012-06-26 22:39:20
Yasha2012-06-26 22:39:28 What can we do next?
Yasha2012-06-26 22:40:13
yingted2012-06-26 22:41:04 expand part of it
AwesomeToad2012-06-26 22:41:04 bash
Yasha2012-06-26 22:41:13 I'd rather do the opposite of expanding :)
Yasha2012-06-26 22:41:31
Yasha2012-06-26 22:41:45 What's something else we can do to get our expression looking more like what we want?
Maz9062012-06-26 22:42:52 rewrite (2m-1)!/(m-1)!^2 as m*(2m-1)C(m-1)
Yasha2012-06-26 22:42:59 Sure, that's one thing we could try.
Yasha2012-06-26 22:43:05 But I want to keep our goal in mind.
Yasha2012-06-26 22:43:41
Yasha2012-06-26 22:43:49 We already have the 2^(2m) part.
Yasha2012-06-26 22:43:56 Let's get the m!^2 part.
Yasha2012-06-26 22:44:12
Yasha2012-06-26 22:44:25 What can we do next?
Yasha2012-06-26 22:44:56
Yasha2012-06-26 22:45:13
Maz9062012-06-26 22:45:38 4m^2(2m-1)! = 2m*(2m)!
Yasha2012-06-26 22:45:48 Yup, that's another way to get to (2m+1)! in the numerator.
Yasha2012-06-26 22:46:12 A good trick for proving that one expression is equal to another is taking out factors from one expression to make it line up with the other one. | 677.169 | 1 |
Pages
Tuesday, September 11, 2012
Math 911
Ahh, another math review. If anyone wants to take any guesses, math is not my strong point, and I can use all the help I can get. So when this product came up for review, I decided to try it out, and see what my older two children thought.
What It Is:
Math 911 is an online math tutorial. The man who thought up this program is Professor Weissman, who has been on the New York City Board of Education, and taught math at Essex Country College in New Jersey. This is a no-nonsense, straightforward program; one of its main features is that there are no distracting games or images, it is just learning math. The courses offered are:
Introductory Algebra
Intermediate Algebra
College Algebra
Trigonometry
Pre-Calculus
Statistics
Your student will begin their course and progress in a methodical way through the topics. A math problem is given at the top of the screen, the instructions are at the bottom, and there is a box in the center where the answer is entered. If your child answers correctly, a big check mark appears with the words RIGHT!!! If they get the problem wrong, the correct answer is shown. On the right side of the screen is a sidebar; this shows the level they are on and has other buttons to choose from which include Calculator, New Problem, Change Level, and See All Steps. It is highly suggested that when your child answers any of the problems wrong, they review and study the See All Steps section. There are also downloadable PDF's to use with the Introductory Algebra course.
What You Get:
You can actually go onto the website and download the Introductory Algebra course right now for free. This is an amazing deal, which would also let you view the program and see how it works. You can upgrade to the Premium Version for $49.95. This never, ever expires, and includes free upgrades from the website and tech support. You can also purchase Algebra In A Flash; this is the flash drive version which is sold on the website for $9.95. Having this program on a flash drive would make it very convenient to take your math homework with you to different computers and laptops. Each course has midterms and a final; you can also print out your child's grades or copy them to a Word document.
How We Used This:
Since I had two children who were going to be using this, I emailed Professor Weissman and he gave me codes to upgrade the program so that I could have more than one user. I had Mr. Lego and Bookworm begin with Introductory Algebra and each day I would set the timer for 45 minutes while they worked through the levels. They both began in the Introductory Algebra Course, which starts with Integers and then moves into Adding and Subtracting Integers, Multiplying and Dividing Integers, and Exponents. We used this for a couple of weeks and then took off a week from school for vacation. When we came back to doing math the next week, the program didn't remember our accounts. I emailed Professor Weissman and was pleased with his prompt response. He had me call him - I didn't realize he was on the East Coast and I called around 4pm Pacific Time - but he did answer the phone and was extremely helpful in figuring out a solution. It was a simple problem of entering the wrong password; he showed me how to figure out what password we had chosen, which we then wrote down next to the computer. (Which is what we should have done in the first place!)
What I Thought:
I liked the direct approach to math in this program; there were truly no frills or animation. This would be very helpful if your child is easily distracted when doing school. And Professor Weissman is very easy to contact and genuinely wants to help if you have any questions. When we started this my children did have trouble getting used to the format; it was confusing for them to have the instructions at the bottom of the page and the answer box towards the top. The style of this program is like the old DOS programs, but you do get used to it pretty quickly. The other thing I noticed is that the program does not "teach" math; the student learns by doing the problems and then studying the step by step solution. For some math learners this is fine; for others they need more instruction and guidance. Since I am not a great math teacher we will continue with our current math curriculums, but I will definitely be using Math 911 when I feel my children need extra practice. And hey, these math courses are for adults too....I might be learning more about Statistics in the near future!
Other members of The Crew reviewed Math 911 and had their students do some of the other math courses....check it out by clicking below.
Disclaimer: As a member of the
TOS Crew, I received this product, at no cost to me, for my
honest and humble review. All
opinions are mine | 677.169 | 1 |
Compass Learning Odyssey Algebra 1 Answer Key
Transcription
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The Mathematics curriculum provides students with the opportunity to achieve their maximum potential in understanding mathematics. An important goal is the development of students problem solving abilityPlease start the slide show from the beginning to use links Click here for active links to various courses CLICK ON ANY COURSE BELOW TO SEE DESCRIPTION AND PREREQUISITES To see the course sequence chart | 677.169 | 1 |
Numerical Analysis: Systems of Linear Equations
In this linear equation worksheet, students use the iterative method to solve systems of equations. Explanations and examples are provided. This four-page worksheet contains approximately three problems. | 677.169 | 1 |
Math Study Skills
Department Link
Creator - Samuel Willis
Samuel Willis
Introduction
Welcome to the Math Study Skills guide. This guide will link you to tutorials, instructional videos, articles, books, web sites, and other resources available through the library or on the web. You will find these helpful in reducing math anxiety and achieving math success! If you still have more questions, check the Help tab to the left to find out how a librarian can help you with your research. | 677.169 | 1 |
ISBN 13: 9780007194940
GCSE Maths 2005 (Exam Practice)
High GCSE grades are gained through a combination of good knowledge, good understanding and good exam technique. Exam Practice is all about exam technique and because it's written by the people who mark the exams, it really will help students improve their performance. The content of Exam Practice GCSE Maths has been thoroughly updated to match the very latest GCSE exams and provides examples of exam questions across all the exam boards. With key skills highlighted, lots of questions to try and tips from top examiners, this book really will give students the confidence to do brilliantly! This new edition includes Tony Buzan's revolutionary Mind Maps which will help to make study and revision easier.
"synopsis" may belong to another edition of this title.
About the Author:
Paul Metcalf if a freelance Consultant in Mathematics and is Principal Moderator for a major examining group. He was formerly a Head of Mathematics and a Deputy Headteacher.
Book Description Collins4858
Book Description Collins 28/014940
Book Description Collins 28/014940
Book Description Collins, 2005. Paperback. Book Condition: Good. Exam Practice - GCSE Maths4940 | 677.169 | 1 |
Mathematics Its Power and Utility, 9th Edition
MATHEMATICS: ITS POWER AND UTILITY, 9e, combines a uniquely practical focus on real-world problem solving with a thorough and effective grounding in basic concepts and skills, allowing even the least-interested or worst-prepared student to appreciate the beauty and value of math while mastering course material. The first section of the book explores the power and historic impact of mathematics and helps you harness that power by developing an effective approach to problem solving. The second section builds on this foundation by applying math concepts to a wide variety of real-life situations--including money management; handling of credit cards; inflation; purchase of a car or home; use of probability, statistics, and surveys; and many more topics of interest. MATHEMATICS: ITS POWER AND UTILITY, 9e, assumes only a basic working knowledge of arithmetic, making it effective even if you have no exposure to algebra or little confidence in your current math | 677.169 | 1 |
Publisher: Wikibooks , 2012 This book is about the topic of mathematical analysis, particularly in the field of engineering. This will build on topics covered in Probability, Algebra, Linear Algebra, Calculus, Ordinary Differential Equations, and others.
In publishing this book the authors have endeavored to supply a handy means of reference to theoretical and applied mathematics used in engineering, and while the first aim has been to make this a mathematical handbook, the book is of greater value because it includes the underlying engineering data and applications as well as the mathematical formulae.
Author: Jonathan M. Borwein Publisher: DocServer, 2010 The desire to understand Pi, the challenge, and originally the need, to calculate ever more accurate values of Pi, has challenged mathematicians for many many centuries, and Pi has provided compelling examples of computational mathematics.
This textbook covers topics such as functions, single variable calculus, multivariate calculus, differential equations and complex functions. The necessary linear algebra for multivariate calculus is also outlined. More advanced topics which have been omitted, but which you will certainly come across, are partial differential equations, Fourier transforms and Laplace transforms.
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This book is a self-contained introduction to interactive proof in higher-order logic, using the proof assistant Isabelle. It is a tutorial for potential users. The book has three parts: Elementary Techniques; Logic and Sets; Advanced Material.
The monograph gives a detailed exposition of the algorithmic real algebraic geometry. It is well written and will be useful both for beginners and for advanced readers, who work in real algebraic geometry or apply its methods in other fields.
The authors offer a tutorial on the new mathematical ideas which underlie their research. Many of the ideas in this book will be accessible to a well-trained undergraduate with a good background in mathematics and computer science.
The main purpose of the book is to acquaint advanced undergraduate and graduate students in computer science, engineering and mathematics with the algorithmic ideas in computer algebra so that they could do research in computational algebra.
The results of cache-simulation experiments with an abstract machine for reducing combinator graphs are presented. The abstract machine, called TIGRE, exhibits reduction rates that compare favorably with previously reported techniques | 677.169 | 1 |
1. Solve for x
x(x+1)a = x2
where a is a given number.
Solution. Clearly, x = 0 is a solution for every a. To find other solutions, we assume that
x is not 0
and divide the equation by x. We obtain a linear equation (x+1)a = x. In standard
form,
x(a-1)=-a
1. Solve for a
a(a-1)x = a2
where x is a given number.
Solution. a = 0 is a solution for every x. To find other solutions, divide the equation
by a.
Then we get a linear equation (a-1)x = a, or (x-1)a= x in standard form.
So a= x/(x-1) when x =/ 1 and the
Linear Programming Advice
Showing 1 to 2 of 2
Dr Byrne has a simple goal. You participate and learn from this class, which is what you'll end up doing. There is no final exam and both midterms are take home. He genuinely just wants you to learn without the pressure of having to memorize endless formulas and methods.
Course highlights:
We learnt about the simplex method which applies to operation research
Hours per week:
3-5 hours
Advice for students:
Go to class! And definitely participate. Wherever you sit on the first day becomes your fixed seat. So sit in the front because participation is a huge part of the grade.
Course Term:Fall 2014
Professor:eli byrne
Course Required?Yes
Course Tags:Great Intro to the SubjectGo to Office HoursParticipation Counts
Nov 22, 2016
| Would recommend.
Pretty easy, overall.
Course Overview:
Jenny Li is very nice and friendly and doesn't overcomplicate the material. She wants everyone to do well, but expects you to put in some effort.
Course highlights:
This course focuses a lot on matrices and linear algebra. I've learned a lot about minimums and maximums of programs.
Hours per week:
6-8 hours
Advice for students:
Do and keep up with the homework. Also read through the book for some helpful tips. | 677.169 | 1 |
Su questo libro:
This is a practical workbook on calculating the prices and returns on complex financial instruments. Designed for working professionals on Wall Street, this workbook is sprinkled with "fast facts" about the inner workings of the capital markets. Its lively and accessible style communicates math concepts accurately on the first reading, with plenty of exercises integrated into the text so you can test your understanding as you go. Depending on your personal knowledge base, you can work through each bite-sized chapter or skip a topic, knowing you have a quick reference.
Who needs this workbook? Professionals working on trading desks and students of money and capital markets, investments, financial institutions, and corporate finance. Excellent math supplement to traditional finance text books.
Appendices: Conventions used in the capital markets for settlement, day counts and bad dates; Newton-Raphson Search; Excel functions useful for capital markets math; regression analysis; the top 100+ formulas; a glossary; and solutions to all the exercises.
About the Author:
Dr. Norman Toy is one of Wall Street's best-known and most beloved instructors. Dr. Toy has taught financial math, capital markets, corporate finance and financial analysis at investment banks for over 30 years. His students range from beginning analysts to executives and directors of major financial institutions.
A member of the faculty of Columbia Graduate School of Business since 1969, he teaches advanced corporate finance, debt markets and investment strategies in the school's Executive MBA Program. His experience includes management posts at Columbia University's Health Sciences Division and its College of Physicians and Surgeons, and he was a director and principal of the financial counseling firm of Brownson, Rehmus & Foxworth, Inc.
Dr. Norman Toy was educated at the University of Florida, where he earned a BA in mathematics, and the Harvard Business School, where he earned an MBA and Doctorate in managerial economics (management science and operations research. Dr. Toy's research interests include capital markets, valuation and corporate finance and he is the author of articles on finance, marketing and statistics.
His books include Introduction to Financial Math and Capital Markets Math, both published by Adkins Matchett & Toy, which delivers practical, hands-on training to investment banks, financial institutions, corporations, asset managers and government agencies. | 677.169 | 1 |
Vedic Mathematics Course
Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). The aim of such sutra is to simplify the entire calculations and arithmetical operations.
Vedic Maths Course
There are 16 sutras in vedic mathematics as per Sri Bharati Krsna Tirthaji Maharaj. The below mentioned list of topics (course content) is how I look at it.
Through (QM) I'll try to cover all the above vedic maths related topics. Lot of other topics, techniques, shortcuts will be part of QM. Mathematical games, puzzles, riddles, etc. are also included to make every kid sharp.
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About the Author
admin
Hi, I'm Vineet Patawari. I fell in love with numbers after being scared of them for quite some time. Now, I'm here to make you feel comfortable with numbers and help you get rid of Math Phobia!
2 Comments
Vilas ThakurJune 11, 2012 at 7:22 pm
Hello , thanks admin for writing so many methods , tricks , techniques for solving Additions ,Subtractions , Multiplications & Divisions in Quicker math"s ! Interested students are requested to Learn one by one to know Basic Math's & Vedic Maths !! I am also one of the Tutor to teach Basic Math's & Vedic Maths as Home Tuition Classes & through Mail id for Outstation Students .To know more details please do write on my mail id is basic.maths483@gmail.com thanks once again Quicker Math's * I have passed M.Sc. ( STAT ) in Ist class from Nagpur University in 1981 .
Sir, I wanted to solve data interpretation questions (Bank Po Exam) fast. I understand all the questions but get failure at the time of solving. It takes too much of time. Please tell me some tricks to solve such questions quickly. Thanking You, Regards | 677.169 | 1 |
Showing 1 to 3 of 3
Ch 18 19 1) Regarding written contracts, disputes are easier to settle when contractual terms are solidified in writing, the moment of writing allows both parties to reconsider terms and ensure what they desire, in general, written contracts aid in the co
Math 2010 NameSg )lQQ \ l (MASON
The Bee Problem
Female bees are born from fertilized eggs, and male bees are born from unfertilized eggs.
This means that a male bee has only a mother, while a female bee has a mother and a
father. If the ancestry of a m
Keep your head above water. Don't drown in all your other classes and summer schedule.
Course Term:Fall 2017
Professor:AaronB.Willis
Course Tags:Math-heavyBackground Knowledge ExpectedGreat Discussions
Jun 26, 2017
| No strong feelings either way.
Not too easy. Not too difficult.
Course Overview:
It's required but the teacher is great!
Course highlights:
I learned a lot. Mostly I learned how to relearn math, so I can better teach it.
Hours per week:
6-8 hours
Advice for students:
Put the work in! Go to the math lab for help and you will do fine.
Course Term:Spring 2015
Professor:A.B. Adams
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsParticipation Counts
Mar 02, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
I would recommend this course, specifically with Mrs. Adams. I have never had an easy time with math courses, but Lorraine made the content make sense. It was easy to earn a good grade in the class because I finally had a teacher who could explain it well. I would recommend printing off the notes that she posts to the class Canvas page and bringing them to class so that you can follow along and make extra notes.
Course highlights:
I gained a better understanding of how different math concepts work, which made it easier to do well in the class. We learned a lot about math that is taught in younger grades, but it wasn't too easy.
Hours per week:
0-2 hours
Advice for students:
Like I mentioned earlier, bring the notes with you to class. It helps you follow along, you can study them before the class, and sometimes she adds to them so you can take some extra notes. Ask questions if you need to. She has no problem answering questions. | 677.169 | 1 |
The Mathematics department is committed to rigorously preparing students for success in college and beyond by focusing on problem solving, reasoning and proofs, communication, representations, and connections between mathematical disciplines. We offer a progression of Algebra I, II, Geometry, and Pre-calculus courses, as well as Honors and pre-AP courses for advanced students. In their last two years, students have the option of enrolling in Statistics, Regular Calculus, or Advanced Placement AB and BC Calculus McKinley Technology High School requires all students to enroll in at least two AP courses.
Algebra I – 1.0 Credit
This Math course focuses on the understanding of the real number system as a basic structure of algebra and to acquire facility in applying algebraic concepts to problem solving. Students will do operations with real numbers, solving equations and inequalities in one variable, lines and graphing, polynomials and algebraic fractions, factoring, solving systems of equations, and the quadratic formula.
Honors Algebra I – 1.0 Credit
This Math course is designed to challenge gifted math students. All of the topics in Algebra I are taught along with the following additional topics: rational functions, radical Graduation equations, and trigonometric ratios.
Geometry – 1.0 Credit
This Math course studies Euclid's postulates and theorems as the basis for an axiomatic system. Students will explore geometry through inductive and deductive processes using proofs and making algebraic connections where appropriate. Students will do investigations that include logic, angle and line relationships, triangles and other polygons, congruence, similarity, and coordinate geometry.
Honors Geometry – 1.0 Credit
This Math course teaches students complex geometric concepts from algebraic and inductive reasoning perspectives. Students will do challenging activities that require them to identify, analyze, and solve problems involving the following topics: segments, angles, triangles, quadrilaterals, polyhedral, circles, spheres, transformations, if-then statements, and proofs. There is a heavy emphasis on proofs in the honors course. Students will also learn the abstract and symbolic nature of geometry, as well as the application, analysis, and synthesis levels of understanding.
Probability & Stats – 1.0 Credit
Prob./Stat part 1: This Math course covers probability, including applications of the counting theory. Students will learn and be able to perform activities including simulation of probability models using current technology, analysis of games of chance, reliability theory, decision theory, applications of Bayes theorem, and distribution theory. Prob./Stat Part II: This Math course introduces statistics topics that include many real-world applications. Students will learn techniques necessary to run a statistical test for a final project. Students will also gather and organize data, and then analyze and interpret this data incorporating skills learned throughout the semester. This course combines the two half credit Prob./Stat courses I & II.
Algebra II & Trigonometry – 1.0 Credit
This Math course explores functions algebraically, as well as numerically and graphically, using technology and manipulatives. The student will know the concepts of complex numbers, matrices, systems of equations and inequalities, sequences and series, probability using permutations and combinations, and trigonometry.
Honors Algebra II – 1.0 Credit
This Math course is an intensive, accelerated course intended to prepare students for advanced mathematics courses. Students will focus on the use of technology and data analysis to develop reasoning, thinking, problem-solving, and communication skills. Students will learn about families of functions, systems of equations, matrix applications, combinatorics, sequences and series, complex numbers, and trigonometry.
Pre-Calculus – 1.0 Credit
This concept. This concepts, develop multiple strategies for analyzing complex situations, and acquire appropriate technological skills. Students analyze situations verbally, numerically, graphically, and symbolically.
Honors Pre-Calculus -1.0 Credit
This Math course further develops and integrates the major math strands of algebra, geometry, and trigonometry. Students will learn and complete advanced units of study in elementary algebraic and transcendental functions; circle and triangle trigonometry; statistics and probability; two- and three dimensional vectors; matrices, determinants and fractals; polar coordinates and complex numbers; sequences and series; and an introduction to the calculus concepts of limits and derivatives.
Conc & Context of Calculus – 1.0 Credit
This Math course builds on students' experience with functions and introduces the basic concepts and skills of calculus. Students will investigate and apply the properties of polynomial, exponential, and logarithmic functions; broaden their understanding of the mathematics associated with rates of change; and develop facility with the concepts and skills of differential calculus as applied to polynomial, rational, exponential, and logarithmic functions.
AP Calculus ABAP Calculus BC | 677.169 | 1 |
We'll help you get the scores you need!
Varsity Tutors Pre-Algebra mobile app
The Varsity Tutors Pre-Algebra Mobile App
Prior to learning algebra, geometry, trigonometry, or calculus, your child must fully understand the elements taught in pre-algebra. This class can set them up for success in their upper-level math endeavors. The free Varsity Tutors Pre-Algebra app for iPhone, iPad, and Android devices thoroughly explain these math fundamentals. The app can help your student work toward the knowledge they need for better grades and gain a deeper understanding of essential mathematical concepts.
Pre-algebra is a subject generally taught in middle school. As the educational landscape gets more competitive, however, students are being exposed to these more advanced concepts at a younger age. When your student takes pre-algebra, they will learn how numbers work. They will learn to understand whole numbers, prime numbers, mixed numbers, real numbers, and many other concepts of number theory. They will be instructed in how to manipulate the types of numbers that they will need to know in Algebra 1. They will also be taught the properties of numbers, such as the associative, commutative, distributive, additive, and multiplicative properties.
After the basic number concepts are mastered, your student will learn about other important concepts, like absolute value. Your child will review addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals. They will also begin mixing negative and positive numbers into these types of calculations.
Near the end of the course, basic one- and two-step algebraic equations will be introduced. Pre-algebra is the first step into the complex algebraic equations your student may perform in the future. Additionally, they will learn how to graph -- another essential math skill that your child will use for years to come.
If your son or daughter is one of the many students who do not have an easy time with math, studying can seem like a real struggle. With the Varsity Tutors Pre-Algebra app for iOS or Android devices, they will have a great number of practice resources that can help to streamline their review.
Have your child begin by taking one of the full-length, timed practice tests to see where their strengths lie, and where they might need more help. The results will help you to build a custom pre-algebra study plan for your child. For example, if they're having trouble distinguishing between a rational and irrational number, Learn by Concept can help your child master that skill. Learn by Concept provides practice problems with comprehensive explanations that show exactly how the correct answer is reached.
If your student is looking for quick study sessions throughout the day, they can use the app's flashcard section. They can use the wealth of cards already available on the app, or create a custom set with the Flashcard Maker. You and your child can use your own audio, photos, and text to personalize a set of flashcards.
If your child needs a little help with pre-algebra, try the free Varsity Tutors Pre-Algebra app. It's a portable, comprehensive resource that enables your child to review anytime, anywhere. Download it from iTunes or the Google Play Store today!
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If you have a middle school student, they will probably need to complete a course in pre-algebra. This course is usually taught in either sixth or seventh grade, and is designed to introduce the basic concepts that will prepare them for more advanced algebra courses. As your child makes their way through this course, they will begin to get experience with the concept of abstract mathematics.
In pre-algebra, topics include learning other types of numbers, like integers, decimals, negative numbers, and fractions. Additionally, your child will learn about basic powers and roots, how to solve basic equations, and will develop an understanding of manipulated variables. While Common CORE curriculum varies from state to state, pre-algebra examinations should include the aforementioned material.
As your child gets started in pre-algebra, they will get an introduction to negative numbers. They will learn about what happens when you add and subtract negative numbers, and about the sequential order of negative numbers on the number line. After being familiarized with the basic concepts of negative numbers, your child will be required to put these concepts into action. By doing this, they will learn about what happens when you add and subtract two negative numbers, as well as what happens when you do the same actions with one negative and one positive number. Furthermore, they will be required to learn the rules of multiplication and division of negative numbers, as well as the concept of absolute value.
Understanding factors and multiples are important concepts that will aid your child as they progress through pre-algebra. In this area, they will learn about divisibility, and which numbers are divisible by other numbers. They will also learn about which factors make up various numbers, and how to find these factors. They will be exposed to prime numbers, and how they can't have a positive divisor aside from the number one, and will learn how to use prime factorization to determine a set of prime numbers.
Rational numbers are an important component of algebraic mathematics. To become familiar with this concept, your child will be required to explore two types of numbers: decimals and fractions. By virtue of spending money and understanding the concept of dollars and cents, they may have a good understanding of the decimal system. In pre-algebra, however, they will need to know how to add, subtract, multiply, and divide decimals. Moreover, they will learn how to convert decimals to fractions and vice versa, as well as how to turn both decimals and fractions into a percentage. To better understand these concepts, you can practice with your child by applying them in a real-world setting. Try having them calculate the percentage of a tip for a restaurant bill, for instance.
Moving into fractions, you and your child can practice using visual activities in order to gain an understanding. Imagine a pizza that is cut into eight pieces. Each slice is considered one-eighth of a pizza. After developing a working knowledge of fractions, your child will learn how to incorporate them into math problems. Furthermore, they will learn about improper fractions and mixed numbers, how you can convert one to the other, and how you multiply and divide improper fractions and mixed numbers.
As pre-algebra progresses, your child will have the chance to get familiar with the concepts of proportions and ratios. Learning about ratios can be easy when applied in realistic terms. Have your child try and imagine the ratio of boys-to-girls in a classroom, or the student-to-teacher ratio in their favorite class. Along with learning about how a ratio is a comparison of two things, they will also learn how to incorporate them into mathematic problems. Proportions are similar to ratios, except that they are a pair of ratios which are equal to one another. For example, if you are expected to eat two pieces of bacon for every egg, then eating three eggs would mean that you have eaten six bacon strips. This is an example of a proportion.
As your child begins to learn more advanced concepts in science and mathematics, you will notice how they begin to share an overlap. Because of this, it is important that they understand certain aspects that will help them in both subjects. These aspects include rates, and how they're used in word problems, the ability to convert various units of measurement, and the conversion of the three primary temperature units.
Towards the end of the pre-algebra course, your child will start digging deeper into scientific mathematics, like statistics and graphs. These are essential skills to know for any social and natural science course. They will learn about how to interpret data presented in tables, bar graphs, and line graphs; solve rational number word problems; and plot mathematical inequalities. They will also be shown various numerical patterns, as well as binary and hexadecimal number systems.
Algebraic equations really get into the heart of algebra prep. When your child approaches the end of their pre-algebra course, they will start to gain hands-on experience with topics that will be addressed in high school algebra. These topics include exponents, including fractional and negative ones, square roots, and cube roots. They will also be introduced to scientific notation, and will be expected to use it in order to manage large numbers. This is also when your child begins to learn about the order of operations, which has been turned into the mnemonic device "please excuse my dear aunt sally." As soon as they master the order of operations, they can begin solving for mathematical variables like X and Y.
While learning about the order of operations, your child will also begin to experience polynomials for the first time. These are problems that are usually mixed between letters and numbers, where you are required to solve for variables while subtracting, multiplying, and potentially working with exponents. As your child studies pre-algebra, they will learn how to add, subtract, multiply, and divide these problems, as well as learn the rules for exponents.
When your child advances to high school, they will be required to have a working knowledge of all the concepts covered in pre-algebra. They will need to apply these concepts in math courses and some science ones, like physics, chemistry, and earth science. Moreover, they will also find the mathematic concepts mentioned have practical uses in the real world. By studying these concepts, you and your child will be working to prepare them for any upcoming pre-algebra tests, and for their mathematical future. | 677.169 | 1 |
Mathway
Mathway is a helpful math app for solving all kinds of mathematics problems. The app will solve basic math problems, calculus, linear algebra, statistics and even math problems in Chemistry. The great thing about Mathway is that it will show users the steps to solve the problem so they can check their answers and make sure that their mathematical thinking process is correct. Mathway is available for iOS users, Android users, or can be accessed on the Mathway website. This app can be really helpful for students to reinforce mathematical thinking and problem solving. | 677.169 | 1 |
Graphing Basic Logarithmic Functions Worksheet!
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this file type before downloading and/or purchasing.
245 KB|2 pages
Product Description
On this worksheet you will find basic practice of graphing logarithmic functions where b > 1 and 0 < b < 1. Students will conceptualize what happens to the graph as the base changes and identify a student error. You can use this worksheet as classwork or a homework assignment on graphing logarithms! | 677.169 | 1 |
Polynomials Interactive Flip Book
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this file type before downloading and/or purchasing.
1 MB|40 pages
Product Description
Polynomials Interactive Flip Book
Teachers- gone are the days of providing multiple copies for lost worksheets! The Interactive Flip Book is a simple way of keeping students organized with everything they need for a Polynomials unit in one place.
Absolutely no teacher prep needed- just print and distribute!
This interactive flip book contains 18 pages of Guided Notes and Practice Problems, along with answer keys, for the following concepts: | 677.169 | 1 |
MATH 180 The Art of Mathematics (3 credits)
Overview
Mathematics & Statistics (Sci) : An overview of what mathematics has to offer. This course will let you discover the beauty of mathematical ideas while only requiring a high school background in mathematics. The topics of the course may include: prime numbers, modular arithmetic, complex numbers, matrices, permutations and combinations, probability, set theory, game theory, logic, chaos. Additional topics may be covered depending on the instructor. | 677.169 | 1 |
When you have completed the practice exam, a green submit button willappear.Click it to see your results. Question 2 2. Question 5 5. The topics are definitely more detailed and complex and students are expected to keep up with long classes which never seem to end. The good news for all the college students looking for a way out of their mathematics woes is that online help is available for college level math problems, anywhere, anytime.
College algebra math problems can be especially difficult. One of the best ways to improve is to practice college math problems online. Math help websites have a repository of college math problems and answers which students can keep solving and figure out how to tackle different problems. College math word problems cover a variety of topics and use familiar situations and scenarios that aid the learning process. College Level Math Problems College math practice problemsA set of college algebra problems on multiplying,dividing and finding polynomials are presented.
Find the other zeros Most of the classes have practice problems with solutions available on the practice problems pages. Notes File Size: 2.14 MB Last Updated: Thursday December 1, 201Practice Problems File Size: 28 KB Last Updated: Monday May 30, 201Practice Problems Solutions File Size: 1.8 MB Last Updated: Monday May 30, 201Assignment Problems File Size: 370 KB Last Updated: Monday June 27, 201.
If the equations are overlapping the text (they are probably all shifted downwards from where they should be) then you are probably using Internet Explorer 10 or Internet Explorer 11. T. | 677.169 | 1 |
Showing 1 to 6 of 6
Foundations of Mathematics 12 5.6
5.6 - INDEPENDENT EVENTS
Goal: Understand and solve problem: that involve independent events.
Independent Events
Two events are independent if the knowledge Ihal one event has occurred has no effect on the probability of
Lesson Notes 4-3
Compound Interest Part II
The compound interest formula has several equivalent forms that you can use to solve
compound interest problems.
To determine the future value, use A = P(1 + i)n.
To determine the present value, or principal, use
Year End Review: Set Theory (Unit 1)
It is important to be able to determine the inverse, converse, contrapositive, and
biconditional of statements.
Inverse
Contrapositive
Converse
Biconditional
Example 1: Determine the inverse, converse, and contrapo
Lesson Notes 8-1
Understanding Angles
A central angle of a circle is an angle that is contained between two radii. For example,
in the following diagram angle A is the central angle.
A
A radian is a measure of the central angle of a circle subtended by an
Lesson Notes 3-2
Probabilities Using Counting Methods
Recall, the probability is defined as the number of favorable outcomes divided by the
total number of outcomes. Sometimes we will have to determine the number of
outcomes using combinatorics.
Example 1
Lesson Notes 6-4
Modeling Data With a Line of Best Fit
A scatter plot is a set of points on a grid, used to visualize a relationship or possible trend
in the data. If the points on a scatter plot seem to follow a linear trend (ie. form
approximately a lin | 677.169 | 1 |
This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modernAn innovative and appealing way for the layperson to develop math skills--while actually enjoying it Most people agree that math is important, but few would say it's fun. This book will show you that the subject you learned to hate in high school can be as entertaining as a witty remark, as engrossing as the mystery novel you can't put down--in... more...
Handy compilation of 100 practice problems, hints and solutions indispensable for students preparing for the William Lowell Putnam and other mathematical competitions. Preface to the First Edition. Sources. 1988 edition. more...
Any child can overcome the disadvantages of mediocre math teaching in school and parental math anxiety at home. Math Power offers easy-to-follow and concrete strategies for teaching math concepts. These lively techniques — including games, questions, conversations, and specific math activities — are suitable for children from preschool to age 10.... more...
Over 300 challenging problems in algebra, arithmetic, elementary number theory and trigonometry, selected from Mathematical Olympiads held at Moscow University. Only high school math needed. Includes complete solutions. Features 27 black-and-white illustrations. 1962 edition. more... | 677.169 | 1 |
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