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Course Overview
In this 6th Grade math course, students enhance computational and problem-solving skills while learning topics in algebra, geometry, probability, and statistics. They solve expressions and equations in the context of perimeter, area, and volume problems while further developing computational skills with fractions and decimals. The study of plane and solid figures includes construction and transformations of figures. Also in the context of problem solving, students add, subtract, multiply, and divide positive and negative integers and solve problems involving ratios, proportions, and percents, including simple and compound interest, rates, discount, tax, and tip problems. They learn multiple representations for communicating information, such as graphs on the coordinate plane, statistical data and displays, as well as the results of probability and sampling experiments. They investigate patterns involving addition, multiplication, and exponents, and apply number theory and computation to mathematical puzzles.
Course Outline
SEMESTER 1
Unit 1: Problem Solving
Mountain climbing involves solving different kinds of problems. Just like solving math problems, climbing requires tools and a solid strategy. In this unit, you will learn about number lines, the order of operations, and problem solving. To solve problems, you will learn how to translate between words and math symbols, and you will use strategies such as drawing figures, estimating, and breaking a problem down into smaller parts. You will also learn how to handle precision and reasonableness.
Semester 1 Introduction
Foundations
On the Number Line
Order of Operations
Number Properties
Translating Between Words and Math
Translating Mixed Operations
Problem-Solving Strategies
Getting to the Core: Problem Solving
Identifying Information in Word Problems
Precision
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 2: Distance: Addition and Equations
If a farmer has painted part of a fence, how much more does she need to paint? Addition equations can help the farmer solve a problem like that one. In this unit, you'll learn how to use units to measure distance and perimeter. You'll also solve addition and subtraction equations and discover how those equations can give rise to the idea of negative numbers. Finally, you will use absolute value and operations with positive and negative numbers to solve problems.
Foundations
Units of Distance
Polygons and Perimeter
Addition and Subtraction Equations
Applications of Addition and Subtraction Equations
Getting to the Core: Addition and Subtraction
Your Choice
Negative Numbers
Absolute Value and Distance
Addition and Subtraction with Negative Numbers
Getting to the Core: Negative Numbers
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit 3: Area: Multiplication Equations
A general contractor needs to calculate area to determine the amount of wood for a floor. In this unit, you'll learn how to compute the areas of squares, triangles, rectangles, and other polygons. You will also learn how to divide to find an unknown side length and how a square root relates a side length to the area of a square.
Units of Area
Areas of Rectangles
Special Quadrilaterals
Getting to the Core: Similar Parallelograms
Your Choice
Areas of Triangles
Figures Made Up of Triangles and Parallelograms
Unknown Side Lengths: Division
Getting to the Core: Modeling by Restructuring
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 4: Working with Rational Numbers
Most two-by-fours are actually about 1-1/2 inches by 3-1/2 inches. Any carpenter working with lumber is also working with rational numbers.In this unit, you will learn how to change between various representations of rational numbers including equivalent fractions and decimals. You'll also add, subtract, multiply, and divide rational numbers and use these skills to solve practical problems.
Foundations
Primes and Composites
Using Prime Factorization
Equivalent Fractions
Representing Rational Numbers
Comparing Rational Numbers
Your Choice
Perimeters with Fractions
Areas with Fractions
Dividing Fractions
Solving Problems with Fraction Division
Getting to the Core: Fractions
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 5: Solids
When shipping merchandise, you need to know the volume of the container to determine how much it will hold. In this unit, you will learn how to find the volume and surface area of shapes such as prisms and pyramids. You will also find out how a cube root connects the volume of a cube to its side length.
Foundations
Cubes and Cube Roots
Volumes of Prisms
Nets of Solids
Getting to the Core: Measuring Volume
Your Choice
Surface Area: Prisms and Pyramids
Properties of Volume and Surface Areas
Getting to the Core: Volumes and Surface Areas
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 6: Comparisons: Ratios
In southern Asia and South America, some mosquitoes carry a disease called malaria. How can you compare how efforts to fight the disease are progressing in various countries? Scientists and doctors use ratios to understand many problems. In this unit, you will use ratios and proportions to solve many different problems. For instance, you will compute interest on loans, as well as calculate taxes, tips, and discounts.
Foundations
Ratios as Comparisons
Percent
Finding Percents of Numbers
Your Choice
Getting to the Core: Understanding Ratio and Percent
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 7: Semester Review and Test
Semester Review
Semester Test
SEMESTER 2
Unit 8: Statistics
Every jelly bean can be described. Each one has color, flavor, mass, and number of calories. The language and tools of statistics help to describe buckets full of data. In this unit, you will learn how to create and interpret statistical graphs including circle graphs, bar graphs, line plots, line graphs, box-and-whisker plots, and histograms. You'll also learn how to calculate and interpret measures of center and variation. Finally, you will learn how sampling can help you make decisions about a population.
Semester 2 Introduction
Foundations
More Statistical Graphs
Histograms
Getting to the Core: Understanding Data Displays
Your Choice
Measures of Center
Box-and-Whisker Plots
Getting to the Core: Distribution of Data
Measures of Variation
Statistical Claims
Getting to the Core: Interpreting Data Sets
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 9: The Second Dimension
Scientists can use data to figure out how tall someone was from a single bone. When you have two variables, such as femur length and overall height, a two-dimensional plot can help you see patterns and make predictions. In this unit, you will learn how to identify and plot points on a coordinate plane. You'll then identify points that are solutions to equations with two variables and create and interpret scatter plots.
Foundations
Points on a Coordinate Plane
Using Points to Solve Problems
Equations with Two Variables
Getting to the Core: Reflecting Points on a Coordinate Plane
Getting to the Core: Coordinate Plane
Your Choice
Scatter Plots
Interpreting Scatter Plots
Figures on a Coordinate Plane
Getting to the Core: Polygons on the Coordinate Plane
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 10: Rates
On average, about 1088 cubic meters of water flow over southern Africa's Victoria Falls every second. That's more than 1,000,000 liters, or enough to fill 26 Olympic-sized swimming pools every minute! In this unit, you will calculate and use rates to solve many types of problems including pricing, speed, and work problems. You'll also use direct variation and see how rates affect graphs of relationships.
Foundations
Rates as Comparisons
Unit Rates
Solving Unit-Rate Problems
Getting to the Core: Another Look at Unit Rates
Your Choice
Average-Speed Problems
Constant-Rate Problems
Getting to the Core: Another Look at Constant Rates
Direct Variation
Interpreting Direct Variation
Getting to the Core: Another Look at Direct Variation
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 11: Working with Positives and Negatives
In the stock market positive and negative numbers are key to understanding how companies stocks are valued. In this unit, you will learn how to add, subtract, multiply, and divide positive and negative numbers including decimals. You'll also work with inequalities.
Foundations
Adding and Subtracting Signed Numbers
Net Gains and Losses
Getting to the Core: Addition/Subtraction of Signed Numbers
Your Choice
Multiplying Signed Numbers
Dividing Signed Numbers
Exponents and Patterns
Getting to the Core: Multiplication/Division of Signed Numbers
Properties of Signed Numbers
Inequalities
Getting to the Core: Number Properties and Inequalities
Unit Review 1
Unit Review 2
Unit Checkpoint 1
Unit Checkpoint 2
Unit 12: Probability
People who play on and coach sports teams, like baseball, as well as those who follow the teams, deal with uncertainty all the time. Probability provides the tools to understand and communicate this uncertainty. In this unit, you will learn how to use Venn and tree diagrams to count the number of ways a trial can be conducted. You can use a diagram to calculate a theoretical probability. You'll also learn how to use experimental probability and the law of large numbers. Finally, you'll learn about independent, dependent, and complementary events.
Foundations
Counting
Probability and Experiments
Experimental Probability
Theoretical Probability
Your Choice
The Law of Large Numbers
Independent and Dependent Events
Complementary Events
Unit Review
Unit Checkpoint
Unit 13: Making and Moving Figures
Two men from Southampton, England, say that they used only planks, rope, hats, and wire to make the first crop circles in the 1970s. Crop circle designs range from the simple to the complex, but anyone who makes crop circles needs to know about circles and transformations. In this unit, you will construct and transform figures. For constructions, you will use paper folding as well as a compass and a straightedge. For transformations, you will use coordinates and other methods.
Foundations
Folded-Paper Construction
Compass and Straightedge Construction
Your Choice
Translation
Reflection
Rotation
Translating with Coordinates
Reflecting with Coordinates
Unit Review
Unit Checkpoint
Unit 14: Semester Review and Test
Semester Review
Semester Test | 677.169 | 1 |
This booklet is a Mathematica notebook. It can be found online at
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È Outline of the course
Session 1: common and useful built-in Mathematica functions; variable assignment and function definition; the Front End and the Kernel; Notebooks. Session 2: organisation of data in Mathematica; lists and expressions; simple programming; functions; nesting. Session 3: the opportunity to develop your proficiency as a Mathematica user through work on an extended problem; Case Studies; Mathematica resources on the Internet.
È Rationale
We're not attempting an exhaustive survey of Mathematica's capabilities: we couldn't come close to doing justice to that task in the time we have. Equally, there are dozens of specialised uses for Mathematica (in pure and applied mathematics, physical science, engineering etc.) that we can't hope to address here (though some are touched on in our "Case Studies": see below). Instead, we focus on the key elements of the Mathematica system and how the system is used. These course notes are not intended as a substitute for the manual, which is The Mathematica Book (Cambridge University Press, Third Edition, 1996), by Stephen Wolfram. The entire contents of the manual, and more, are available on Mathematica's extensive online Help system, which you should certainly take time to explore. In addition to these course notes we have prepared some Case Studies, or "common tasks in Mathematica for the academic user". These are an attempt to address just a few of the more specialised roles in which Mathematica is used. This booklet includes information about the various free sources of Mathematica information on the Internet, and how to get in touch with the world-wide Mathematica user community. There are numerous other books besides the Wolfram "manual" about Mathematica itself, and its use in mathematics, science, engineering and finance (and some of these are available in other languages).
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È Session 1
In this session, we explore the arithmetical, symbolic and graphical capabilities of Mathematica. We cover global variable assignment and local variable substitution, and introduce you briefly to the idea of defining your own Mathematica functions (to be covered in more depth in Session 2). We also explore some key characteristics of Mathematica's user interface. The final section (1.6) on Notebooks is optional: you may prefer to skip it for now and come back to it at a later time. Each section consists of a piece of text followed by some exercises. Exercises marked with a five-pointed star (à) are central, and you should do all of these if you have the time. The other exercises, while useful, are more peripheral and can be skipped if necessary. All Mathematica code is printed in these notes in Courier Font. We have used the ellipsis mark ". . ." to indicate where code has been missed out.
ì Getting help
If you get stuck, here are some ways to recover:
ì Use the Help systems, which are especially useful for finding out about Mathematica functions.
There is the system activated from the Help item in the menu (this gives access, amongst other things, to the entire Mathematica user manual), or you can use the special query character ? to get information about any function for example:
? Sqrt
ì You can do "wildcard" searches as well. The following queries ask Mathematica to list all the
function names beginning with, or ending with, Plot, respectively:
? Plot* ? *Plot
ì If all you need is a reminder of the syntax of a command, type the command name, then hold down
the shift and control keys and type K, for example
Plot < shift - control - K >
ì If a command doesn't work properly check the error messages (in blue text). ì If your input is simply returned unchanged, with no error messages to help, it means that
Mathematica is unable to do anything with what you have typed. Check that you have spelt the command correctly, that the number of inputs is correct, that you haven't left out any commas, and that the types of the inputs (integer, real number, symbol, and so on) are appropriate. These are the most common causes of this error ì If Mathematica seems to have stopped, Abort the calculation or (more drastic) Quit the Kernel, using the Kernel menu.
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ì If everything seems to have gone wrong Quit or Exit from Mathematica (via the File menu) and start
again. It's a good idea to Save your work as you go along so that you can recover from these situations.
ì 1.1 Arithmetic
At its simplest, Mathematica can be thought of as a highly sophisticated calculator. Like a calculator, it does arithmetic, for example:
2+5 2*5 2 5 2^5 100! Sin@Pi 3D Sqrt@50D 2 ^ H1 + 4L Log@2, %D
etc. To get Mathematica to perform a calculation, hold down the shift key and press return (on some keyboards called enter or ↵). The shift-return operation sends "instructions" from the interface where you' re typing to the "engine" of Mathematica for processing: see Sections 1.6–1.7 for more about this. So to lay out code more clearly on the screen you can use "return" characters. You'll notice right away two peculiarities of the syntax. One is that the names of all Mathematica functions, variables and constants begin with capital letters. This is important: Mathematica is completely case-sensitive, and it will simply be unable to interpret, for instance:
sin@pi 3D
The other is that square brackets, [. . .] and round parentheses, (. . .) are both used in Mathematica, but not interchangeably. The former are always used to enclose the arguments of functions such as Sin. The latter are used only for the purpose of grouping expressions, either algebraically, as here, or procedurally, as you'll see in Session 2. Otherwise, the notation for arithmetic is straightforward, except to note that a space can implicitly mean multiplication. The percent sign, %, is used to mean "the last output" (so the final expression in the above list will calculate the logarithm to base 2 of 32). You may have noticed that all inputs and outputs are numbered and can be referred back to using those numbers (see Section 1.6).
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Note, too, that some of the above calculations can be laid out in a way that corresponds more closely to conventional mathematical notation, by using the Basic Input palette. This will probably have appeared automatically on the right of your screen. If it hasn't, find it in the File menu under Palettes. For example:
25
p SinA €€€€€ E 3 "!!!!! ! 50
21+4
and so on. This is especially useful when you need to build up large expressions. All the above are examples of exact arithmetic, carried out using whole numbers, rationals, surds or rational multiples of constants such as Pi. Mathematica will also perform approximate, floating-point arithmetic of the sort that conventional calculators can do. Numbers entered with a decimal point are interpreted as approximations, even if they're integers, and all other numbers in the same expression (with the exception of symbolic constants such as Pi) will be converted to this latter form. For example:
100.0!
"!!!!!!!!!! 50.0
3.35759100
To force Mathematica to convert exact expressions to decimal ones, you can use the N command, as in:
"!!!!! ! NA 50 E
N@Sin@p 3DD
"!!!!! ! NA 50 , 25E
N@p , 200D
The last two cases illustrate one way in which Mathematica can be made to work to arbitrary precision. Mathematica will handle complex numbers as well as real ones: see the exercises for some examples.
ì Exercises 1.1
à 1. Type in and test all the code in this section. à 2. Try the following:
ì 1.2 Algebra and Calculus
As well as being an arithmetical calculator, Mathematica is also an algebraic one. For example:
Expand@Hx + 2 yL2 Hx - 3 yL5D Factor@%D
For more on the manipulation of algebraic expressions, see the exercises. Equations in Mathematica are set up using a double equals sign, "==": this is because the single equals sign has a different meaning, which we introduce later on. The Solve command tries to find exact solutions to algebraic equations:
Solve@x2 - 3 x + 2 == 0, xD Solve@x4 - 3 x3 + 5 x2 - 11 x + 2 == 0, xD Solve@8x + 4 y == 5, 2 x - y == 8<, 8x, y<D
Notice the use of curly brackets — braces — in the last Solve command. Curly brackets are used in Mathematica to group pieces of data together, forming structures called lists. These are studied in more depth in Session 2. For the moment, it is enough to note the kinds of circumstances when lists crop up. Here, we need to group the two equations, x + 4y = 5 and 2x – y = 0, and the two unknowns, x and y. For equations that do not have exact solutions, or for those whose exact solutions are unwieldy (such as quartic polynomials), there is the NSolve command which operates using a sophisticated repertoire of numerical methods :
NSolve@x7 + 3 x4 + 2 == 0, xD NSolve@x4 - 3 x3 + 5 x2 - 11 x + 2 == 0, xD
2. Use Mathematica to find all the solutions in the complex plane of the equation cos z = 2.
1 3. Use Mathematica to express €€€€€€€€€€€€€ in terms of its real and imaginary parts, where z = x + i y, and x z+1 and y are real.
Then select the fraction inside the integral, and click on the Apart @ôD button. With the same piece of text selected, click on Together @ôD . Try using Expand @ôD on the numerator, and so on. Explore further. Investigate, too, the use of the Evaluate in Place instruction (under Evaluation in the Kernel menu). 6. Type
Sum@1 r ^ 2, 8r, 1, 6<D
or
6 1 j z Ê i €€€€€€€ y 2 r=1 k r {
Try summing from 1 to 20. Express the sum as a decimal. Try summing from 1 to n, and from 1 to infinity (Infinity in Mathematica, or use the symbol from the Basic Input palette). 7. Solve the ordinary differential equation
ì 1.3 Assignment, substitution and function definition
As you've seen, the percent sign, %, gives us a useful way of referring to earlier output. And in fact you can refer to any output in this way using its "In/Out" number—see Section 1.6. However, it's inadvisable to rely on % in this way. The principal drawback is that if you save your work and call it up again, or even if you need to edit or debug work you've already done, the sequencing on which % depends can be disrupted. It's better instead to get into the habit of naming things which it's likely you'll want to use again, like this:
2x expression1 = €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ € H1 + x2 L H1 + xL expression2 = Apart@expression1D expression3 = Together@expression2D TrueQ@expression1 == expression3D
An important thing to note is the use of the single equals sign, =, in commands such as
2x expression1 = €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ € 2 L H1 + xL H1 + x
2x which means "let the symbol expression1 have value €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ". This is to be distinguished from H1 + x2 L H1 + xL
the double equals sign, ==, which, as you've seen, is used to set up equations. The final command,
TrueQ@expression1 == expression3D
means "test whether the equation expression1 == expression3 is true for all values of the variable or variables". Notice the final Q in the function name: this is a convention for logical functions (those whose output is True or False). Assigning values to symbols in this way is clearly very useful: indispensable, in fact. But one thing that's a disadvantage in some circumstances is that these assignments are completely global: unless we take steps, 2x the symbol expression1 will continue to call up the value €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ in whatever future context we H1 + x2 L H1 + xL use it. This is not irreversible: we can make expression1 into an unassigned symbol again by clearing its value:
Clear@expression1D
We could also quit our Mathematica session: that will clear all assignments pretty effectively, and leave everything clear for our next go! But these approaches are all fairly cumbersome, and it's sometimes more appropriate to avoid global assignments of this type and opt for local substitution instead.
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Compare the following pieces of code, each of which aims at finding the value of the expression x2 – 5x + 9 at x = 3. Here's the first one:
x = 3 x2 - 5 x + 9 Clear@xD
Here's the second:
x2 - 5 x + 9 . x -> 3
In the first, it's clear what we've done: the value 3 has been assigned to the symbol x, and the quadratic expression evaluated; finally, the symbol x has been cleared. The second piece of code is more obscure: it means "evaluate the expression x 2 – 5x + 9 subject to the local substitution x = 3". The "/." is a shorthand for the ReplaceAll command. It is not necessary to clear x afterwards, since x has never been assigned any value. Instead, all occurences of x in the expression x 2 – 5x + 9 have simply been replaced by 3, with no permanent effect on x at all. The structure x -> 3 is an example of what's called a rule. You may recall that the output from Solve is generally in the form of a list of rules (more about that in Session 2, and Case Study 6). A related idea to assignment is function definition. Here's an example:
Clear@xD f@x_D := x2 - 5 x + 9
What's happened here is this: the symbol x has been cleared, in case it had any value attached to it, and the function f has been defined, such that f HxL = x2 - 5 x + 9. We can now use this function like any built-in one, for example:
f@3D f@zD D@f@zD, zD f'@xD
Notice that we've used the compound symbol := instead of = in the definition This is almost always appropriate for function definition, and = is almost always appropriate for variable assignment, though this is more complex than it seems and exceptions do exist. Notice, too, that on the right-hand side of the defining statement the underscore symbol, _, has been used. This gives x the status of a placeholder or dummy variable, standing for all possible arguments. If you want to explore what happens when you leave the underscore out, try typing
Clear@x, fD f@xD := x2 - 5 x + 9
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f@xD f@3D f@zD D@f@xD, xD D@f@zD, zD f'@xD
Working with your own functions in Mathematica always involves two distinct stages: first you define the function, using the underscore character and (usually), "colon-equals". After that, Mathematica has "learnt" this new function, and for the rest of your session you can use it in just the same way as inbuilt functions such as Sin and Sqrt. Note that the first step, defining the function, doesn'tgenerate any output; this can be disconcerting the first few times you see it. Our f in the above example corresponds exactly to a "function" in the mathematical sense. But in Mathematica, the term is rather broader. For example, the following is a "function" for comparing two expressions and deciding whether they appear to be algebraically equivalent (as far as Mathematica can make out):
algEquivQ@expr1_, expr2_D := TrueQ@Simplify@expr1 - expr2D == 0D
Notice that we've made all our functions start with lower case letters. This is a good idea in general, to avoid clashes between your own functions and internally defined Mathematica ones and to make it clear, to yourself and other users, which is which.
ì Exercises 1.3
à 1. Type in, and test, the first section of code, which assigns values to the symbols expression1, expression2 and expression 3 and tests the equivalence of expression1 and expression3. Find, in turn, the value of each of these expressions when x is 5: do this by assignment and by local substitution.
Implement expression1 as a function of x, and check that this function evaluates to what you would expect at 5.
à 2. Define the function algEquivQ as in the text. Test it on the pairs:
(i)
x2 + 2 x + 1 and H x + 1L2 ;
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(ii) (iii)
y +5 y+6 €€€€€€€€€€€€€€€€€€€€€€€€€€€€€ and y + 2; y+3
2
cos 2 t and cos2 t - sin2 t.
Try to find an equivalent pair for which algEquivQ fails. How about if you use FullSimplify instead of Simplify? 3. Write, and try out, your own function called equalAtQ, which tests whether two expressions in the same variable have equal value at a given value of the variable. Thus
equalAtQ@2 x2, 6 x, 8x, 3<D
(Notice again the use of curly brackets to form lists.) It's important to bear in mind that Plot always assumes that graphs are continuous. Functions with asymptotes will often come out wrong, therefore, with
These use the substitution rules you met in Section 1.3. For a complete list of options for Plot together with their default settings, type:
Options@PlotD
For all that and more, type:
?? Plot
Note that many Mathematica functions feature option settings in this way: they are by no means confined to graphical functions such as Plot. Options are an important way of building in flexibility, and you can do this with your own functions too (see the manual: Section 2.3.10). The function ParametricPlot can be used to generate plots of pairs of parametric equations, as in
ParametricPlot@8t + Sin@tD, 1 + Cos@tD<, 8t, 0, 4 p<D
The function ListPlot can be used to generate plots of sets of coordinates structured as lists, as in:
ListPlot@ 880.0, 1.2<, 81.0, 2.9<, 82.0, 5.3<, 83.0, 7.0<, 84.0, 8.8<<D
The principal three-dimensional plotting functions are Plot3D, ParametricPlot3D and ContourPlot (the last-named produces a two-dimensional contour plot of a function of two variables). These are explored further in the exercises for this section.
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Notice, by the way, the way one of the inputs above is broken over two lines, to fit within the width of the page. Mathematica does this automatically, or you can override the default line breaks using the "return" key.
ì Exercises 1.4
à 1. Type in, and test, all the code in this section. Note down in particular the effect of all the option settings for Plot. Explore this further if you need to. Test the effect on the ListPlot command of the options PlotJoined -> True and PlotStyle -> PointSize[0.03 ].
2. Use If to define a function called unitStep, which evaluates to 0 for inputs equal to 0 or less, and to 1 otherwise. Generate a plot of this function for - 3 ‹ x ‹ 3. 3. Use the Show command to generate a figure showing a straightforward function plot of the curve y = x2 on the same axes as a parametric plot of the curve x = y2 . The scales should be the same on either axis. 4. Write a function called plotWithInverse, such that, for example
plotWithInverse@x2 - x4, 8x, - 3, 3<D
See Case Study 3 for how contour and surface plots may be combined. 6. Generate a plot of the surface u = x2 + y2 . Generate, too, a parametric plot of the unit sphere. Show these two figures on the same diagram.
ì 1.5 Data Fitting
Although Mathematica is not a dedicated statistical package, it does come with a large set of statistical capabilities. There is not nearly enough time to cover them all on this course, so we focus on one that colleagues find especially useful, namely regression and data fitting. This is used where you have some data (from an experiment, say) and a mathematical model that you are pretty confident describes your data, but that contains some constants (known as parameters) whose values you do not know but wish to estimate. For example, suppose that we have the following data...
Suppose, too, that we believe the data to come from a model of the form y = a + b x + c x2 , but that we do not know the values of the constants a, b and c. Now, experimental data always has random errors associated with it. We're looking, therefore, for an expression of the form a + b x + c x2 that, while it is unlikely to fit the data perfectly, is the best fit available. Mathematica can generate this "best fit" expression:
bestFit1 = Fit@data1, 81, x, x2 <, xD
We can generate a plot of this function...
curvePlot1 = Plot@bestFit1, 8x, 0, 1< D
... and superimpose this on the original data:
Show@dataPlot1, curvePlot1D
ì Exercises 1.5
à 1. Type out and test all the code in this section.
is believed to come from a law of the form y = a x + b x. Use the data to estimate the values of a and b, and generate a plot of the best fit curve superimposed on the data.
à 3. Mathematica has a function called MultipleListPlot that is specially designed for plotting statistical data of this type. However, like many "specialist" commands it is only accessible if you load the library package that contains it, which you can do by typing
Illustrate them both by typing
multPlot = MultipleListPlot@data3, data4D
You might need to enlarge the plot to see the distinct point symbols Mathematica has used to distinguish the two data sets. Find the best fit straight lines for each data set and show graphs of them both on the same diagram as your points. 4. For additional statistical analysis (standard errors, t-statistics and so on) Mathematica has a function called Regress, that you can load by typing
<< Statistics'LinearRegression'
Try
Regress@data1, 81, x, x2 <, xD
You can find out more about this function using the help system if you need to.
à 5. The Fit command works whenever the model we are trying to fit to the data has the form
ì 1.6 The Front End and the Kernel
Mathematica isn't just one program: it's two. When you click on the Mathematica icon, what gets loaded is the Front End: the part of Mathematica that handles things like screen display of input and output, printing and the creation of files. When you do your first calculation, the Kernel gets loaded: the "calculating engine" of Mathematica. For this reason, the first calculation always seems to take a very long time. Many users get into the habit of kicking off with something innocuous like
2+2
then going on to more serious calculations once the Kernel is up and running. It's possible to evaluate code that is already present, or to edit and re-evaluate: simply click in the text, edit if necessary, then "shift-return" in the normal way. The detached relationship between the Front End and the Kernel is not very common for computer programs these days, and can take some getting used to. Those readers who used computers in the pre-micro days of the 1970s-80s will be familiar with the idea of typing locally, sending text down a wire to be processed by a mainframe computer elsewhere, and getting output back through the wire. Indeed, you could think of "shift-return" as a "send" instruction. So, what happens when you press "shift-return" is the following: the text (Mathematica command) that you' ve typed is sent to the Kernel as input, and it sends back the output from processing your command, which appears just underneath the input, and a "line number" which appears as a label for both input and output (the In[. . .] and Out[. . .]). It's worth knowing that whereas the percent sign, %, means "the last output", %61, say, means "output number 61" of the current Kernel session. The Front End-Kernel split may at first sight seem to be no more than a complication. But it makes Mathematica very powerful and useful in a number of ways. First, the Front End and Kernel need not be located on the same machine, so you can use Mathematica in the comfortable environment of your personal PC or Macintosh whilst exploiting the computational power of a remote workstation. Instructions for setting up remote links like this depend on what platform (computer system) you're using. Secondly, the Front End is designed to offer a sophisticated document interface, and this is claimed (by the developers, and with a certain amiunt of justice) to be of professional word processor quality. In the next section we describe the most basic features of "notebook" documents. We've labelled the section "advanced", meaning that it's not essential to go through it now, but it's advisable to do so at some point.
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ì 1.7 Advanced Topic: Notebooks
Figure 1: excerpt from a Mathematica notebook As you work in Mathematica, a document is built up. Known as a notebook, this is a bit like a word processing document in a Windows or Mac application such as Word or WordPerfect: you can save it, you can select, cut and paste within it, you can mouse to different points in the text and edit in place, etc. But notebooks are in some ways more complex than word-processor documents, because of the different roles text can play. Often, perhaps most of the time, you'llsimply want to be typing code to evaluate. But you may also wish to add annotations or explanations, or to set up titles and section headings, and Mathematica needs a way of distinguishing these "inactive" forms of text both from one another and from "active" code. It does this by dividing the text in the notebook into disjoint cells, marked by square brackets in the right margin. In this way, you can build up complex documents of the type shown in Figure 1. By default, Mathematica assumes that anything you type is code. To change that assumption, click on the cell bracket in the right margin and choose Style from the Format menu. You can then choose an appropriate style. If you wish to change the size, font or alignment associated with a cell style, or even within a particular cell, this is entirely possible. You'll notice from Figure 1 that brackets around brackets exist, covering more than one cell. These define what are called cell groups. In all versions, a piece of input will be automatically grouped with its output, and later versions allow even more automatic grouping. To manually group a collection of cells (or of already existing cell groups), select all their cell brackets and choose Cell Grouping (or, in earlier versions,
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Group Cells) from the Cell menu. There's no limit to the depth of the hierarchies you can build up in this way.
Figure 2: Excerpt from a Mathematica notebook, partially closed A group can be closed by double-clicking on its grouping bracket; this hides all the cells except the first. It' s often helpful to hide large collections of cells behind section headings: this allows documents to be skim-read for contents without scrolling right through them. A closed cell group can be opened by double-clicking on the grouping bracket (which is distinguishable by a small hook). Figure 2 shows a partially closed version of the notebook in Figure 1, with the grouping bracket selected. Notebooks, then, are complex documents. Their management is the task of the Front End, which therefore has to handle multiple types of text organised in complicated hierarchical ways. By contrast, all the Kernel does is keep a strictly chronological record of your calculations, whose ordering is reflected in the In[. . .] and Out[. . .] messages you see. So you have to be careful: just because a certain calculation comes last in the notebook doesn't mean it's the latest one as far as the Kernel's concerned.
ì Exercises 1.7
1. Start a fresh Mathematica notebook and reproduce Figure 1. By closing the appropriate cell group, reproduce Figure 2 as well. 2. Experiment with local style changes: reproduce a Text cell that looks more or less like this: This is a Text cell, but one in which I have experimented with various IRQWV, sizes and typefaces. 3. Experiment with style sheets: with (say) your "Figure 1" notebook on screen, find Style Sheet in the Format menu, and try several options. 4. Experiment with the various options under Format – Screen Style Environment . 5. The following excerpt (Figure 3) from a Mathematica notebook seems to show something going wrong. Examine it carefully, and explain why the Clear[x] command doesn't seem to have worked. What has really happened here?
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Figure 3: Excerpt from a Mathematica notebook
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È Session 2
In this session, we introduce the expression, the principal Mathematica data structure, and the list, one of its most useful manifestations. We examine the way Mathematica handles matrices and vectors. We look again at the idea of defining your own Mathematica functions, and explore different approaches to that task, focussing on the commands Do, While, Map, Apply, Nest and FixedPoint. There is optional material in Sections 2.5–2.7 on pure functions, local scoping of variables and recursive function definition. As in Session 1, each section consists of a piece of text followed by some exercises. Exercises marked by a à are especially important or central, and you should do all of these if you get the time. The other exercises, while useful, are more peripheral and can be skipped if necessary.
ì 2.1 Lists and expressions
The standard way of storing multiple items of data in Mathematica is the list. An example might be {1, x2 , 0.937, 3 + 2I, Factorial} (admittedly a rather artificial one). Notice that there are no restrictions on the type of data you can hold in a list: here, for example, each data item is of a different type. Defining your own lists is easy. You can, for example, type them in full, like this:
oddList = 81, 3, 5, 7, 9, 11, 13, 15, 17<
Alternatively, if (as here) the list elements correspond to a rule of some kind, the command Table can be used, like this:
oddList = Table@2 n + 1, 8n, 0, 8<D
We can pick out, say, the 5th element in oddList by typing:
oddList@@5DD
(Note the double square brackets. This is a shorthand notation; the function we've used here is indexed in the manual under its "full" name, Part). We can generate a list containing the elements of oddList in reverse order by typing:
Reverse@oddListD
We can add new elements to lists by using Append or Prepend, as in:
Append@oddList, 19D Prepend@oddList, - 1D
See the exercises for some more commands that are useful when handling lists. Lists are important things, but there's a sense in which there's nothing very special about them: they're simply an example of what's called an expression. Internally, Mathematica represents oddList not the way we see it on the screen but like this:
List@1, 3, 5, 7, 9, 11, 13, 15, 17D
You can see this internal representation if you type
FullForm@oddListD
In a similar way, the internal representation of the equation
x == y == z
is
Equal@x, y, zD
So the essential structure of equations is exactly the same as that of lists. The same is true of virtually anything we can type into, or get out of, Mathematica. As it says in the manual: "everything is an expression" (section 2.1.1). This enables us to use some of the commands you've just met on things that aren't lists, as in:
eqn1 = Hx == y == zL eqn1@@3DD Reverse@eqn1D Append@eqn1, 0D FullForm@eqn1D
Lists proper have a variety of uses in Mathematica. Most simply, they are a way of grouping together data we want to keep in one place, or refer to by one name. An example of a specialised use is to represent vectors referred to a Cartesian basis. Then the scalar product of two vectors is represented, as in standard mathematical notation, by a dot, as in:
This is a good example of where Mathematica's flexibility comes in handy. It doesn't matter at all that the elements of oddList are numeric whereas those of xPowers are symbolic expressions. A matrix is represented as a list of lists, that is, as a list of the rows, each row itself a list. This is covered more fully in the exercises.
ì Exercises 2.1
à 1. As above, define
oddList = Table@2 n + 1, 8n, 0, 8<D
and try out the commands [[. . .]], Reverse, Append, Prepend and Join on it in the ways suggested. Repeat for
eqn1 = Hx == y == zL
ì 2.2 One-time code versus reusable functions
The following is some Mathematica code for calculating the population variance of a large list of randomly generated data.
thisData = Table@Random@RealD, 81000<D;
H* sample size *L n = Length@thisDataD; H* calculate mean *L total = 0; Do@total = total + thisDatathisData@@iDD - meanL2 , 8i, 1, n<D; H* return the population variance *L total €€€€€€€€€€€€€€€€€ n-1
Notice that in the first line we have suppressed the output by using a semicolon. This saves time, and the assignment still takes place. Notice, too, that any text between the symbols (* and *) is a comment, ignored by Mathematica. Finally, note the use of the "looping" command Do to make Mathematica perform the same operation several times. This code works well enough as far as it goes. But what if we wanted to calculate the variance of several sets of data? It's possible to use the above code more than once (for example, by mousing back to the relevant bit of the notebook, or by using the Cut and Paste facilities you met in Session 1). But it is awkward to do so. A much better approach is to define a function which takes a data set as input and outputs the variance (note: the following is all one command, so be sure not to "shift-return" until you get to the end!):
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myVariance@data_ListD := i j k H* sample size *L n = Length@dataD;
H* calculate mean *L total = 0; Do@total = total + datadata@@iDD - meanL2 , 8i, 1, n<D; H* return the population variance *L total €€€€€€€€€€€€€€€€€ n-1 y z {
(the brackets will size themselves automatically). Notice, again, the use of the underscore character to make the variable name data a placeholder, standing for any possible input. Three syntactical features of this definition need special remark. (1) The use of the word List after the underscore is a type declaration; it instructs Mathematica to expect data to be in the form of a list (if it isn't, Mathematica will return the expression unevaluated). (2) The use of semicolons: previously, we've introduced the semicolon as a means of suppressing output, but here its role is rather different. Put simply, when a function consists of more than one command, the commands must be separated by semicolons. It's as though the commands are being "strung together" into a single larger command with just one final output . (3) The reason for the outermost pair of parentheses (. . .) is not obvious. What they are doing is to group all the function code on the right-hand side of the :=. The myVariance function can be used like this:
thisData = Table@Random@RealD, 81000<D; myVariance@thisDataD thatData = Table@Random@Real, 100D, 8500<D; myVariance@thatDataD
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It's good to get into the habit of incorporating code you intend to reuse into function definitions. In long, complex chunks of code, this has the added advantage of making dependencies explicit, allowing you to keep track of what quantities depend on what other quantities.
ì Exercises 2.2
à 1. Type in, and test, the two versions of the variance code above.
2. Write a Mathematica function called myMax, which takes as its argument a list of numbers, and returns the maximum number in the list. Test this function.
à 3. Write a function called tangent. Your function definition should begin
tangent@expression_, x_, a_D :=
The function should return an expression in x corresponding to the tangent to the graph of expression at the point x = a. 4. The following is an attempt to write two functions: myMean, which calculates the mean of a list of data, and myVariance2, which calculates the variance of the data and calls myMean in the process.
myMean@data_ListD := in = Length@dataD; j k total = 0; Do@total = total + data@@iDD, 8i, 1, n<D; total z €€€€€€€€€€€€€€€€€ y n { myVariance2@data_ListD := in = Length@dataD; j k total = 0; Do@total = total + Hdata@@iDD - myMean@dataDL2 , 8i, 1, n<D; total z €€€€€€€€€€€€€€€€€ y n-1 {
Both of the functions work, but one of them is very inefficiently written. Identify the inefficient one, and alter the code so that myVariance2 still calls myMean, but the inefficiency has been removed.
ì 2.3 Apply and Map
The various pieces of variance code that you met in the last section all work upon lists of data. Although they differ in their details, what they have in common is that they all use the Do command to pick out elements of the list in turn and perform actions of some kind using them or upon them. But this turns out to be a pretty inefficient way of dealing with lists in Mathematica: by using "whole list" operations we can often improve execution time appreciably (by a factor of 2 or 3), and have more compact,
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elegant code . The keys to handling lists, and indeed expressions in general, as single entities (rather than picking them apart and handling their elements separately) are the commands Apply and Map. For example, suppose we want to add all the numbers in the list
oddList = 81, 3, 5, 7, 9, 11, 13, 15, 17<
One way of doing this is to do what we did in the last section, namely
total = 0; Do@total = total + oddList@@iDD, 8i, 1, 9<D; total
Or we might try:
Sum@oddList@@iDD, 8i, 1, 9<D
or
9 i=1
Ê oddList@@iDD
But the most efficient approach of all is this:
Apply@Plus, oddListD
This simply generates the expression Plus[1,3,5,7,9,11,13,17], and then evaluates it. Apply replaces the head of the expression oddList, namely List, with Plus.
Apply is the best thing to use, then, when you have a list of things that you want to combine in some way.
But what if you have a list of things that you want to treat separately, doing the same thing to each? For example, suppose we want to generate a list called squareList consisting of all the squares of the elements of oddList, in order. One way to do this is to use the Do command, like we did in the last section:
squareList = 8<; Do@squareList = Append@squareList, oddList@@iDD2 D, 8i, 1, 9<D; squareList
But it's most efficient of all to do this:
square@x_D := x2 squareList = Map@square, oddListD
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This sets up a function called square which is then applied to each of the elements of oddList separately. Here's a more complex example, where a "magnitude" function is mapped over a list of coordinate pairs:
ptsList = Table@8Random@D, Random@D<, 810<D "!!!!!!!!!!!!! mag@8x_, y_<D := x2 + y2 Map@mag, ptsListD
Using Map gives us a way of processing each of the elements in a list "simultaneously", without needing to trawl through the list element by element.
ì Exercises 2.3
à 1. Type in each of the three sets of code for adding all the elements of a list. Test them on the list
bigOddList = Table@2 n + 1, 8n, 0, 9999<D;
Type in each of the three sets of code for squaring the elements of a list. Test them on bigOddList .
bigOddList is sufficiently large that the differences in execution time are noticeable for the different codes. However for precise comparison Mathematica provides the Timing command:
2. Write a function called derivativeList, which takes as its arguments a list of expressions in x, and returns a list consisting of the derivatives of the expressions with respect to x. Thus, the input
derivativeList@8x2 , Log@xD, Cos@xD<D
should return
1 92 x, €€€€€ , - Sin @xD= x
à 3. Write a function called myVariance3, which calculates the variance of a list of data. This time, your function should make use of Map and Apply, instead of using Do to iterate through the list. Use the Timing command and the lists thisData and thatData from Section 2.2 to compare the execution time of your new function with myVariance and myVariance2.
4. Write a function called squareBothSides, which squares both sides of an equation. Thus
squareBothSides@x - y == 4D
should return
Hx - yL2 == 16
(Remember that we can treat all Mathematica expressions like lists.) Go on to write a function called doToBothSides, which performs any given operation on both sides of an equation. Thus
doToBothSides@Sin, x - y == p 2D
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should return
Sin @x - yD == 1
ì 2.4 Nesting functions
Clearly Do is a useful Mathematica command. As you've seen, we can use it to process lists of data, either by combining all the data elements or by doing the same thing to each one. However, as you've also seen, there are more efficient and more economical ways of doing both those things. Another use of the Do command is when we want to apply the same operation repeatedly to one piece of data, as in this implementation of a (well-known, and rather inefficient) iterative algorithm for finding the square root of 5.0€€ , 8n<E; x + 1.0 z xy {
But even in this case there's an alternative. It involves no great saving in execution time, but it is, perhaps, a little more economical and elegant as far as the code is concerned. This is it:
x + 5.0 g@x_D := €€€€€€€€€€€€€€€€€€€€€€ ; x + 1.0 sqrt5Iterate2@n_D := Nest@g, 0.0, nD
What's happened is that a function g has been defined, and the Mathematica command Nest applies g repeatedly, using a starting value of 0.0. This has exactly the same effect as before, but avoids the use of Do. Notice that Nest automatically returns the final value as output. Do, by contrast, doesn't return anything: it has no output. That is why we have to finish sqrt5Iterate1 by explicitly calling the value of x. Perhaps it's best to see Do as a utility, "multi-purpose" command, and things like Table, Map, Apply and Nest as more finely-tuned, specialised tools. It's rare that an application of Do will be the best way to get what you want out of Mathematica, but it may often be the first that springs to mind. Suppose that we want to carry out our square root iteration not for a fixed and predicted number of steps but for as many steps as necessary until it has converged . One way to do this is using the While command, as follows:
The iteration is carried out until the condition "the current and previous iterates are different (within the precision of machine accuracy)" ceases to be true. "Machine accuracy" is, as the name suggests, machine-dependent and depends on the quality of the arithmetical processing hardware. On most machines it's around 16 digits . This kind of iteration, too, can be more elegantly done. The command FixedPoint is exactly like Nest, except that it returns not the nth iterate but the final one after convergence has been established. To get our iterative approximation to ,5, all we need to do is type:
x + 5.0 g@x_D := €€€€€€€€€€€€€€€€€€€€€€ x + 1.0 FixedPoint@g, 0.0D
Some iterative algorithms may take a long time to converge to machine accuracy, but may converge perfectly satisfactorily for practical purposes long before that. We can still use FixedPoint; all we have to do is change the SameTest option, like this:
x + 5.0 g@x_D := €€€€€€€€€€€€€€€€€€€€€€ x + 1.0 prettyDamnCloseQ@x_, y_D := TrueQ@Abs@x - yD < 0.0001D FixedPoint@g, 0.0, SameTest -> prettyDamnCloseQD
ì Exercises 2.4
à 1. Test out the sqrt5Iterate1 and sqrt5Iterate2 functions. Examine the effect of replacing the Nest command with NestList.
Compare, too, the two approaches to conditional stopping: the one that uses While and the one that uses FixedPoint. Examine the effect of replacing FixedPoint with FixedPointList.
à 2. Write a function called sqrtIterate such that sqrtIterate[a, n] will return the nth iterated square root approximation for any number a, again using the algorithm
x+a x – €€€€€€€€€€€€€€€€€€€€€€€€€€€ x + 1.0 with starting value 0.0. Use Nest rather than Do. Write a function called sqrtApprox such that sqrtApprox[a] returns, as a decimal approximation to ,a, the final iterate of the above algorithm once convergence has occurred. Use FixedPoint rather than While.
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3. Write a function called derivs which takes as its argument an expression, a variable name and an integer n, and returns a list containing the expression, its first derivative, its second derivative, and so on up to its nth derivative, all with respect to the variable. Thus
derivs@Log@1 + xD, x, 3D
This is a pretty efficient way of squaring all the elements of oddList, but not the most efficient—quite. What's wasteful about it is that we seem to have to define a whole new Mathematica function, square: that's because the first argument of Map must always be a the name of a function. Well, it's not quite true that Map demands the name of a function as its first argument. It will also accept what's known as a pure function, and it's this that gives us a way round the problem. The same piece of code, recast in pure function form, looks like this:
squareList = Map@H#2L&, oddListD
Instead of going to the trouble of defining the square function separately, we've used the rather odd-looking expression (#2 )&. This is an example of a pure function; its key characteristics are the following:
ì the use of the hash mark, # , to stand for the argument of the function; ì the use of the ampersand, &, at the very end of the expression to signify that it is a pure function.
"################################### # MapA #@@1DD2 + #@@2DD2 &, ptsListE
This uses the Part command, written in short form as [[. . .]], to extract coordinate values. Pure functions behave exactly like ordinary functions, and can be used in the same ways—as inputs to Apply and Nest, for example.
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Functions of more than one variable can also be dealt with in this way, using the symbols #1, #2, etc. As the notation suggests, you can think of these symbols as "argument number 1", "argument number 2", . . . . Here's an example:
ApplyA
Here, both the iterated numerical function g and the convergence test prettyDamnCloseQ have been replaced by pure functions, the latter being a function of two variables.
ì Exercises 2.5
1. Test the code in this section. 2. Rewrite the functions myVariance3 and squareBothSides (both from Exercises 2.3) and sqrtIterate and sqrtApprox (from Exercises 2.4) so that they use pure functions. 3. Write a function magnitude that calculates the magnitude (square root of the sum of the squares of the elements) of a list of any length.
This is probably the most efficient form of the variance code so far, but it does have a problem, and one that you'll find in many of the functions you've met in this section. Every time the function is called, a value gets attached to the symbol n. This value is global: it applies outside the context of the function, and will be used every time the symbol n occurs subsequently, until n is reassigned or cleared or until the session is terminated. This can cause serious problems: interference between different functions and so on. Unless you specifically want your function to make global assignments like this, it's best to override the default and make n local. We do that by means of the command With, as follows:
neatVariance2@data_D := WithA8n = Length@dataD<, 1 i 1 z €€€€€€€€€€€€€ jApply@Plus, Map@H#2 L&, dataDD - €€€€€ HApply@Plus, dataD2L y € n-1 k n { E
This tells Mathematica to replace all occurrences of the symbol n with Length[data] where those occurrences are inside the function, but to leave n unassigned, or assigned as it was before, elsewhere. As rewritten, the function will neither change any already existing global value of n nor use that value during the calculation. There are two sets of circumstances in which local scoping is important but With won't work. One is when we want to make changes to the value of a symbol during the execution of the function: in other words, when the symbol represents a variable rather than a constant. The other is when we want the symbol not to have a value: when we want it to be purely symbolic. In either case, we can use the Module command. Here's an example of the first case, in which the value attached to the symbol changes during the execution of the function. Consider the following code, which you first met in Section 2.4 , 8n<E; x + 1.0 z xy {
In order that values of x assigned during the execution of this function should not clash with other occurrences of that symbol, this should really be rewritten
sqrt5Iterate1@n_D := ModuleA8x = 0.0<, x + 5.0 DoAx = €€€€€€€€€€€€€€€€€€€€ , 8n<E; x + 1.0 xE
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Here, 0.0 is used merely as the initial value of x. (Of course, you'll recall that this whole function can be rewritten using Nest!) For an example of the "purely symbolic" case, consider the following code for calculating derivatives from first principles:
expr . Hx -> x + hL - expr firstPrincD1@expr_, x_D :=E h
This works fine provided h hasn't already been given a global value; if it has, the code fails badly. To make it work even in those circumstances, it should be written like this:
firstPrincD2@expr_, x_D := expr . Hx -> x + hL - expr ModuleA8h<,EE h
ì Exercises 2.6
and comment on what you observe. Clear any global value for n by typing
Clear@nD
and then test neatVariance2. Type
n
again, and comment on what you find. 2. Test both forms of the sqrtIterate function in the same way. 3. Begin by making sure that no value is attached to the symbol h by typing
Clear@hD
Now test the firstPrincD1 function on some suitable expression. Test it again, but this time assign a value to h first by typing
h = 0.3
Clear the symbol h and test the firstPrincD2 code in the same way. 4. Review your code from earlier exercises in this session. Rewrite some of it with local scoping of variables and constants where appropriate.
ì 2.7 Advanced Topic: iteration and recursion
Suppose we wanted to rewrite Mathematica's Factorial function from scratch— for non-negative integers, at any rate. One way would be the following:
myFactorial1@n_D := Product@i, 8i, 1, n<D
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or
n
myFactorial1@n_D := Ì i
i=1
This is an example of iterative code: the value of i is made to increase sequentially, and the product is built up as it does so . But there is another possible approach. Consider the following code:
myFactorial2@0D = 1; myFactorial2@n_D := n * myFactorial2@n - 1D
This may look like a circular definition. However, it is rescued from circularity by two things: the fact that n - 1 is less than n and the fact that a simple non-circular definition exists for one case, namely n = 0. When functions call themselves in this way, we have what's called recursion. Many problem situations present us with a choice between iteration and recursion, and some seem tailor-made for the latter. Recursion, though it's often elegant and pleasing, is rather inefficient in many computer languages and impossible in some (older dialects of Fortran, for example). In Mathematica, though, it usually works rather well.
ì Exercises 2.7
Can you explain what's going on here? What are the advantages and disadvantages of this approach? (Hint: when you've done some preliminary testing of each, compare the outputs of ??myFactorial2 and ??myFactorial3 .) 2. Write and test a recursive function for the Fibonacci sequence 1, 1, 2, 3, 5, 8, ... . (Each term of this sequence is generated by adding the previous two.) Thus
fibonacci@6D
should return the 6th term in the sequence, and so on. 3. Write a recursive function called myDet, which must not call Mathematica's Det function, for calculating matrix determinants by cofactor expansion.
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È Session 3
In this session, there are three things you can opt to do. Option 1: a major programming exercise with the idea of putting into practice the Mathematica you'vebeen learning in Sessions 1 and 2. This forms Section 3.1 Option 2: one or more of the various Case Studies which are available separately: a list of these forms Section 3.2. Option 3: Mathematica surgery. You bring along any problems in your own work for which Mathematica might be useful, and we'll try to help you implement Mathematica appropriately. In addition, we have described a number of the Internet resources available to Mathematica users. These form Section 3.3.
ì 3.1 Extended exercise: the Logistic Map
The logistic map is one of the simplest, and most famous, of nonlinear dynamical systems. We won't cover any of the theory here (which has been described in a vast number of books, articles and backs of breakfast cereal packets) beyond mentioning a few interesting things to look at. We are interested in the map x – axH1 - xL which is equivalent to the iterative equation: xn+1 = axn H1 - xn L . The parameter range of interest is 0 ‹ a ‹ 4, because for that range if 0 ‹ x0 ‹ 1 then 0 ‹ xn ‹ 1 for all n. Before starting, you should read Section 2.5 on "pure functions": it may help you to program more elegantly. 1. Write a function called logisticIterates that outputs a list containing the iterates from 0 to n of xn+1 = axn H1 - xn L, starting from a suitable x0 . x0 , a and n should be inputs to the function, and thus
logisticIterates@0.3, 3.1, 4D
should return
80.3, 0.651 , 0.704317 , 0.645589 , 0.709292 <
Generate a list of the iterates from 0 to 21 of the logistic map with x0 = 0.1, a = 2.75. Use ListPlot with appropriate option settings to generate a "time series" plot of the type shown in Figure 4.
generates the above figure automatically. Investigate the behaviour of the logistic map for different values of a, focussing on the intervals 0 ‹ a ‹ 1; 1 ‹ a ‹ 2; 2 ‹ a ‹ 3; 3 ‹ a ‹ 3.569946; 3.569946 ‹ a ‹ 4. 2. When looking at convergent behaviour such as that described in the table above it's helpful to discard the early iterates: write an "attractor" function that takes x0, a, n and m as inputs, and outputs a list of m iterates beginning with the nth (so it has to calculate, but not output, iterates 1 up to n–1). Write another time series function to plot the attractor data. 3. Write a function to produce bifurcation diagrams for the logistic map (a bifurcation diagram is a plot of the limit points—long-term x-values—of the map against the parameter a). This is a non-trivial programming task; it may be helpful to recall the way the Flatten function works (see Exercises 2.1).
Course.nb
39
1 0.8 0.6 0.4 0.2
1.5
2
2.5
3
3.5
4
Figure 5: Bifurcation diagram generated with Mathematica: the interval between values of a is 0.1; for the attractor the first 40 iterates are discarded, 10 retained. 4. Write a function to produce cobweb diagrams for the logistic map, such as the one on the front cover of this booklet. For this task, which is again non-trivial, you'll need to think about what needs to be done to your logisticIterates data so that the appropriate coordinate points are generated in the right order. Further ideas you might explore: Generalize the code you have written so far so that it can be applied to any map, not just the logistic map. Use this new code to investigate the complex map z – z2 + c, where c is a complex parameter and z0 = 0. Write Mathematica code for plotting the Julia Sets and the Mandelbrot Set in the Argand Diagram.
ì 3.2 Case studies
Separate from this booklet we have prepared a number of Case Studies, or "common tasks for the academic user of Mathematica". Here are the current titles: 1. 2. 3. 4. 5. 6. Handling experimental data. Animations and movies. Contour and surface plotting. Moving data between Mathematica and Excel. Exporting graphics to other applications. Equations and rules.
ì 3.3 Resources for Mathematica users on the Internet
Besides numerous books about Mathematica, there is a great deal of information freely available, and an active user community, on the Internet. For these links go to the WWW page at
The central information point for Mathematica is the WWW server at Wolfram Research Inc. (Champaign, Illinois, USA), creators of the Mathematica system:
It's a good idea to browse around that site, check out the latest books and order your Mathematica baseball caps, T-shirts and coffee mugs. You can even interact with a Mathematica program over the Web, also known as "The Integrator":
Maintained on the Wolfram server is a very large collection called MathSource of Mathematica programs and notebooks contributed by Mathematica users world-wide:
If you have a specialist interest you may well find some useful stuff there. Another useful section is the Technical Support Frequently Asked Questions (FAQ):
There are two useful news/email discussion groups for Mathematica users: MathGroup is dedicated to Mathematica, and is accessible via both email and the newsgroup comp.soft-sys.math.mathematica . For details see:
MathGroup is moderated, which means less junk than usual and its archives are available online:
Also there is a (prototype) search engine for the archive at:
The newsgroup sci.math.symbolic carries discussions about Maple, Mathematica and other symbolic software. It isn't moderated and you get what you pay for: for example it is a host to frequent discussions of that burning question: "which program is best?". Finally, remember that with the Web, often the best way to track something down is to use a search engine. Digital's "AltaVista" is currently one of the best around:
Course.nb
41
Last but not least, do visit the WWW server of the METRIC project: | 677.169 | 1 |
50 Mathematical Puzzles and Problems: Orange Collection
Categories
Keywords
Additional Images
Product Details
Author Name:
Cohen, Gilles
Binding:
Paperback
Book Condition:
Good
Publisher U.S.A. Key Curriculum Press 2001
ISBN Number
1559534990 / 9781559534994
Seller ID
122779
An Engaging Series of Problems More than a decade's worth of puzzles and problems from the International Championship of Mathematics are found in this three-volume set. The problems are organized by mathematical themes-geometry and symmetry, arithmetic and number theory, logic and algorithmic process-for easy adaptation to your mathematics curriculum. Orange Collection The Orange Collection (Grades 9-12) is the intermediate set and will challenge most high school students. Students divide polygons into tilings of congruent shapes; they encounter knots, chains, and networks; they decipher messages and break codes. A few solutions are facilitated with algebra and trigonometry. | 677.169 | 1 |
Learning with Graphing Calculators
With the graphing calculator, you are able to have the calculator create graphs which are typically used to solve mathematical equations. Each type of graph which this calculator can solve are typically solved using a graphing calculator, although there are manual ways for them to be solved.
The majority of graphing calculators out there are programmable which allows for the user to create a custom program which can be used for a variety of different reasons. The most common reason for this is for either a scientific, engineering, or education reasons. Furthermore, because of how big the screen provided is, these calculators can be used for storing vast amounts of text or calculators at one time.
In terms of the more modern graphing calculators, colour and interactive graphics have become a possibility. On top of that, there are even some calculator manufacturers who offer a range of software allowing for graphing calculators to become so much more. There have been so many advancements made with graphing calculators which have allowed for many schools to offer a wider range of mathematic courses.
With some of the more advanced graphing calculators, thermometers and such can be connected along with other devices, allowing for them to be used as data collection devices.
Students who have access to a graphing calculator are presented with so much more potential in regards to their work, than if they didn't have access to one. With a graphing calculator they are able to make an intelligent partner to work with.
Studies have shown how with the use of a graphing calculator; you are able to:
Improve the self-confidence of students
Boost the dynamics of a classroom
Improve students understanding of certain math concepts
Improve students problem solving abilities.
Don't interpret this wrongly, the pen and paper approach is still a viable option. However, by combining a graphing calculator with the pen and paper approach, a balance is created in which both approaches can be used advantageously.
If students use their graphing calculators appropriately, the calculator should not be seen as an "assistant" as such but in fact, should be seen as something used to enhance the students knowledge.
Graphing calculators assist students in:
The visualization of their problems
Discovering and learning new mathematical functions which they can use on their own accord
Not only check their answers, but validate their own formulae
Discover different methods to calculate problems
Provide students with the tools to educate themselves on more math topics which they have yet to begin study in class
Graphing calculators ease the process of active learning. Rather than students simply sitting and staring mindlessly as their teacher explains new concepts to them, with the use of a graphing calculator they are able to discover these new concepts on their own and learn about them in ways which they are comfortable with. On top of that, they are able to come up with their own solutions and formulae which they are confident with.
Graphs can often be difficult to draw and are a very time consuming thing to put on paper. With a graphing calculator, students can get more work done and can produce their graphs at a much faster rate. This is efficient and allows them to focus on more aspects of their work.
Previously, advanced math classes were required for students to learn graph drawing skills. Now, a graphing calculator can be used instead. With a cheap graphing calculator, all students are able to take their graph skills to a higher level.
Graphing calculators provide students with so much more than the simple pen and paper method could ever offer, and that's very important when it comes to mathematics. With the data that can be obtained from these calculators and how that data can be manipulated, students can get on with their work at a faster rate than otherwise.
In terms of cost, graphing calculators are cheap to the point that all classrooms are able to purchase them for their students to take advantage of without having to spend too much. For an average sized class, you would be able to purchase a set of these calculators for around the same price as you would purchase two school computers. The newer the graphing calculator, the more functionality you are given access to.
The algebra aspect of mathematics can be very difficult for both teachers and students and with the use of graphing calculators, the learning process becomes a lot easier for both sides.
Rather than students spending several minutes on just one section of an equation, they are able to use a graphing calculator to speed up the process and get straight to the real calculation. The tedious steps involved in graphing can slow down a student's learning cycle and make it undesirable for the student to continue. The faster that they get the equation done, the more equations they can complete, and the more that they can learn.
Graphing calculators are a fantastic piece of technology that without a doubt, make the mathematical learning process a lot easier for everyone.
Free online graphing calculator from goodcalculators.com
Considering how big a part your graph can play when it comes to your equations, making sure that the graph is drawn properly is very important. Sure, you could take the manual approach. On the other hand, using the Good Calculators Online Graphing Calculator makes everything so much easier.
Buying a graphing calculator isn't the easiest thing to do. Not only are they more expensive than your regular calculator, but they simply aren't something we can all afford. This Online Graphing Calculator is completely free therefore you don't need to worry about the cost!
Upon first stumbling upon this free online graphing calculator, I thought that I'd be at a disadvantage to those who used a regular graphing calculator. I was very wrong about that. This calculator is just as good as a regular online graphing calculator, if not better. If anything, I'd say I had an advantage over those who used any other calculators.
The thing that I like most about this calculator is that I don't need to have another piece of equipment taking up space on my desk. I already have my computer open, and I can simply switch to the calculator when needed! If you haven't already tried it out, this calculator is worth giving a shot. | 677.169 | 1 |
Clear structure, closely mirroring the Student's Book content New listening tasks per unit, with all audio contained on the CD Lots of exam practice throughout the pages and in the Get Ready for your Exam sections Plenty of revision, reviews for each pair of units, and a self-check page (and answers provided) at the end of every unit …
Eureka Math is a comprehensive, content-rich PreK-12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. | 677.169 | 1 |
comprehensive, detailed reference to Mathematica provides the reader with both a working knowledge of Mathematica programming in general and a detailed knowledge of key aspects of Mathematica needed to create the fastest, shortest, and most elegant implementations possible to solve problems from the natural and physical sciences. The Guidebook gives the user a deeper understanding of Mathematica by instructive implementations, explanations, and examples from a range of disciplines at varying levels of complexity. | 677.169 | 1 |
comprehensive discussion of numerical computing techniques with an emphasis on practical applications in the fields of civil, chemical, electrical, and mechanical engineering. It features two software libraries that implementMore...
This book provides a comprehensive discussion of numerical computing techniques with an emphasis on practical applications in the fields of civil, chemical, electrical, and mechanical engineering. It features two software libraries that implement the algorithms developed in the text - a MATLAB toolbox, and an ANSI C library. This book is intended for undergraduate students. Each chapter includes detailed case study examples from the four engineering fields with complete solutions provided in MATLAB and C, detailed objectives, numerous worked-out examples and illustrations, and summaries comparing the numerical techniques. Chapter problems are divided into separate analysis and computation sections. Documentation for the software is provided in text appendixes that also include a helpful review of vectors and matrices. The Instructor's Manual includes a disk with software documentation and complete solutions to both problems and examples in the book.
Sandra L. Harris is Associate Professor of Chemical Engineering at Clarkson University. Dr. Harris teaching interests include process control, thermodynamics, and biochemical engineering. Her research interests are centered on periodic processing, control of systems having varying dead times, and generation of input signals for efficient process identification | 677.169 | 1 |
Bundles, connections, metrics & curvature are the lingua franca of modern differential geometry & theoretical physics. Supplying graduate students in mathematics or theoretical physics with the fundamentals of these objects, & providing numerous examples, the book would suit a one-semester course on the subject of bundles & the associated geometry more...
This book highlights important developments on artinian modules over group rings of generalized nilpotent groups. Along with traditional topics such as direct decompositions of artinian modules, criteria of complementability for some important modules, and criteria of semisimplicity of artinian modules, it also focuses on recent advanced results on... more...
A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics, including commutative algebra, geometry, and knot theory. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object.... more...
Get ready to master the principles and operations of algebra! Master Math: Algebra is a comprehensive reference guide that explains and clarifies algebraic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced topics that will help prepare you for pre-calculus... more... | 677.169 | 1 |
See the original Waec Syllabus for Further Mathematics and Recommended TextBooks for this year. Note that this syllabus is for both Internal May/June and external Nov/Dec GCE candidates.
WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)
233
AIMS OF THE SYLLABUS
The aims of the syllabus are to test candidates on:
(i) further conceptual and manipulative skills in Mathematics;
(ii) an intermediate course of study which bridges the gap between Elementary
Mathematics and Higher Mathematics;
(iii) aspects of mathematics that can meet the needs of potential Mathematicians,
Engineers, Scientists and other professionals.
EXAMINATION FORMAT
There will be two papers both of which must be taken.
PAPER 1: (Objective) – 1½ hours (50 marks)
PAPER 2: (Essay) – 2½ hours (100 marks)
PAPER 1 (50 marks) – This will contain forty multiple-choice questions, testing the
areas common to the two alternatives of the syllabus, made up
of twenty-four from Pure Mathematics, eight from Statistics and
Probability and eight from Vectors and Mechanics. Candidates
are expected to attempt all the questions.
PAPER 2 – This will contain two sections – A and B.
SECTION A (48 marks) – This will consist of eight compulsory questions that are
elementary in type, drawn from the areas common to both
alternatives as for Paper 1 with four questions drawn from Pure
Mathematics, two from Statistics and Probability and two from
Vectors and Mechanics.
SECTION B (52 marks) – This will consist of ten questions of greater length and difficulty
consisting of three parts as follows:
PART I (PURE MATHEMATICS) – There will be four questions with two drawn from the
common areas of the syllabus and one from each
alternatives X and Y.
PART II (STATISTICS AND PROBABILITY) – There will be three questions with two drawn
from common areas of the syllabus and one
from alternative X.
PART III (VECTORS AND MECHANICS) – There will be three questions with two drawn
from common areas of the syllabus and one from
alternative X.
Candidates will be expected to answer any four questions with at least one from each part.
WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)
234
Electronic calculators of the silent, cordless and non-programmable type may be used in these papers.
Only the calculator should be used; supplementary material such as instruction leaflets, notes on
programming must in no circumstances be taken into the examination hall. Calculators with paper
type output must not be used. No allowance will be made for the failure of a calculator in the
examination.
A silent, cordless and non-programmable calculator is defined as follows:
(a) It must not have audio or noisy keys or be operated in such a way as to disturb other
candidates;
(b) It must have its own self-contained batteries (rechargeable or dry) and not always be
dependent on a mains supply;
(c) It must not have the facility for magnetic card input or plug-in modules of programme
instructions.
DETAILED SYLLABUS
In addition to the following topics, harder questions may be set on the General Mathematics/
Mathematics (Core) syllabus.
In the column for CONTENTS, more detailed information on the topics to be tested is given while
the limits imposed on the topics are stated under NOTES.
NOTE: Alternative X shall be for Further Mathematics candidates since the topics therein are
peculiar to Further Mathematics.
Alternative Y shall be for Mathematics (Elective) candidates since the topics therein
are peculiar to Mathematics (Elective).
WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION
FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE)
Friction:
Distinction
between smooth
and rough planes.
Determination of
the coefficient of
friction required.
3. Dynamics
(a) (i) The concepts of
Motion, Time and
Space.
(ii) The definitions of
displacement,
velocity,
acceleration and
speed.
(iii) Composition of
velocities and
accelerations.
(b) Equations of motion
(i) Rectilinear motion;
(ii) Newton's Law of
motion.
(iii) Consequences of
Newton's Laws:
The impulse and
momentum
equations:
Conservation of Linear
Momentum.
(iv) Motion under
gravity.
Application of the
equations of motions:
V = u + at;
S = ut + ½ at²;
V² = u² + 2as.
Motion along
inclined planes | 677.169 | 1 |
Essential Mathematical Skills Author: S Barry
ISBN: 9781921410338 Format: Paperback Number Of Pages: 212 Published: 1 January 2008 Country of Publication: AU Dimensions (cm): 24.0 x 17.6 x 1.3 Description: This book covers the main areas of mathematics used in the first years of a typical engineering, science or applied mathematics degree. This is not a textbook. It is a concise guide to what the important skills in mathematics are: the ones that need to be remembered.
This second edition also includes the essential elements of MATLAB and Maple - the two most common computer tools used by students at university | 677.169 | 1 |
College Math Homework Help: How To Memorize Formulas
Math is difficult, or at least it can be for those of us liberal arts majors who want to forget all the hurt (physical and mental) it caused throughout high school. So, what does math do for an encore? By the numbers (pun intended) it numerically-oops, I did it again-eliminated you from the requisite SAT score needed to gain admission at your school of choice. I'm assuming you have a letter that confirms this.
But the pain lingers. Not only do you find yourself in a state university auditorium having to fulfill a general education requirement at a safety school surrounded by those you sought protection from, you still suck at math, too.
A semester abroad isn't an option yet, because I just said 'yet', and anyways, if you can't get through a couple of math courses, you will never be eligible to go anywhere. Tough love, folks. What's the plan? The biggest difference between high school and college is college students WANT to be there. Usually. Unless it's math class.
A semester really isn't a long period of time to suffer through, even if it's math. Let's check out one of ways to make math tolerable. Memorizing formulas.
Get it done
Learn how a formula works. Admittedly, this sounds like learning math. But it's your ability to understand as opposed to learn is what's important here.
If you are able to figure out why the formula is created, how it works suddenly becomes far easier to understand.
The next step to understanding a formula is picturing it in your head.
Symbols
It doesn't take a math major to realize the importance of symbols in mathematics. Let the symbols work for you! Math is a language, and just like Latin, it is really boring and you can't speak it. However, their importance is immeasurable. Just as Latin is the root of so many languages the world uses today, the symbols used in mathematics have been understood by thousands for thousands (of years).
For us liberal arts majors, creativity is the only trick we have up our sleeve. To help memorize formulas, make it personal. Create a story.
What is it about these numbers or letters that draws your ire? What makes these square roots, plus/minus signs, and theorems so important that it is necessary to carve them into your brain? | 677.169 | 1 |
Grade 12 Math made completely easy!
Aligned with your class or textbook, you will get grade 12 math help on topics like Trigonometric identities, Vectors, Permutations and Combinations, Matrices, Polynomial functions and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest grade 12 math questions. Then, strengthen your understanding with tons of grade 12 math practice.
All our lessons are taught by experienced Grade 12 math teachers. Let's finish your homework in no time, and ACE that final.
I'm taking MCV4U and MHF4U with VHS (Virtual High School). Which course should I sign up for?
Doesn't matter. J Your subscription gives you unlimited access to all math help in all courses. Our Grade 12 math help should cover all you need to know for MHF4U (Advanced Functions). For calculus topics in MCV4U (Calculus and Vectors), you can check out our Calculus 1 help too. So, we got you all covered!
Students and parents love our math help
Study Pug is a great supplementary resource for students to reinforce the math they learn in class at home. The video lessons allow an easy review on previous topics.
Terry Anderson
Grade 12 math student, Hill Park Secondary, Hamilton, ON
I thought I was very good at math until I started taking MHF4U. Those hard-to-understand concepts, like log and trig, were my worst nightmares. Dennis explains even the hardest things so well and right to the point. He always shows the easiest and fastest ways to tackle a difficult question. My math marks improved a lot because of you Dennis!
Cathy Aquino
Grade 12 math student, Britannia School, Edmonton, AB
I was thinking about applying for 2 of the Ivy League schools in the States, so I had to ace my Math 30-1. Your site is such a great resource that supplements things that my math teacher missed in class. I also use StudyPug to get a head start so that I can have more time to work on my other subjects rather than spending nights and days on getting the math homework done. | 677.169 | 1 |
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About this product
Description
Description
Many mathematicians have been drawn to mathematics through their experience with math circles: extracurricular programs exposing teenage students to advanced mathematical topics and a myriad of problem-solving techniques and inspiring in them a lifelong love for mathematics. Founded in 1998, the Berkeley Math Circle (BMC) is a pioneering model of a U.S. math circle, aspiring to prepare our best young minds for their future roles as mathematics leaders. Over the last decade, 50 instructors - from university professors to high school teachers to business tycoons - have shared their passion for mathematics by delivering more than 320 BMC sessions full of mathematical challenges and wonders.Based on a dozen of these sessions, this book encompasses a wide variety of enticing mathematical topics: from inversion in the plane to circle geometry; from combinatorics to Rubik's cube and abstract algebra; from number theory to mass point theory; from complex numbers to game theory via invariants and movariants. The treatments of these subjects encompass every significant method of proof and emphasize ways of thinking and reasoning via 100 problem-solving techniques.Also featured are 300 problems, ranging from beginner to intermediate level, with occasional peaks of advanced problems and even some open questions. The book presents possible paths to studying mathematics and inevitably falling in love with it, via teaching two important skills: thinking creatively while still obeying the rules, and making connections between problems, ideas, and theories. The book encourages you to apply the newly acquired kwledge to problems and guides you along the way, but rarely gives you ready answers. Learning from our own mistakes often occurs through discussions of n-proofs and common problem-solving pitfalls.The reader has to commit to mastering the new theories and techniques by getting your hands dirty with the problems, going back and reviewing necessary problem-solving techniques and theory, and persistently moving forward in the book. The mathematical world is huge: you'll never kw everything, but you'll learn where to find things, how to connect and use them. The rewards will be substantial. | 677.169 | 1 |
Solución gráfica de ecuaciones linales
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings. | 677.169 | 1 |
MA 15200 I
Lesson 4
Section P.2 (part 2) and Section P.3 (part 1)
Writing Numbers in Scientific Notation
A number is written in scientific notation when it is in the form a 10n where 1 a < 10 and n is an integer. If the original number is 10 or greater, t
MA 15200 I
Lesson 5
P.3 (part 2)
Simplifying Square Root Radicals
A square root is simplified when its radicand has no factors other than 1 that are perfect squares. Remember: a 2 = a , if a is assumed to be positive. We will assume all variables represen
Lesson 2
Section P.1 (part 2)
I More on Evaluating Algebraic Expressions Reminder: The Order of Operations Agreement 1. Perform operations within the innermost grouping and work outwards. If the expression involves a fraction, treat the numerator CLASS PERIOD
Ground Rules
Spring 2012
Students are expected to attend every class meeting and to read the appropriate sections of the text before coming to class. Instructors often may not have time to cover every topic in class. Refer to your t
GS 29000 (CRN 33158) Study Skills for College Algebra (paired with MA 15200)
2 credit hours must be enrolled in MA 15200 to take this course Meets T TH 1:30 2:20 BRNG B280 This is a course for those who: o Hate math o Are afraid of math o Have poor math s Lesson 7 Section P.5 *There are several good study tips in this lesson on the textbook pages.
Factoring Polynomials
1st factoring method
If two binomials are multiplied together, each polynomial is a factor of the product. The first factoring met
Math 13900
Lesson 31 Section 25.1 Today we will use the program Geometer's Sketchpad to investigate the ideas of area and perimeter as well as review transformations. We will just be scratching the surface of what Geometer's Sketchpad can do, but this sho
Online Homework Information To register for the first time (if you've never used MyMathLab for math online homework), go to the Registration for MyMathLab PowerPoint presentation found on the course web page and follow the steps. The first slide describes | 677.169 | 1 |
for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point.
Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure.
After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas.
Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction
Explains identification of techniques and how they are applied in the specific problem
Illustrates how to read written proofs with many step by step examples
Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter
Recommendations:
Save
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Pre-Algebra A
ID : 10182
Pre-Albrebra is for students who need to strnghen their mathematical skills prior to taking algebra. Fundamental skills of arithmetic are expanded and problem-solving skills are practiced. Elementary algebra and geometry topics, including equations, inequalities, data analysis, graphing, probability, areas and volumes are introduced. Emphasis is placed on real-world applications and making connections to other disciplines.
Featured Classes | 677.169 | 1 |
Introduction to Lambda Trees [Hardcover]
Item description for Introduction to Lambda Trees by Ian Chiswell...
Introductory text for mathematicians and research students in algebra and topology, introducing the fundamental concepts and theory of A-Trees, including the origins and history of the theory. Discusses connections with other theories such as model theory and R-Trees.
Promise Angels is dedicated to bringing you great books at great prices. Whether you read for entertainment, to learn, or for literacy - you will find what you want at promiseangels.com!
More About Ian Chiswell
Ian Chiswell is Emeritus Professor in the School of Mathematical Sciences at Queen Mary, University of London. His main area of research is geometric group theory, especially the theory of -trees. Other interests have included cohomology of groups and ordered groups.
Ian Chiswell was born in 1948 and has an academic affiliation as follows - Queen Mary, University of London.
Reviews - What do customers think about Introduction to Lambda Trees | 677.169 | 1 |
Inexpensive but comprehensive two-volume set. Covers the history of Hindu mathematics far more comprehensively than any other source I've found. WellInexpensive but comprehensive two-volume set. Covers the history of Hindu mathematics far more comprehensively than any other source I've found. Well written, doesn't suffer from the language and printing errors ocasionally seen in older Indian mathematics publications. The Chakravala algorithm and its relationship to continued fractions is presented particularly well with excellent detail. Recommended....more | 677.169 | 1 |
Research
VanDieren's mathematics research is in Model Theory, a branch of Mathematical Logic. Her specialization is in abstract elementary classes. In her Ph.D. thesis she introduced the concept of tameness for abstract elementary classes and began the development for a classification theory for these classes.
In addition to model theory research, VanDieren is interested in pedagogical research, particularly on interdisciplinary honors courses and on student understanding of multi-variable calculus concepts.
A Pittsburgh Post Gazette article describes her research, outreach activities and how she finds beauty in mathematics.
M. VanDieren. Visualizing and Estimating the Mass of a Solid Using Multi-colored Blocks. In J. Libertini and J. Barnes (Eds). Tactile Learning Activities in Undergraduate Mathematics: A Recipe Book for the Classroom. To appear. Published by the MAA.
M. VanDieren, D. Moore-Russo, J. Wilsey, and P. Seeburger. Students' Understanding of Vectors and Cross Products: Results from a Series of Visualization Tasks. To appear in Proceedings of the 20th Annual Conference on Research on Undergraduate Mathematics Education.
Amazon Web Services in Education Grant to house a WeBWorK server for RMU students. 2014-2015. $6000.
Professional Experience Program (PREP) Grant to fund an undergraduate mathematics and computer science research assistant to aid in converting CalcPlot3D exploratory labs into WeBWorK using the Perl programming language. 2014-15 academic year. | 677.169 | 1 |
th
Binding:
Paperback Textbook
Publisher:
Wiley
Supplemental materials are not guaranteed for used textbooks or rentals (access codes, DVDs, workbooks).,...
Show More, this edition of the manual is three-hole punched for easy customization. When students truly understand the mathematical concepts, it's magic. Students who use Mathematics for Elementary Teachers: A Contemporary Approach are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement of the mathematical world. That's why the new Seventh Edition of Musser, Burger, and Peterson's best-selling textbook focuses on one primary goal: helping students develop a true understanding of central concepts using solid mathematical content in an accessible and appealing format. The components in this complete learning program--from the textbook, to the eManipulative activities, to the online problem-solving tools and the resource-rich website--work in harmony to help achieve this goal | 677.169 | 1 |
Course Summary
Get reacquainted with algebra, geometry, calculus and more as you study for the NYSTCE Mathematics assessment. This comprehensive study guide offers fun video lessons, short quizzes and practice exams to enhance your preparations for the test. | 677.169 | 1 |
This best-selling, calculus-based text is recognized for its carefully crafted, logical presentation of the basic concepts and principles of physics. PHYSICS FOR SCIENTISTS AND ENGINEERS, Sixth Edition, maintains the Serway traditions of concise writing for the students, carefully thought-out problem sets and worked examples, and evolving...
Solomon/Berg/Martin, BIOLOGY is often described as the best majors text for LEARNING biology. Working like a built-in study guide, the superbly integrated, inquiry-based learning system guides you through every chapter. Key concepts appear clearly at the beginning of each chapter and learning objectives start each section. You can quickly...
Success in your calculus course starts here! James Stewart's CALCULUS texts are world-wide best-sellers for a reason: they are clear, accurate, and filled with relevant, real-world examples. With CALCULUS, Sixth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an...
The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. Each title is just one component in a comprehensive calculus...
The new edition calculus textbook has been thoroughly revised. It continues to embrace the best aspects of reform by combining the traditional theoretical aspects of calculus with creative teaching and learning techniques. This is accomplished by a focus on conceptual understanding, the use of real-world data and real-life applications,...
Algebra is accessible and engaging with this popular text from Charles "Pat" McKeague! ELEMENTARY AND INTERMEDIATE ALGEBRA is infused with McKeague's passion for teaching mathematics. With years of classroom experience, he knows how to write in a way that you will understand and appreciate. McKeague's attention to detail and...
The fourth edition of Algebra: Introductory and Intermediate examines the fundamental
ideas of algebra. Recognizing that the basic principles of geometry are a
necessary part of mathematics, we have also included a separate chapter on
geometry (Chapter 3) and have integrated geometry topics, where appropriate,
throughout the...
Designed specifically for biology and life/social sciences majors, this applied calculus program motivates students while fostering understanding and mastery. The authors emphasize integrated and engaging applications that show students the real-world relevance of topics and concepts. Several pedagogical features--from algebra review to study...
Intermediate algebra is a bridge course. The course and its syllabus bring the student
to the level of ability required of college students, while getting them ready to make a
successful start in college algebra or precalculus.
Most students study calculus for its use as a tool in areas other than mathematics. They desire information about why calculus is important, and where andhow it can be applied. I kept these facts in mind as I wrote this text. In particular, when introducing new concepts I often refer to problems that are familiar to students and that require...
As the discipline of computer science has matured, it has become clear that a study of discrete mathematical topics is an essential part of the computer science major. The course in discrete structures has two primary aims. The first is to introduce students to the rich mathematical structures that naturally describe much of the content of... | 677.169 | 1 |
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Description
Description
ELEMENTARY LINEAR ALGEBRA's clear, careful, and concise presentation of material helps you fully understand how mathematics works. The author balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system. To engage you in the material, a new design highlights the relevance of the mathematics and makes the book easier to read. Data and applications reflect current statistics and examples, demonstrating the link between theory and practice. The companion website LarsonLinearAlgebra.com offers free access to multiple study tools and resources. CalcChat.com offers free step-by-step solutions to the odd-numbered exercises in the text.
Author Biography | 677.169 | 1 |
Algebra. CC: HSN-RN.A.1. This is a great book mark that shows all the properties of numbers and some valuable equations. It would be great for students at the abstract level with these concepts who just need a quick reminder. Simply print the book mark, cut it out, and hand out!
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Here is a FREE (for now:) poster linking the parts of a parabola to the questions students will be asked when solving quadratic word problems...
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FOIL method Poster for multiplying binomials. I am a big fan of the FOIL method for multiplying binomials. Although I know some educators use the box method, my students find the FOIL method easier and much faster with a little practice.
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Elementary Algebra is generalized form of arithmetic. It provides a language to represent problems and functions. Algebraic thinking is also one of the first forms of abstract thinking that students develop in mathematics. Lets look at some of the common gotchas of algebraic learning. | This is Excellent!! I'm loving the explanations!!! | 677.169 | 1 |
Preparation for Teaching Middle School Mathematics • Nearly three-fourths of middle school mathematics teachers received their undergraduate degree in areas other than mathematics or mathematics education. • Roughly two-thirds have taken 8 or more courses in mathematics (the equivalent of a minor.) • Most have had college coursework in general teaching methods (92%) and methods of teaching mathematics (78%).
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Preparation for Teaching Middle School Mathematics NCTM recommends that middle school teachers have college coursework in the following areas: • Abstract Algebra • Geometry • Calculus • Probability and statistics • Applications of mathematics/ problem solving • History of Mathematics
Preparation for Teaching Middle School Mathematics Here's how current middle school teachers stack up with these courses:
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GACE Middle Grades Mathematics 013 eBook
This digital study guide is available for immediate download and use. Its two 60-question practice exams include question rigor, skill reference, as well as full answer rationales. Aligned specifically to standards prescribed by the Georgia Department of Education, this guide covers the subareas of numbers and operations; measurement and geometry; patterns, algebra, and functions; data analysis and probability; and mathematical processes and perspectives | 677.169 | 1 |
Showing 1 to 30 of 638
Section 2.6: Limits at infinity & Horizontal Asymptotes
<Note: For definitions and more examples, dont forget to also read the textbook!>
Count Lev Tolstoy (1828-1920)
A man is like a fraction whose numerator is what he is and whose denominator is what heMath 124 Midterm 2 Main Topics for Review
This midterm will be based on the material from sections: 3.3, 3.4, 3.5, 3.6, 3.9, 3.10 and 10.2 (tangents only).
First, know well all the rules of differentiation: what they are and how to apply them (alone or inWorksheet
Math 124
Week 10
Worksheet for Week 10: Maximizing functions
The best way to study for the Math 124 nal is to solve challenging problems. In
this worksheet, youll solve a dicult problem that incorporates several topics from the
class: position a
Worksheet
Math 124
Week 4
Worksheet for Week 4: Velocity and parametric curves
In this worksheet, youll use dierentiation rules to nd the vertical and horizontal
velocities of an object as it follows a parametric curve. Youll also get a preview of how
toWorksheet
Math 124
Week 2
Worksheet for Week 2: Graphs and limits
In this worksheet, youll practice using the graph of an objects position to learn about
its velocity. Youll also learn a useful technique for computing limits of certain types of
functionsShowing 1 to 3 of 4
For me, this was a required class as one of the pre reqs for my major. I'm being completely honest when I say this was the HARDEST class I ever took. It's not for the faint of heart. If it's not a required course for you and you're thinking about taking it, I'd listen to the rest of this. This class requires a lot of work. There is a lot of homework and it's definitely not easy. Most of the homework is harder than what you learn in class, so unless you've taken the class before, you will need to read the textbook, go to study sessions, or go to office hours for extra help. It's a very analytical kind of math that gets very complicated. The reason I do recommend this class though, is that it really opened my perspective to new ways of thinking and studying. I really had to change my typical habits to accommodate for this class. I don't "strongly recommend" because it is a very tough class, and it definitely kicked my butt, so I think anyone who isn't required to take this class and is thinking about it, should really think about if it's worth it. It is likely to affect your GPA negatively if you don't put in your best effort. The lectures are really just a couple example problems, with little to no explanation. You are expected to read the textbook and understand the concepts on your own. Dr. Pezzoli strongly believes in struggling through problems on your own, and offers very limited help. She also gives harder problems on tests and homework, than she does in class so I highly recommend going over past exams and going to review sessions. They will be your lifesavers!
Course highlights:
Honestly, for me, this class was a nightmare and I couldn't wait for it to be over. It kicked my butt. It was a whole new way of thinking for me and it took time to adjust. Time I didn't have. I do think it definitely opened my eyes to changing certain habits and forcing me to look at things with a new perspective. It also showed me that you can't just skate by in all your classes. I had been so used to getting all A's with barely any work, and if you think that will be you in this class, think again.
Hours per week:
9-11 hours
Advice for students:
Study study study! You will not pass this class if you don't put in extra time outside of lectures and quiz sections. You may think you have it all nailed from what you do in class, but it is much harder on the exams and homework. I strongly recommend going to tutoring and review sessions. They were my lifesavers. Dr. Pezzoli was not very good at explaining the concepts, if she even bothered to at all, so going to get help from graduates students who had been in the class before was one of the most helpful tools. I also highly recommend finding websites that offer step by step solutions to your problems. I found that if I could see it solved the right way, I use it as a guideline for the rest of my problems and just tweek a few things here and there to fit each individual problem.
Course Term:Winter 2017
Professor:Elena Pezzoli
Course Required?Yes
Course Tags:Math-heavyGo to Office HoursMeetings Outside of Class
Feb 22, 2017
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
Toro was a fairly good teacher. She knew what she was talking about and moved at a reasonable pace. She was more than happy to answer questions, and got to know some of the students, which was nice. The only downside was sometimes, if left uninterrupted, she may start to drone. Overall a solid class though.
Course highlights:
Highlights of the course, for me, definitely came from working one on one with Toro. When not in a lecture context, she is very very efficient at communicating and making sure you're following along. There were plenty of times, after working with her, she just let me take her notes as a study guide, which was incredibly useful.
Hours per week:
9-11 hours
Advice for students:
Make use of the math study center! People there are always willing to help, and it's useful to have multiple people working on the same problem. Also your TA's office hours. And make sure to talk to Toro one on one, if you have the chance.
Course Term:Fall 2016
Professor:tatianatoro
Course Tags:Math-heavyMany Small AssignmentsParticipation Counts
Nov 30, 2016
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
I came into class knowing a lot about calculus and having taken 2 years of it in high school. If I didn't have that, I definitely would be struggling a lot more. The material isn't too difficult and lot of it is intuitive but it's also a lot of material in not a lot of time. It's helpful to organize your thoughts everyday after lecture and attend quiz section! The homework is also way harder than the tests will ever be so don't get discouraged.
Course highlights:
I solidified a lot of concepts I already knew. None of it was new material for me. It's a good refresher course but not a great first time with calculus class because it is so quick and nuanced.
Hours per week:
6-8 hours
Advice for students:
REVIEW YOUR NOTES AFTER CLASS. Make sure the day you learn something that you try the homework and make sure you get the big picture of what the heck calculus is all about. It's easy to keep your head down and just barrel through the class but the concepts won't connect or make sense to you the way they need to. Also definitely attend quiz section because there are some problems that make no sense that will be explained to you. | 677.169 | 1 |
This text, intended for a graphing calculator required precalculus course, shows students when and how to use concepts, and promotes real understanding not just rote memorization. In addition, the graphing calculator is used as a tool to help explain ideas rather than merely to find answers. The book reflects AMATYC, MAA, and NCTM guidelines, and makes use of real world data in presenting a balanced algebraic and graphical approach to understanding precalculus concepts. The result is a thorough preparation for the calculus course.
Book Description Addison Wesley Longman, 2003321132033-N | 677.169 | 1 |
How to Ace the Multiple-Choice Mathematics Test
How to Ace the Multiple-Choice Mathematics Test is my third book. This book discusses multiple-choice mathematics tests (including standardized tests, mathematics sections of college admissions tests, mathematics competitions, and others) and how to improve your score on these tests.
This book is divided into several chapters, each with a different focus. The second chapter discusses strategies for finding solutions to problems. The fourth discusses mathematical concepts frequently found on multiple-choice tests geared towards high school students. There are also chapters on mental arithmetic, a glossary of terms seen frequently on multiple-choice tests, advanced concepts, aids (calculators, rulers, compasses, etc.), study tips, and what to do before, during, and after the test. There are also 40 practice questions included at the end of the book with full solutions.
Last updated November 3, 2014.
URL:
For questions or comments, e-mail James Yolkowski (mathlair@allfunandgames.ca). | 677.169 | 1 |
An application for math plot.Can be used arithmetic operations, trigonometric functions (angles measured in radians), decimal, natural logarithms, the logarithm to an arbitrary ground, whole and fractional parts of numbers | 677.169 | 1 |
״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a...
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״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject. We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student's level of mathematical maturity will increase as the course progresses. Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Elementary Number Theory
Select this link to open drop down to add material Elementary Number Theory to your Bookmark Collection or Course ePortfolio
This is a free textbook offered by BookBoon.'In this book, which is basically self-contained, the following topics are...
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This is a free textbook offered by BookBoon.'In this book, which is basically self-contained, the following topics are treated thoroughly: Brownian motion as a Gaussian process, Brownian motion as a Markov process, and Brownian motion as a martingale. Brownian motion can also be considered as a functional limit of symmetric random walks, which is, to some extent, also discussed. Related topics which are treated include Markov chains, renewal theory, the martingale problem, Itô calculus, cylindrical measures, and ergodic theory. Convergence of measures, stochastic differential equations, Feynman-Kac semigroups, and the Doob-Meyer decomposition theorem theorem are discussed in the second part of stochastic processes: Part II to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Advanced stochastic processes: Part II
Select this link to open drop down to add material Advanced stochastic processes: Part II to your Bookmark Collection or Course ePortfolio
״This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties,...
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״This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text Theory of Numbers to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material An Introduction to the Theory of Numbers
Select this link to open drop down to add material An Introduction to the Theory of Numbers to your Bookmark Collection or Course ePortfolio
This is a free, online textbook. According to the author, "This free online textbook (e-book in webspeak) is a one semester...
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This is a free, online textbook. According to the author, "This free online textbook (e-book in webspeak) is a one semester course in basic analysis. These were my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in fall 2009. The course is a first course in mathematical analysis aimed at students who do not necessarily wish to continue a graduate study in mathematics. A Sample Darboux sums prerequisite for the course is a basic proof course. The course does not cover topics such as metric spaces, which a more advanced course would. It should be possible to use these notes for a beginning of a more advanced course, but further material should be added Analysis: Introduction to Real Analysis to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Basic Analysis: Introduction to Real Analysis
Select this link to open drop down to add material Basic Analysis: Introduction to Real free, online textbook is from a complex analysis project that "started in 2003, and since that early beginning it...
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This free, online textbook is from a complex analysis project that "started in 2003, and since that early beginning it has been replaced several times." According to the authors, "We find that there is a tendency to teach complex analysis with pencil and paper, and it is the hope of this web site to overcome this inertia. Perhaps you will find useful computer illustrations that will encourage students to explore complex analysis using Mathematica and Maple. Since we are a bellwether in this movement we hope that you will follow us through the pages in this web site and see the beautiful explorations that can be made. Hopefully you will gain some insight as to how this can help you teach complex analysis and the benefits your students can gain by making their own computer explorations. Finally, we would like to emphasize that this web site is a complementary supplement that is coordinated with the current version of our textbook Complex Analysis for Mathematics and Engineering, Sixth Ed., 2012 Complex Variables - Complex Analysis to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Complex Variables - Complex Analysis
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'DataAnalysis is a general purpose iPad App for the plotting and fitting of all types of data that can be formulated as x,y...
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'DataAnalysis is a general purpose iPad App for the plotting and fitting of all types of data that can be formulated as x,y pairs. The program can be used easily by both students and professionals. It is particularly useful for quick analyses of various types of data by curve fitting, value prediction via a standard curve, and analysis of kinetic data including enzyme kinetics.Data can be entered directly via the keyboard or imported from the Mail App as a text file (.txt), and comma separated value file (.csv). In addition, data can be imported via drag and drop within iTunes. Data files in txt, csv and native formats can be imported directly from Dropbox.After data entry the user has the option to average, baseline correct, and/or normalize the data before plotting and fitting.Plotting of the data is as simple as pressing the Plot button. The axes are autoscaled and the graph is immediately ready to add to a document or presentation. The user has a variety of options to customize the graph by altering the axes, axes labels, graph title, and changing the symbol, its size and color.The data can be fit to a variety of mathematical equations by non-linear regression including simple functions such as linear, 2nd and 3th order polynomials, power, exponential, log functions. Data can also be fit to more complex equations for such processes as radioactive decay, enzyme kinetics, 1st and 2nd order chemical reactions, as well as gaussian, logistics and surge functions. The fit can be overlaid on the data with different line widths, types and thicknesses.For many analytical situations, the data fit can be used as a standard curve to determine the value of unknowns. This analytical procedure is completely automated within DataAnalysis.The program output, graphics and text, can be copied, emailed, or uploaded to Dropbox in a variety of formats including PDF, PNG and TXT. The graphics in PDF format can be edited with programs like Adobe Illustrator™ as objects.DataAnalysis also supports multitasking and printing of input data as well as all forms of output.DataAnalysis has built in tool tips, comprehensive web based instructions and how-to videos available via the Help MenuAnalysis App for iPad to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material DataAnalysis App for iPad
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'This is a textbook about classical elementary number theory and elliptic curves. The first part discusses elementary topics...
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'This is a textbook about classical elementary number theory and elliptic curves. The first part discusses elementary topics such as primes, factorization, continued fractions, and quadratic forms, in the context of cryptography, computation, and deep open research problems. The second part is about elliptic curves, their applications to algorithmic problems, and their connections with problems in number theory such as Fermats Last Theorem, the Congruent Number Problem, and the Conjecture of Birch and Swinnerton-Dyer. The intended audience of this book is an undergraduate with some familiarity with basic abstract algebra, e.g. rings, fields, and finite abelian groups: Primes, Congruences, and Secrets to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Elementary Number Theory: Primes, Congruences, and Secrets
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This is a free, online textbook. According to Textbook Revolution, "This is an open set of lecture notes on...
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This is a free, online textbook. According to Textbook Revolution, "This is an open set of lecture notes on metaheuristics algorithms, intended for undergraduate students, practitioners, programmers, and other non-experts. It was developed as a series of lecture notes for an undergraduate course I taught at GMU. The chapters are designed to be printable separately if necessary. As it's lecture notes, the topics are short and light on examples and theory. It's best when complementing other texts. With time, I might remedy this Metaheuristics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Essentials of Metaheuristics
Select this link to open drop down to add material Essentials of Metaheuristics to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Introduction to Graph or graduate courses in Graph Theory in departments of mathematics or computer science. This text offers a comprehensive and coherent introduction to the fundamental topics of graph theory. It includes basic algorithms andMore...
For undergraduate or graduate courses in Graph Theory in departments of mathematics or computer science. This text offers a comprehensive and coherent introduction to the fundamental topics of graph theory. It includes basic algorithms and emphasizes the understanding and writing of proofs about graphs. Thought-provoking examples and exercises develop a thorough understanding of the structure of graphs and the techniques used to analyze problems. The first seven chapters form the basic course, with advanced material in Chapter 8 | 677.169 | 1 |
The user reviews definitions of important algebra terms. After viewing further explanations and some examples, users can interactively test their understanding of the definitions of important algebra... More: lessons, discussions, ratings, reviews,...
The user reviews like terms and how to group like terms together in expressions. After viewing examples, users can interactively practice matching expressions on one side with expressions on the othe... More: lessons, discussions, ratings, reviews,...
The 3-D animated video helps students understand the wordings in the following distance, rate, and time word problem: Two space jets named Dragon and Eagle start from Mars and fly in opposite dire | 677.169 | 1 |
mathematical modeling uses elementary functions to describe and explore real-world data and phenomena. nbsp; Helps readers connect math with the world around them through real-world applications of elementary mathematics.More...
This introduction to mathematical modeling uses elementary functions to describe and explore real-world data and phenomena. nbsp; Helps readers connect math with the world around them through real-world applications of elementary mathematics. Shows how to construct useful mathematical models, how to analyze them critically, and how to communicate quantitative concepts effectively. Uses concrete language and examples throughout to foster quantitative literacy. nbsp; For anyone interested in gaining a solid foundation in mathematical concepts.
(Note: Each chapter begins with a real-world vignette that is revisited as the chapter evolves, and concludes with a Review and a project-style Investigation.)
Functions and Mathematical Models
Functions Defined by Tables
Functions Defined by Graphs
Functions Defined by Formulas
Average Rate of Change
Linear Functions and Models
Constant Change and Linear Growth, Linear Functions and Graphs
Piecewise-Linear Functions
Fitting Linear Models to Data
Natural Growth Models
Percentage Growth and Interest
Percentage Decrease and Half-Life, Natural Growth and Decline in | 677.169 | 1 |
Algebra Touch v1.0
Total Downloads: 771
5.0
Paid
Experience algebra like someone who's good at it.
Say you have x + 3 = 5. You can drag the 3 to the other side of the equation.
Enjoy the wonderful conceptual leaps of algebra, without getting ...Read more > | 677.169 | 1 |
books.google.com - P... Beginner's Guide
LaTeX Beginner's Guide
P and practical introduction. Particularly during studying in school and university you will benefit much, as a mathematician or physicist as well as an engineer or a humanist. Everybody with high expectations who plans to write a paper or a book will be delighted by this stable software.
About the author (2011)
Stefan Kottwitz studied mathematics in Jena and Hamburg. Afterwards, he worked as an IT Administrator and Communication Officer onboard cruise ships for AIDA Cruises and for Hapag-Lloyd Cruises. Following 10 years of sailing around the world, he is now employed as a Network & IT Security Engineer for AIDA Cruises, focusing on network infrastructure and security such as managing firewall systems for headquarters and fleet. In between contracts, he worked as a freelance programmer and typography designer. For many years he has been providing LaTeX support in online forums. He became a moderator of the web forum http: //latex community.org/ and of the site http: //golatex.de/. Recently, he began supporting the newly established Q&A site http: //tex.stackexchange.com/ as a moderator. He publishes ideas and news from the TeX world on his blog at http: //texblog.net. | 677.169 | 1 |
Algebra Foldable - Factoring Flipbook
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
1.19 MB | 7 pages
PRODUCT DESCRIPTION
This completely editable flipbook is just what you need to help your students learn the ins and outs of factoring. Everything your students need to know - rules, helpful hints, and examples - can be at their fingertips in this concise toolbox.
Included in the toolbox are five tabs:
Factoring by Greatest Common Factor and Trinomials with a Lead Coefficient of 1
Factoring Grouping
Factoring Trinomials with a Lead Coefficient not equal to 1
Special Binomials (Difference of Perfect Squares, Difference of Two Cubes, and Sum of Two Cubes)
Factoring Higher Order Polynomials
Each tab includes basic instructions to walk students through the process of each type of factoring. The last tab includes helpful guidelines of how to tackle ANY problem.
I have included two versions of the flipbook: one with examples and one without examples. This enables you to decide what examples are best for your students.
If you have any questions, comments, or special requests, feel free to contact me or visit my websiteEverything is Nerdiful | 677.169 | 1 |
Developmental Algebra|2nd Edition
Product Description:
This book is designed to be a combination of a textbook and a notebook. If the instructor chooses to follow the outline of the book, there is space provided in the book for the student to take notes. Students are encouraged to highlight and annotate the book as needed. By using the textbook in this way, the students have their textbook and notes all in one convenient package. The book is bound like a spiral notebook for ease of use in the classroom. Because it is much lighter than most college math books, it is easier to carry around. For instructors who wish to follow the outline in the book, there is much less preparation required to teach the course.
The purpose of Developmental Algebra is skills development. The material in this book is intended to establish a strong foundation of algebra skills that can be built upon, with lessons designed to prepare the student for success in a college-level mathematics course.
21 Day Unconditional Guarantee
REVIEWS for Developmental Algebra | 677.169 | 1 |
Find a Monte SerenoNot being ready, students can't study calculus successfully. Therefore, in Precalculus, students will be introduced to the important and basic mathematical concepts inquired before in algebra with deeper and higher details. They comprise, but not limited in, inequalities, equations, absolute values, and graphs of lines and circles | 677.169 | 1 |
Lorenzo Robbiano
This is a conventional introduction to matrices that the author has attempted to make more accessible by garnishing it with palindromes, jokes, and aphorisms. It is a translation of the 2007 Italian-language work Algebra Lineare per tutti (also published by Springer). The translation is smooth and easy to read, although many of the jokes are left untranslated.
From the title one might guess that this is a text for a hypothetical Linear Algebra Appreciation course, but the author admits on p. x that the focus is much narrower. The motivation for the book was to produce a unified text that all departments would use, rather than have each teach its own flavor of linear algebra. I think this unification is unsuccessful; the applications are very generic and other departments would consider it slanted too much toward pure mathematics and not having anything special that the departmental students need to know.
The book takes a very concrete approach. As motivation, it starts out with a selection of practical problems that can be expressed in terms of systems of linear equations. The treatment is slanted very much toward matrices and numerical work and avoids the linear-spaces interpretation, except in Chapter 4 where the book explores geometric vectors and rotations and projections.
Assuming that the work is in fact aimed at linear algebra students and not at "everyone", its big weakness is that it doesn't have enough exercises and worked examples. This omission would make it unsuitable for most American college-level courses. Other popular textbooks such as Strang's Introduction to Linear Algebra or Lay's Linear Algebra and Its Applications would be better choices for a college course. They are only slightly more expensive than the present book, have many applications from all areas, and have an adequate number of exercises and worked examples. | 677.169 | 1 |
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Summary
This package consists of the textbook plus an access kit for MyMathLab/MyStatLab. #xA0; Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Introductory Algebra, Fourth Editionwas written to provide students with a solid foundation in algebra and to help stuents make the transition to intermediate algebra. The new edition offers new resources like the Student Organizerand now includes Student Resourcesin the back of the book to help students on their quest for success. #xA0; MyMathLabprovides a wide range of homework, tutorial, and assessment tools that make it easy to manage your course online.
Author Biography
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An award-winning instructor and best-selling author, Elayn Martin-Gay has taught mathematics at the University of New Orleans for more than 25 years. Her numerous teaching awards include the local University Alumni Association's Award for Excellence in Teaching, and Outstanding Developmental Educator at University of New Orleans, presented by the Louisiana Association of Developmental Educators.
Prior to writing textbooks, Elayn developed an acclaimed series of lecture videos to support developmental mathematics students in their quest for success. These highly successful videos originally served as the foundation material for her texts. Today, the videos are specific to each book in the Martin-Gay series. Elayn also pioneered the Chapter Test Prep Video to help students as they prepare for a test–their most "teachable moment!"
Elayn's experience has made her aware of how busy instructors are and what a difference quality support makes. For this reason, she created the Instructor-to-Instructor video series. These videos provide instructors with suggestions for presenting specific math topics and concepts in basic mathematics, prealgebra, beginning algebra, and intermediate algebra. Seasoned instructors can use them as a source for alternate approaches in the classroom. New or adjunct faculty may find the videos useful for review.
Her textbooks and acclaimed video program support Elayn's passion of helping every student to succeed.
Table of Contents
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Chapter R
Prealgebra Review
R.1 Factors and the Least Common Multiple
R.2 Fractions
R.3 Decimals and Percents
Chapters R Group Activity: Interpreting Survey Results
Chapter R Vocabulary Check
Chapter R Highlights
Chapter R Review
Chapter R Test
Chapter 1 Real Numbers and Introduction to Algebra
1.1 Tips for Success in Mathematics
1.2 Symbols and Sets of Numbers
1.3 Exponents, Order of Operations, and Variable Expressions
1.4 Adding Real Numbers
1.5 Subtracting Real Numbers
1.6 Multiplying and Dividing Real Numbers
1.7 Properties of Real Numbers
1.8 Simplifying Expressions
Chapter 2 Equations, Inequalities, and Problem Solving
2.1 The Addition Property of Equality
2.2 The Multiplication Property of Equality
2.3 Further Solving Linear Equations
2.4 An Introduction to Problem Solving
2.5 Formulas and Problem Solving
2.6 Percent and Mixture Problem Solving
2.7 Linear Inequalities and Problem Solving
Chapter 3 Exponents and Polynomials
3.1 Exponents
3.2 Negative Exponents and Scientific Notation
3.3 Introduction to Polynomials
3.4 Adding and Subtracting Polynomials
3.5 Multiplying Polynomials
3.6 Special Products
3.7 Dividing Polynomials
Chapter 4 Factoring Polynomials
4.1 The Greatest Common Factor
4.2 Factoring Trinomials of the Form x^2 + bx + c
4.3 Factoring Trinomials of the Form ax^2 + bx + c
4.4 Factoring Trinomials of the Form ax^2 + bx + c by Grouping
4.5 Factoring Perfect Square Trinomials and the Difference of Two Square
4.6 Solving Quadratic Equations by Factoring
4.7 Quadratic Equations and Problem Solving
Chapter 5 Rational Expressions
5.1 Simplifying Rational Expressions
5.2 Multiplying and Dividing Rational Expressions
5.3 Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominator
5.4 Adding and Subtracting Rational Expressions with Different Denominators | 677.169 | 1 |
Introduction to Number Theory
Introduction to Number Theory
15.00
The aim of this 200 page book is to enable talented students to tackle the sort of problems on number theory which are set in mathematics competitions. Topics include primes and divisibility, congruence arithmetic and the representation of real numbers by decimals. A useful summary of techniques and hints is included. This is a fully revised and extended edition of a book which was12-X
ISBN 13: 978-1-906001-12 | 677.169 | 1 |
Literal Equations Notes & Practice
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3.95 MB | 13 pages
PRODUCT DESCRIPTION
Solving literal equations can be a difficult topic for students to understand, especially if there is variety in the types of equations/formulas they are asked to solve. These notes and practice are appropriate for middle or high school students in pre-algebra and above. I have included some introductory problems as well as some that are more challenging. (See the preview to see actual examples.)
These versatile, detailed notes and examples will provide structure to your lessons. Choose to use all or some of the three pages of notes and examples to tailor your lesson to the needs of your students. Follow up with one or both of the worksheets (18 questions each | 677.169 | 1 |
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About this product
Description
Description
Written to cover the AQA GCSE Mathematics A modular specification (Unit 3, Foundation), our student book targets the skills and kwledge required for the exam. Functional mathematics is integrated throughout providing an emphasis on applying mathematics in a real-life context. | 677.169 | 1 |
There are 38
books, if you include Answer Keys/Teachers Editions. The Teachers Editions are simply the same
book as their counterpart, with the answers included next to most of the problems.
If you would like to see sample pages from the books, those can be
viewed by clicking here.
Basic Math
"An
effort has been made throughout the work to observe a natural and strictly logical connection between the different
parts, so that the learner may not be required to rely on a principle, or employ a process, with the rationale of
which he is not already acquainted"
Ray's Primary Arithmetic. 95 pages. The first book in the Ray's Series, Primary Arithmetic starts at
the very beginning of mathematics by teaching the children to
count. The book then slowly progresses with simple problems, first with addition, then subtraction, then
multiplication and division. The problems are very simple, so that the child can learn the concepts involved.
Ray's Intellectual Arithmetic. 141 pages. Intellectual
Arithmetic begins by reviewing the basic concepts covered in Ray's Primary Arithmetic, adding more problems and
raising the difficulty by increasing the quantities involved. The book then adds the new concept of Fractions
and percentages. As these concepts are introduced, applications for the problems are given, showing the child
how to solve increasingly complex problems.
Ray's
Elementary Arithmetic. 192 pages. Ray's New Elementary
Arithmetic is designed to lead straight into Practical Arithmetic, by providing an extensive amount of drill
work for the student, in order to allow the student ample time and problems to thoroughly master the
fundamentals before moving forward.
Ray's Practical Arithmetic. 337 pages. Practical
Arithmetic starts by quickly reviewing the basics of addition, subtraction, multiplication and division covered
previously in Elementary Arithmetic, and then moves into a study of different types of measurement, followed by
factoring, and a more involved study of fractions and percentages. After these mathematical bases have been
studied and mastered, real world applications for these mathematics are introduced. These include Transactions,
Commissions, Stock values and investments,
Interest,Discounts, Monetary exchange, Insurance and Taxes.
Practical Arithmetic then concludes by introducing basic geometry.
Intermediate Math
"To fix the principles in the mind of the
student, and to show their bearing and utility, great attention has been paid to the preparation of practical
exercises."
This continual grounding in practicality is a peculiar aspect of
Ray's Arithmetic, one which is very rare and very valuable. Modern math books have focused so much on the
abstract that the student is left to wonder how he will ever be able to use it in his day to day life; a feeling
that easily makes mathematics seem like a chore or useless exercise.
Ray's Higher Arithmetic. 409 pages. A very complete study of Arithmetic, this is the last book in the
Ray's series before the introduction of Algebra. All of the basic mathematical methods are reviewed thoroughly; and
more complicated applications and uses are explored. Finally the book begins the study of Geometry, and the
fundamentals of Trigonometry are introduced.
Ray's New Elementary Algebra. 241 pages. "In
introducing Algebra to the student with Elementary Algebra, great care has been taken to make the student feel
that he is not operating with unmeaning symbols, by means of arbitrary rules; that Algebra is both a rational
and practical subject, and that he can rely on his reasoning, and the results of his operations with the same
confidence as in arithmetic. For this purpose, he is furnished, at almost every step, with the means of testing
the accuracy of the principles on which the rules are founded, and of the results which they
produce."
I cannot stress highly enough the importance the above paragraph
has to a student embarking on a study of Algebra. From personal experience while studying with Saxon Algebra I
know just how frustrating it is to be told what to do, while not given any reasons for why we are doing it or
how it works. I didn't want to simply take their word for it but prove it for myself, a thought process most
students share. And while I did manage to work these things out eventually, it was a slow and painful process.
One of the things that make Ray's Arithmetic such an excellent series is the attention given to the student.
Instead of neglecting the reasoning and deducing ability of the students themselves, they are instead encouraged
to think on their own. This increases the students interest, his understanding of the material, as well as his
recollection of studies later on.
Ray's New Elementary Algebra focuses on the basic forms of Algebra.
Algebraic Fractions, Simple Equations, Powers, Roots, Radicals, and finally Quadratic Equations are among the
concepts explored. As always, after a concept has been taught, real-world applications for the process are given
to the student.
Ray's New Higher Algebra. 407 pages. After
reviewing the fundamentals, Higher Algebra then moves on to Theorems, Factoring, Algebraic Fractions, Quadratic
Equations, Ratio, Proportion, Binomial Theorem, etc etc. This book is quite lengthy, thoroughly teaching
algebraic concepts. While there are relatively few problems for the student to work on and solve, these have
been supplied by Test Problems for Ray's New Higher Algebra. 152 pages, as well
as A Complete Algebra. 359 pages.
Advanced Math
Ray's Treatise on Geometry and Trigonometry. 421
pages. Begins by giving definitions for some basic geometrical terms, then begins Geometry, starting with
parallel lines, then continuing with Arcs and Radii, the properties of triangles, Parallelograms, measuring
area, Polygons, the geometry of space, Pyramids, Prisms, etc. The book then continues on into the subject of
Trigonometry, and supplies logarithmic tables. No problems are supplied for the student in this book, which
instead gives all it's attention to teaching the concepts. It is suggested that the student use another book to
provide problems while learning the mathematical processes from this book.
Ray's Analytic Geometry. 608 pages. Equations to
the Right Line, the Plane, Quadrics, The Ellipse, The Hyperbola, and Properties of Conics discussed with great
fullness. Abridged Notation is introduced in this book. This book does not supply problems for student work,
focusing on teaching the concepts.
Ray's Differential and Integral Calculus. 442
pages. Begins with definitions. Careful attention has been given to the teaching of the doctrine of limits,
which has been made the basis of both the Differential and Integral Calculus. Problems are supplied in the
book.
Extra-Curricular Texts
In order to provide students with examples of the interesting
fields mathematical studies opens, several books of ranging difficulties have been supplied.
Complete Book Keeping. 161 pages. An often
over-looked area of study, book-keeping will always be an important area of expertise for anyone who earns or
spends money. As the author states: "Book-keeping... cultivates the judicial powers of the mind... contributes
to private and public virtue.. leads to economy and thrift... and it's practice will reduce pauperism and
crime." Beginning with the basic form of double entry book-keeping, Debit, Credit, and all areas of accounting
are taught.
Norton's Elements of Physics. 269 pages. This
incredibly well-written and intriguing book is so well written that it does not feel so much like a dry text on
the mathematics of physics as an exploration of physical laws, thus allowing someone well acquainted with
physics or not at all to read this book with great enjoyment. Carefully illustrated, this book begins by
introducing the student first to general notions of matter and force, and then introducing new elements one by
one. Friction, adhesion, fluids, sound, light, heat, and electricity are all explored, with careful explanations
of experiments and studies done by the scientists who explored these properties.
Schuyler's Principles of Logic. 169 pages. Logic
is a mathematical pursuit. How can we tell? Through logic.... Because math is based on logical premises
(induction), and then followed through in a logical working out of the premises (deduction). The author explains
this more thoroughly, but you'll have to read the book.
Ray's Elements of Astronomy. 342 pages. Taking a
different approach to astronomy, rather than focusing on the Greek names for constellations this book focuses on
the movements of heavenly bodies and the science of astronomy.
Ray's Surveying and Navigation. 492 pages. While
this book thoroughly covers the old art of Surveying (the same business George Washington was in for a number of
years) this book is especially useful because of it's great attention to the field of Plane and Spherical
Trigonometry and Mensuration, and may because of this be used as a textbook for those fields | 677.169 | 1 |
College Algebra: Graphs&models
The Barnett Graphs & Models series in college algebra and precalculus maximizes student comprehension by emphasizing computational skills, real-world data analysis and modeling, and problem solving rather than mathematical theory. Many examples feature side-by-side algebraic and graphical solutions, and each is followed by a matched problem for the student to work. This active involvement in the learning process helps students develop a more thorough understanding of concepts and processes.
A hallmark of the Barnett series, the function concept serves as a unifying theme. A major objective of this book is to develop a library of elementary functions, including their important properties and uses. Employing this library as a basic working tool, students will be able to proceed through this course with greater confidence and understanding as they first learn to recognize the graph of a function and then learn to analyze the graph and use it to solve the problem. Applications included throughout the text give the student substantial experience in solving and modeling real world problems in an effort to convince even the most skeptical student that mathematics is really useful.
"synopsis" may belong to another edition of this title.
About the Author:
I was born and raised in Cleveland, and started college at Bowling Green State University in 1984 majoring in creative writing. Eleven years later, I walked across the graduation stage to receive a PhD in math, a strange journey indeed. After two years at Franklin and Marshall College in Pennsylvania, I came home to Ohio, accepting a tenure-track job at the Hamilton campus of Miami University. I've won a number of teaching awards in my career, and while maintaining an active teaching schedule, I now spend an inordinate amount of time writing textbooks and course materials. I've written or co-authored either seven or twelve textbooks, depending on how you count them, as well as several solutions manuals and interactive CD-ROMS. After many years as developmental math coordinator at Miami Hamilton, I share the frustration that goes along with low pass rates in the developmental math curriculum. Far too many students end up on the classic Jetson's-style treadmill, with the abstract nature of the traditional algebra curriculum keeping them from reaching their goals. Like so many instructors across the country, I believe the time is right to move beyond the one-size-fits-all curriculum that treats students the same whether they hope to be an engineer or a pastry chef. "Because we've always done it that way" is NOT a good reason to maintain the status quo in our curriculum. Let's work together to devise alternate pathways that help students to learn more and learn better while hastening their trip into credit-bearing math courses. Since my book (Math in Our World) is written for the Liberal Arts Math and Quantitative Literacy market, I think I'm in the right place at the right time to make a difference in the new and exciting pathways course. I'm in a very happy place right now: my love of teaching meshes perfectly with my childhood dream of writing. (Don't tell my publisher this – they think I spend 20 hours a day working on textbooks – but I'm working on my first novel in the limited spare time that I have.) I'm also a former coordinator of Ohio Project NExT, as I believe very strongly in helping young college instructors focus on high-quality teaching as a primary career goal. I live in Fairfield, Ohio with my lovely wife Cat and fuzzy dogs Macleod and Tessa. When not teaching or writing, my passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel, golf, and home improvement.
About the Author:
Dave Sobecki was born and raised in Cleveland, and started college at Bowling Green State University in 1984 majoring in creative writing. Eleven years later, he walked across the graduation stage to receive a PhD in math, a strange journey indeed. After two years at Franklin and Marshall College in Pennsylvania, he came home to Ohio, accepting a tenure-track job at the Hamilton campus of Miami University. Dave has won a number of teaching awards in his career, and more recently has turned his attention to writing textbooks. Dave is in a happy place where his love of teaching meshes perfectly with his childhood dream of writing. He lives in Fairfield, Ohio with his lovely wife Cat, and fuzzy dogs Macleod and Tessa. When not teaching or writing, Dave's passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel, golf, and home improvement. | 677.169 | 1 |
Algebra 1: Learning in Context
Many of the tools of today's technology--computer chips, lasers, and HDTV--were invented by people who are highly skilled in number sense and operations. As technology has become a major part of our everyday lives, these skills are in ever increasing demand. | 677.169 | 1 |
SCI - Scientific Calculator Perfect for all kind of activities, this scientific calculator includes trigonometric functions such as sine and cosine, exponential functions such as square root, and several math | 677.169 | 1 |
Simplifying Radical Functions Practice KAGAN Sage and Scribe
PDF (Acrobat) Document File
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0.28 MB | 4 pages
PRODUCT DESCRIPTION
This activity is for students to work in partner pairs based on KAGAN best practices. The students will practice Simplifying Radical Functions with Radical and Rational exponents. Answers are included.
Directions: Each pair will receive one worksheet.
1 Student A is the Scribe and Student B is the Sage.
The Sage gives the Scribe step-by-step instructions how to answer questions.
2 The Scribe records the Sage's responses step-by-step, coaching and clarifying if necessary.
3 The Scribe praises the Sage
4. Students switch roles for the next problem | 677.169 | 1 |
Best Reference Books – Systems Theory
Book Review: This book combines the concepts of algebra and geometry to discuss the three dimensional area wherein the vectors can be plotted. The book provides an introduction to linear algebra that deals with vectors and vector spaces. The book very nicely demonstrates how linear algebra is used in the fields of natural and social sciences apart from mathematics for studying weather problems, traffic flow, electronic circuits and population genetics.
>
2. "Elementary Topics in Differential Geometry" by J.A. Thorpe
Book Review: This book provides an introduction to linear algebra thereby also teaches the concepts of differential equations and multivariate calculus. This book helps the students in understanding spaces of many dimensions. The chapters included in the book are finite dimensional vector spaces, measure theory, ergodic theory and Hilbert spaces. This book is suitable for the courses at the junior level and provides attention to 2 dimensional surfaces rather than the surfaces of arbitrary dimension | 677.169 | 1 |
Overview of Maths Science and Social Studies for Class VII CBSE by carveNiche
The product 'Maths, Science and Social Studies for Class VII CBSE exhaustively covers the topics of Class VII CBSE Maths, Science and Social Studies through the use of animation videos in English.The product is delivered in Pen Drive and can be used in windows PC or Android Tablets.
Table of contents
Maths
Integers
Fractions and Decimals
Data Handling
Simple Equations
Lines and Angles
The Triangles and its Properties
Visualising Solid Shapes
Congruence of Triangle
Comparing Quantities
Rational Numbers
Practical Geometry
Perimeter and Area
Algebraic Expressions
Exponents and Powers
Symmetry
Visualising Solid Shapes.
Science
Nutrition in Plants
Nutrition in Animals
Respiration in Organisms
Transportation in Animals and Plants
Reproduction in Plan With Weather, Climate and Adaptations of Animal and Climate | 677.169 | 1 |
9780849339882 textbook thoroughly outlines combinatorial algorithms for generation, enumeration, and search. Topics include backtracking and heuristic search methods applied to various combinatorial structures, such as:
Combinations
Permutations
Graphs
Designs
Many classical areas are covered as well as new research topics not included in most existing texts, such as:
Group algorithms
Graph isomorphism
Hill-climbing
Heuristic search algorithms
This work serves as an exceptional textbook for a modern course in combinatorial algorithms, providing a unified and focused collection of recent topics of interest in the area. The authors, synthesizing material that can only be found scattered through many different sources, introduce the most important combinatorial algorithmic techniques - thus creating an accessible, comprehensive text that students of mathematics, electrical engineering, and computer science can understand without needing a prior course on combinatorics | 677.169 | 1 |
Browse by
(iii
)
á6mææq g¸± mugg Oaud æ¸gg m¤gqgg3æ!
-kndh‹kÙa« Rªjudh®
The Government of Tamil Nadu has decided to evolve a uniform system of school education in
the state to ensure social justice and provide quality education to all the schools of the state. With due
consideration to this view and to prepare the students to face new challenges in the feld of Mathematics,
this text book is well designed within the frame work of NCF2005 by the textbook committee of
subject experts and practicing teachers in schools and colleges.
Mathematics is a language which uses easy words for hard ideas. With the aid of Mathematics
and imagination the nano or the googolplex all things may be brought within man's domain. This laptop
of handbook is an important collection of twelve topics. A brief and breezy explanation of each chapter
proceeds with an introduction to the topics, signifcant contributions made by the great Mathematicians,
concise defnitions, key concepts, relevant theorems, practice problems and a brief summary at the end of
the lesson written with wit and clarity to motivate the students. This book helps the student to complete
the transition from usual manipulation to little rigorous Mathematics.
Real life examples quoted in the text help in the easy grasp of meaning and in understanding
the necessity of mathematics. These examples will shape the abstract key concepts, defnitions and
theorems in simple form to understand clearly. But beyond fnding these examples, one should examine
the reason why the basic defnitions are given. This leads to a split into streams of thought to solve the
complicated problems easily in different ways.
By means of colourful visual representation, we hope the charming presents in our collection will
invite the students to enjoy the beauty of Mathematics to share their views with others and to become
involved in the process of creating new ideas. A mathematical theory is not to be considered complete
until it has been made so clear that the student can explain it to the frst man whom he or she meets on
the street. It is a fact that mathematics is not a mere manipulation of numbers but an enjoyable domain
of knowledge.
To grasp the meaning and necessity of Mathematics, to appreciate its beauty and its value, it is
time now to learn the depth of fundamentals of Mathematics given in this text. Any one who penetrates
into it will fnd that it proves both charming and exciting. Learning and creating Mathematics is indeed
a worthwhile way to spend one's life.
Mathematics is not a magic it is a music ; play it, enjoy! bloom!! and fourish!!!
E. Chandrasekaran
and writting team
Preface
(iv)
SYMBOLS
= equal to
! not equal to
1
less than
#
less than or equal to
2 greater than
$ greater than or equal to
. equivalent to
j union
k intersection
U universal Set
d belongs to
z does not belong to
1 proper subset of
3 subset of or is contained in
1 Y not a proper subset of
M not a subset of or is not contained in
( ) or A A
c
l complement of A
Q (or) { } empty set or null set or void set
n(A) number of elements in the set A
P(A) power set of A
P(A) probability of the event A
T symmetric difference
N natural numbers
R real numbers
W whole numbers
Z integers
3 triangle
+ angle
= perpendicular to
|| parallel to
( implies
` therefore
a since (or) because
absolute value
-
approximately equal to
| (or) : such that
/ (or) , congruent
/ identically equal to
r pi
! plus or minus
Y end of the proof
(v)
CONTENT
1. THEORY OF SETS 1-32
1.1 Introduction 1
1.2 Description of Sets 1
1.3 Representation of a Set 3
1.4 Different kinds of Sets 7
1.5 Set Operations 17
1.6 Representation of Set Operations using Venn Diagram 25
12. PROBABILITY 247-262
12.1 Introduction 247
12.1 Basic Concepts and Definitions 248
12.3 Classification of Probability 250
12.4 Probability - An Empirical Approach 250
Theory of Sets
1
GEORG CANTOR
(1845-1918)
The basic ideas of set
theory were developed by
the German mathematician
Georg Cantor (1845-1918).
He worked on certain kinds
of infinite series particularly
on Fourier series. Most
mathematicians accept
set theory as a basis of
modern mathematical
analysis. Cantor's work was
fundamental to the later
investigation of
Mathematical logic.
1.1 I ntroduction
The concept of set is vital to mathematical thought
and is being used in almost every branch of mathematics. In
mathematics, sets are convenient because all mathematical
structures can be regarded as sets. Understanding set theory
helps us to see things in terms of systems, to organize
things into sets and begin to understand logic. In chapter 2, we
will learn how the natural numbers, the rational numbers
and the real numbers can be defined as sets. In this
chapter we will learn about the concept of set and some
basic operations of set theory.
1.2 Description of Sets
We often deal with a group or a collection of objects,
such as a collection of books, a group of students, a list
of states in a country, a collection of coins, etc. Set may
be considered as a mathematical way of representing a
collection or a group of objects.
● To describe a set
● To represent sets in descriptive form, set builder form
and roster form
● To identify different kinds of sets
● To understand and perform set operations
● To use Venn diagrams to represent sets and set operations
● To use the formula involving ( ) n A B , simple word
problems
Mai n Tar get s
No one shall expel us from the paradise that
Cantor has created for us
- DAVID HILBERT
THEORY OF SETS
1
Chapter 1
2
Key Concept Set
A set is a collection of well-defned objects. The objects of a set are called
elements or members of the set.
The main property of a set in mathematics is that it is well-defned. This means that given
any object, it must be clear whether that object is a member (element) of the set or not.
The objects of a set are all distinct, i.e., no two objects are the same.
Which of the following collections are well-defned?
(1) The collection of male students in your class.
(2) The collection of numbers 2, 4, 6, 10 and 12.
(3) The collection of districts in Tamil Nadu.
(4) The collection of all good movies.
(1), (2) and (3) are well-defned and therefore they are sets. (4) is not well-defned
because the word good is not defned. Therefore, (4) is not a set.
Generally, sets are named with the capital letters A, B, C, etc. The elements of a set are
denoted by the small letters a, b, c, etc.
Reading Notation
! 'is an element of' or 'belongs to'
If x is an element of the set A, we write x A ! .
g 'is not an element of' or 'does not belong to'
If x is not an element of the set A, we write x A g .
For example,
Consider the set A = , , , 1 3 5 9 " ,.
1 is an element of A, written as 1 A !
3 is an element of A, written as 3 A !
8 is not an element of A, written as 8 A g
Theory of Sets
3
Example 1.1
Let A = , , , , , 1 2 3 4 5 6 " ,. Fill in the blank spaces with the appropriate symbol or g ! .
(i) 3 ....... A (ii) 7 ....... A
(iii) 0 ...... A (iv) 2 ...... A
Solution (i) 3 A ! (a 3 is an element of A)
(ii) 7 A g (a 7 is not an element of A)
(iii) 0 A g (a 0 is not an element of A)
(iv) 2 A ! (a 2 is an element of A)
1.3 Representation of a Set
A set can be represented in any one of the following three ways or forms.
(i) Descriptive Form
(ii) Set-Builder Form or Rule Form
(iii) Roster Form or Tabular Form
1.3.1 Descriptive Form
Key Concept Descriptive Form
One way to specify a set is to give a verbal description of its elements.
This is known as the Descriptive form of specifcation.
The description must allow a concise determination of which elements
belong to the set and which elements do not.
For example,
(i) The set of all natural numbers.
(ii) The set of all prime numbers less than 100.
(iii) The set of all letters in the English alphabets.
Chapter 1
4
1.3.2 Set-Builder Form or Rule Form
Key Concept Set-Builder Form
Set-builder notation is a notation for describing a set by indicating the
properties that its members must satisfy.
Reading Notation
'|'or ':' such that
A = : x x is a letter in the word CHENNAI " ,
We read it as
"A is the set of all x such that x is a letter in the word CHENNAI"
For example,
(i) N = : x x is a natural number " ,
(ii) P = : x x is a prime number less than 100 " ,
(iii) A = : x x is a letter in the English alphabet " ,
1.3.3 Roster Form or Tabular Form
Key Concept Roster Form
Listing the elements of a set inside a pair of braces { } is called the roster form.
For example,
(i) Let A be the set of even natural numbers less than 11.
In roster form we write A = , , , , 2 4 6 8 10 " ,
(ii) A = : 1 5 x x x is an integer and 1 # - " ,
In roster form we write A = 1, , , , , , 0 1 2 3 4 - " ,
(i) In roster form each element of the set must be listed exactly once. By convention,
the elements in a set should NOT be repeated.
(ii) Let A be the set of letters in the word "COFFEE", i.e, A={ C, O, F, E }. So, in
roster form of the set A the following are invalid.
O C, , E " , (not all elements are listed)
C, O, F, F, E " , (element F is listed twice)
(iii) In a roster form the elements in a set can be written in ANY order.
R
e
m
a
r
k
Theory of Sets
5
The following are valid roster form of the set containing the elements 2, 3 and 4.
, , 2 3 4 " , , , 2 4 3 " , , , 4 3 2 " ,
Each of them represents the same set
(iv) If there are either infnitely many elements or a large fnite number of elements,
then three consecutive dots called ellipsis are used to indicate that the pattern of
the listed elements continues, as in , , , 5 6 7 g " , or , , , , , , 3 6 9 12 15 60 g " ,.
(v) Ellipsis can be used only if enough information has been given so that one can
fgure out the entire pattern.
Representation of sets in Different Forms
Descriptive Form Set - Builder Form Roster Form
The set of all vowels in
English alphabet
{ x : x is a vowel in the
English alphabet}
{a, e, i, o, u}
The set of all odd positive
integers less than or
equal to 15
{ x : x is an odd number
and 0 15 x 1 # }
{1, 3, 5, 7, 9, 11, 13, 15}
The set of all positive cube
numbers less than 100
{ x : x is a cube number
and 0 100 x # # }
{1, 8, 27, 64}
Example 1.2
List the elements of the following sets in Roster form:
(i) The set of all positive integers which are multiples of 7.
(ii) The set of all prime numbers less than 20.
Solution (i) The set of all positive integers which are multiples of 7 in roster form is
{7, 14, 21, 28,g}
(ii) The set of all prime numbers less than 20 in roster form is
{2, 3, 5, 7, 11, 13, 17, 19}
Example 1.3
Write the set A ={ x : x is a natural number # 8} in roster form.
Solution A ={ x : x is a natural number 8 # }.
So, the set contains the elements 1, 2, 3, 4, 5, 6, 7, 8.
Hence in roster form A ={1, 2, 3, 4, 5, 6, 7, 8}
Chapter 1
6
Example 1.4
Represent the following sets in set-builder form
(i) X ={Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
(ii) A = , , , , , 1
2
1
3
1
4
1
5
1
g
$ .
Solution (i) X ={Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
The set X contains all the days of a week.
Hence in set builder form, we write
X ={x : x is a day in a week}
(ii) A = , , , , , 1
2
1
3
1
4
1
5
1
g
$ .
. The denominators of the elements are 1, 2, 3, 4, g
` The set-builder form is : , A x x
n
n N
1
! = = $ .
1.3.4 Cardinal Number
Key Concept Cardinal Number
The number of elements in a set is called the cardinal number of the set.
Reading Notation
n(A) number of elements in the set A
The cardinal number of the set A is denoted by n(A).
For example,
Consider that the set , , , , , , A 1 0 1 2 3 4 5 = - # -. The set A has 7 elements.
` The cardinal number of A is 7 i.e., ( ) 7 n A = .
Example 1.5
Find the cardinal number of the following sets.
(i) A ={x : x is a prime factor of 12}
(ii) B ={ : , 5} x x x W ! #
Solution (i) Factors of 12 are 1, 2, 3, 4, 6, 12. So, the prime factors of 12 are 2, 3.
We write the set A in roster form as A ={2, 3} and hence n(A) = 2.
(ii) { : , 5} B x x x W ! # = . In Tabular form, B ={0, 1, 2, 3, 4, 5}.
The set B has six elements and hence n(B) =6
Theory of Sets
7
1.4 Different Kinds of Sets
1.4.1 The Empty Set
Key Concept Empty Set
A set containing no elements is called the empty set or null set or void set.
Reading Notation
Q or { } Empty set or Null set or Void set
The empty set is denoted by the symbol Q or { }
For example,
Consider the set : , A x x x 1 < N ! = # -.
There are no natural numbers which is less than 1.
{ } A ` =
The concept of empty set plays a key role in the study of sets just like the role
of the number zero in the study of number system.
1.4.2 Finite Set
Key Concept Finite Set
If the number of elements in a set is zero or fnite, then the set is called a
fnite set.
For example,
(i) Consider the set A of natural numbers between 8 and 9.
There is no natural numbers between 8 and 9. So, A ={ } and n(A) =0.
` A is a fnite set
(ii) Consider the set X ={x : x is an integer and x 1 2 # # - }.
X ={ 1 - , 0, 1, 2} and n(X) =4
` X is a fnite set
The cardinal number of a fnite set is fnite
Think and Answer !
What is ( ) n Q ?
Note
Note
Chapter 1
8
1.4.3 InfniteSet
Key Concept Infnite Set
A set is said to be an infnite set if the number of elements in the set is not fnite.
For example,
Let W =The set of all whole numbers. i. e., W ={0, 1, 2, 3, g}
The set of all whole numbers contain infnite number of elements
` W is an infnite set
The cardinal number of an infnite set is not a fnite number.
Example 1.6
State whether the following sets are fnite or infnite
(i) A ={x : x is a multiple of 5, x N ! }
(ii) B ={x : x is an even prime number}
(iii) The set of all positive integers greater than 50.
Solution (i) A ={x : x is a multiple of 5, x N ! } ={5, 10, 15, 20, ...}
` A is an infnite set.
(ii) B ={x : x is even prime numbers}. The only even prime number is 2
` B ={ 2 } and hence B is a fnite set.
(iii) Let X be the set of all positive integers greater than 50.
Then X ={51, 52, 53, ...}
` X is an infnite set.
1.4.4 Singleton Set
Key Concept Singleton Set
A set containing only one element is called a singleton set
For example,
Consider the set A ={x : x is an integer and 1 <x <3}.
A ={ 2 } i. e., A has only one element
` A is a singleton set
Note
Theory of Sets
9
It is important to recognise that the following sets are not equal.
(i) The null set Q
(ii) The set having the null set as its only element {Q}
(iii) The set having zero as its only element { 0 }
1.4.5 Equivalent Set
Key Concept Equivalent Set
Two sets A and B are said to be equivalent if they have the same number
of elements
In other words, A and B are equivalent if n(A) =n(B).
Reading Notation
. Equivalent
A and B are equivalent is written as A B c
For example,
Consider the sets A ={ 7, 8, 9, 10 } and B ={ 3, 5, 6, 11 }.
Here n(A) =4 and n(B) =4 ` A B .
1.4.6 Equal Sets
Key Concept Equal Sets
Two sets A and B are said to be equal if they contain exactly the same
elements, regardless of order. Otherwise the sets are said to be unequal.
In other words, two sets A and B, are said to be equal if
(i) every element of A is also an element of B and
(ii) every element of B is also an element of A.
Reading Notation
= Equal When two sets A and B are equal we write A =B.
!
Not equal When they are unequal, we write A B ! .
R
e
m
a
r
k
Chapter 1
10
For example,
Consider the sets
A ={ a, b, c, d } and B ={ d, b, a, c }
Set A and set B contain exactly the same elements ` A =B
If two sets A and B are equal, then n(A) =n(B).
But, if n(A) =n(B), then A and B need not be equal
Thus equal sets are equivalent but equivalent sets need not be equal
Example 1.7
Let A ={2, 4, 6, 8, 10, 12, 14} and
B ={x : x is a multiple of 2, x N ! and x 14 # }
State whether A =B or not.
Solution A ={2, 4, 6, 8, 10, 12, 14} and
B ={x : x is a multiple of 2, x N ! and x 14 # }
In roster form, B ={2, 4, 6, 8, 10, 12, 14}
Since A and B have exactly the same elements, A =B
1.4.7 Subset
Key Concept Subset
A set A is a subset of set B if every element of A is also an element of B.
In symbol we write A B 3
Reading Notation
3 is a subset of (or) is contained in
Read A B 3 as 'A is a subset of B' or 'A is contained in B'
M
is not a subset of (or) is not contained in
Read A B M as 'A is not a subset of B' or
'A is not contained in B'
For example,
Consider the sets
A ={7, 8, 9} and B ={ 7, 8, 9, 10 }
Theory of Sets
11
We see that every element of A is also an element of B.
` A is a subset of B.
i.e. A B 3 .
(i) Every set is a subset of itself i.e. A A 3 for any set A
(ii) The empty set is a subset of any set i.e., A Q 3 , for any set A
(iii) If A B 3 and B A 3 , then A =B.
The converse is also true i.e. if A =B then and A B B A 3 3
(iv) Every set (except Q) has atleast two subsets, Q and the set itself.
1.4.8 Proper Subset
Key Concept Proper Subset
A set A is said to be a proper subset of set B if A B 3 and A B ! . In symbol
we write A B 1 . B is called super set of A.
Reading Notation
1 is a proper subset of
Read A B 1 as, A is a proper subset of B
For example,
Consider the sets A ={5, 7, 8} and B ={ 5, 6, 7, 8 }
Every element of A is also an element of B and A B !
` A is a proper subset of B
(i) Proper subsets have atleast one element less than its superset.
(ii) No set is a proper subset of itself.
(iii) The empty set Q is a proper subset of every set except itself
(Q has no proper subset). i.e., Q1A if A is a set other than Q.
(iv) It is important to distinguish between and ! 3. The notation x A ! denotes
x is an element of A. The notation A B 3 means A is a subset of B.
Thus { , , } a b c Q 3 is true, but { , , } a b c Q ! is not true.
It is true that { }, x x ! but the relations { } x x = and { } x x 3 are not correct.
Note
R
e
m
a
r
k
Chapter 1
12
Example 1.8
Write or M 3 in each blank to make a true statement.
(a) {4, 5, 6, 7} ----- {4, 5, 6, 7, 8} (b) {a, b, c} ----- {b, e, f, g}
Solution (a) {4, 5, 6, 7} ----- {4, 5, 6, 7, 8}
Since every element of {4, 5, 6, 7} is also an element of {4, 5, 6, 7, 8},
place 3 in the blank.
` {4, 5, 6, 7} {4, 5, 6, 7, 8} 3
(b) The element a belongs to {a, b, c} but not to {b, e, f, g}
So, place M in the blank
{ , , } { , , , } a b c b e f g ` M
Example 1.9
Decide whether , 1 3 or both, can be placed in each blank to make a true statement.
(i) {8, 11, 13} ----- {8, 11, 13, 14}
(ii) {a, b, c} ------ {a, c, b}
Solution (i) Every element of the set {8, 11, 13} is also an element in the set {8, 11, 13, 14}
So, place 3 in the blank
{ , , } { , , , } 8 11 13 8 11 13 14 ` 3
Also, the element 14 belongs to {8, 11, 13, 14} but does not belong to {8, 11, 13}
` {8, 11, 13} is proper subset of {8, 11, 13, 14}.
So, we can also place 1 in the blank. ` {8, 11, 13} {8, 11, 13, 14} 1
(ii) Every element of {a, b, c} is also an element of {a, c, b}
and hence they are equal. So, {a, b, c} is not a proper subset of {a, c, b}
Hence we can only place 3 in the blank.
1.4.9 Power Set
Key Concept Power Set
The set of all subsets of A is said to be the power set of the set A.
Reading Notation
P(A) Power set of A
The power set of a set A is denoted by P(A)
Theory of Sets
13
For example,
Let A ={ , 3 4 - }
The subsets of A are , { 3}, {4}, { 3, 4} Q - - .
Then the power set of A is ( ) , { }, { }, { , } P A 3 4 3 4 Q = - - " ,
Example 1.10
Write down the power set of {3, {4, 5}} A =
Solution {3, {4, 5}} A =
The subsets of A are
, , , , {3, {4, 5}} 3 4 5 Q " " " , ,,
( ) P A ` = , 3 , 4, 5 , {3, {4, 5}} Q " " " " , ,, ,
Number of Subsets of a Finite Set
For a set containing a very large number of elements, it is diffcult to fnd the number of
subsets of the set. Let us fnd a rule to tell how many subsets are there for a given fnite set.
(i) The set A Q = has only itself as a subset
(ii) The set { } A 5 = has subsets Q and {5}
(iii) The set { , } A 5 6 = has subsets , {5}, {6}, {5, 6} Q
(iv) The set { , , } A 5 6 7 = has subsets , {5}, {6}, {7}, {5, 6}, {5, 7}, {6, 7} {5, 6, 7} and Q
This information is shown in the following table
Number of Elements 0 1 2 3
Number of subsets 1 2
0
=
2 2
1
= 4 2
2
= 8 2
3
=
This table suggests that as the number of elements of the set increases by one, the
number of subsets doubles. i.e. the number of subsets in each case is a power of 2.
Thus we have the following generalization
The number of subsets of a set with m elements is 2
m
The 2
m
subsets includes the given set itself.
` The number of proper subsets of a set with m elements is 2 1
m
-
( ) [ ( )] 2 n A m n P A 2
( ) n A m
& = = =
Chapter 1
14
Example 1.11
Find the number of subsets and proper subsets of each set
(i) { , , , , } A 3 4 5 6 7 = (ii) { , , , , , , , } A 1 2 3 4 5 9 12 14 =
Solution (i) {3, 4, 5, 6, 7} A = . So, ( ) n A 5 = . Hence,
The number of subsets = ( ) 2 32 n P A
5
= = 6 @ .
The number of proper subsets =2 1
5
- 32 1 = - 31 =
(ii) {1, 2, 3, 4, 5, 9, 12, 14} A = . Now, ( ) n A 8 = .
` The number of subsets 2 2 2 32 2 2 2
8 5 3
# # # # = = = =256
The number of proper subsets =2 1
8
- =256 1 - =255
Exercise 1.1
1. Which of the following are sets? Justify your answer.
(i) The collection of good books
(ii) The collection of prime numbers less than 30
(iii) The collection of ten most talented mathematics teachers.
(iv) The collection of all students in your school
(v) The collection of all even numbers
2. Let A ={0, 1, 2, 3, 4, 5}. Insert the appropriate symbol or g ! in the blank spaces
(i) 0 ----- A (ii) 6 ----- A (iii) 3 ----- A
(iv) 4 ----- A (v) 7 ----- A
3. Write the following sets in Set-Builder form
(i) The set of all positive even numbers
(ii) The set of all whole numbers less than 20
(iii) The set of all positive integers which are multiples of 3
(iv) The set of all odd natural numbers less than 15.
(v) The set of all letters in the word 'TAMILNADU'
4. Write the following sets in Roster form
(i) { : , 2 10} A x x x N 1 ! # =
(ii) : , B x x x
2
1
2
11
Z 1 1 ! = - $ .
Theory of Sets
15
(iii) C ={x : x is a prime number and a divisor of 6}
(iv) { : 2 , 5} X x x n n and N
n
! # = =
(v) { : 2 1, 5, } M x x y y y W # ! = = -
(vi) { : 16} P x x x is an integer,
2
# =
5. Write the following sets in Descriptive form
(i) A ={a, e, i, o, u}
(ii) B ={1, 3, 5, 7, 9, 11}
(iii) C ={1, 4, 9, 16, 25}
(iv) P ={x : x is a letter in the word 'SET THEORY'}
(v) Q ={x : x is a prime number between 10 and 20}
6. Find the cardinal number of the following sets
(i) { : 5 , 5} A x x n n and N
n
1 ! = =
(ii) B ={x : x is a consonant in English Alphabet}
(iii) X ={x : x is an even prime number}
(iv) P ={x : x <0, x W ! }
(v) Q ={ : , x x x 3 5 Z # # ! - }
7. Identify the following sets as fnite or infnite
(i) A ={4, 5, 6, ...}
(ii) B ={0, 1, 2, 3, 4, ... 75}
(iii) X ={x : x is a even natural number}
(iv) Y ={x : x is a multiple of 6 and x >0}
(v) P =The set of letters in the word 'KARIMANGALAM'
8. Which of the following sets are equivalent?
(i) A ={2, 4, 6, 8, 10}, B ={1, 3, 5, 7, 9}
(ii) X ={ : , 1 6}, { : x x x Y x x N 1 1 ! = is a vowel in the English Alphabet}
(iii) P ={x : x is a prime number and x 5 23 1 1 }
Q ={ : , 0 5 x x x W 1 ! # }
9. Which of the following sets are equal?
(i) A ={1, 2, 3, 4}, B ={4, 3, 2, 1}
Chapter 1
16
(ii) A ={4, 8, 12, 16}, B ={8, 4, 16, 18}
(iii) X ={2, 4, 6, 8}
Y ={x : x is a positive even integer 0 <x <10}
(iv) P ={x : x is a multiple of 10, x N ! }
Q ={10, 15, 20, 25 30, .... }
10. From the sets given below, select equal sets.
A ={12, 14, 18, 22}, B ={11, 12, 13, 14}, C ={14, 18, 22, 24}
D ={13, 11, 12, 14}, E ={ , 11 11 - }, F ={10, 19}, G ={ , 11 11 - }, H ={10, 11}
11. Is { } Q Q = ? Why ?
12. Which of the sets are equal sets? State the reason.
0, , {0}, { } Q Q
13. Fill in the blanks with or M 3 to make each statement true.
(i) {3} ----- {0, 2, 4, 6} (ii) {a } ----- {a, b, c}
(iii) {8, 18} ----- {18, 8} (iv) {d} ----- {a, b, c}
14. Let { , , , , , } X 3 2 1 0 1 2 = - - - and { : 3 2} Y x x x is an integer and 1 # = -
(i) Is X a subset of Y ? (ii) Is Y a subset of X ?
15. Examine whether A ={x : x is a positive integer divisible by 3} is a subset of
B ={x : x is a multiple of 5, x N ! }
16. Write down the power sets of the following sets.
(i) A ={x, y} (ii) X ={a, b, c} (iii) A ={5, 6, 7, 8} (iv) A Q =
17. Find the number of subsets and the number of proper subsets of the following sets.
(i) A ={13, 14, 15, 16, 17, 18}
(ii) B ={ , , , , , , } a b c d e f g
(iii) X ={ : , } x x x W N g !
18. (i) If A Q = , fnd ( ) n P A 6 @ (ii) If ( ) 3, ( ) n A n P A find = 6 @
(iii) If ( ) n P A 512 = 6 @ fnd n(A)
(iv) If ( ) n P A 1024 = 6 @ fnd n(A)
19. If ( ) n P A 1 = 6 @ , what can you say about the set A?
Theory of Sets
17
20. Let A ={x : x is a natural number <11}
B ={x : x is an even number and 1 <x <21}
C ={x : x is an integer and x 15 25 # # }
(i) List the elements of A, B, C
(ii) Find n(A), n(B), n(C).
(iii) State whether the following are True (T) or False (F)
(a) B 7 ! (b) 16 A g
(c) { , , } C 15 20 25 1 (d) { , } B 10 12 1
1.5 SET OPERATI ONS
1.5.1 Venn Diagrams
We use diagrams or pictures in geometry to
explain a concept or a situation and sometimes we also
use them to solve problems. In mathematics, we use
diagrammatic representations called Venn Diagrams
to visualise the relationships between sets and set
operations.
1.5.2 The Universal Set
Sometimes it is useful to consider a set which contains all elements pertinent to a
given discussion.
Key Concept Universal Set
The set that contains all the elements under consideration in a given
discussion is called the universal set. The universal set is denoted by U.
For example,
If the elements currently under discussion are integers, then the universal set U is the set
of all integers. i.e., { : } U n n Z d =
The universal set may change from problem to problem.
Remark
J ohn Venn (1834-1883)
a British mathematician used
diagrammatic representation as an
aid to visualize various relationships
between sets and set operations.
John Venn
(1834-1883)
Chapter 1
18
In Venn diagrams, the universal set is generally represented
by a rectangle and its proper subsets by circles or ovals inside the
rectangle. We write the names of its elements inside the fgure.
1.5.3 Complement of a Set
Key Concept Complement Set
The set of all elements of U (universal set) that are not elements
of A U 3 is called the complement of A. The complement of A is
denoted by Al or A
c
.
Reading Notation
In symbol, { : } A x x U x A and g ! =
l
For example,
Let U ={ , , , , , , , } a b c d e f g h and A ={ , , , } b d g h .
Then { , , , } A a c e f =
l
In Venn diagram Al, the complement of
set A is represented as shown in Fig. 1.2
(i) ( ) A A =
l l (ii) U Q =
l (iii) U Q =
l
1.5.4 Union of Two Sets
Key Concept Union of Sets
The union of two sets A and B is the set of elements which are in A or in B
or in both A and B. We write the union of sets A and B as A B , .
Reading Notation
, Union
Read A B , as 'A union B'
In symbol, { : } A B x x A x B or , ! ! =
Note
Al
3
1 5 6
4 7
8
A
U
Fig. 1.1
A
U
Al (shaded portion)
Fig. 1.2
a c
e f
g
b
d h
Theory of Sets
19
For example,
Let A ={11, 12, 13, 14} and
B ={9, 10, 12, 14, 15}.
Then A B , ={9, 10, 11, 12, 13, 14, 15}
The union of two sets can be represented by a
Venn diagram as shown in Fig. 1.3
(i) A A A , = (ii) A A , Q = (iii) A A U , =
l
(iv) If A is any subset of U then A U U , =
(v) A B 3 if and only if A B B , = (vi) A B B A , , =
Example 1.12
Find the union of the following sets.
(i) {1, 2, 3, 5, 6} {4, 5, 6, 7, 8} A B and = =
(ii) {3, 4, 5} X = and Y Q =
Solution (i) A ={1, 2, 3, 5, 6} and B ={4, 5, 6, 7, 8}
1, 2, 3, 5, 6 ; 4, 5, 6, 7, 8 (repeated)
{ , , , , , , , } A B 1 2 3 4 5 6 7 8 ` , =
(ii) ={3,4,5}, Y Q = . There are no elements in Y
{ , , } X Y 3 4 5 ` , =
1.5.5 I ntersection of Two Sets
Key Concept Intersection of Sets
The intersection of two sets A and B is the set of all elements common to
both A and B. We denote it as A B + .
Reading Notation
+ Intersection
Read A B + as 'A intersection B'
Symbolically, we write { : } A B x x A x B and + ! ! =
Note
Think and Answer !
Can we say
( ) A A B and , 1
( ) B A B , 1 ?
B
U
A
A B , (shaded portion)
Fig. 1.3
U
11
13
9
10
15
12
14
Chapter 1
20
For example,
Let A ={ , , , , } a b c d e and B ={ , , , } a d e f .
{ , , } A B a d e ` + =
The intersection of two sets can be
represented by a Venn diagram as
shown in Fig. 1.4
(i) A A A + = (ii) A + Q Q =
(iii) A A + Q =
l (iv) A B B A + + =
(v) If A is any subset of U, then A U A + =
(vi) If A B 3 if and only if A B A + =
Example 1.13
Find A B + if (i) A ={ 10, 11, 12, 13}, B ={12, 13, 14, 15}
(ii) A ={5, 9, 11}, B =Q
Solution (i) A ={10, 11, 12, 13} and B ={12, 13, 14, 15}.
12 and 13 are common in both A and B.
` A B + ={12, 13}
(ii) A ={5, 9, 11} and B =Q .
There is no element in common and hence A B + Q =
When B A 3 , the union and intersection of two sets A and B are represented
in Venn diagram as shown in Fig.1.6 and in Fig.1.7respectively
1.5.6 Disjoint Sets
Key Concept Disjoint Sets
Two sets A and B are said to be disjoint if there is no element common to
both A and B.
In other words, if A and B are disjoint sets, then A B + Q =
Note
Remark
A
B B
A
B
A
B A 3
Fig.1.5
A B , (shaded portion)
Fig.1.6
A B + (shaded portion)
Fig.1.7
Think and Answer !
Can we say
( ) A B A and + 1
( ) A B B + 1 ?
B
U
A
A B + (shaded portion)
Fig. 1.4
b
f
c
a
d
e
Theory of Sets
21
For example,
Consider the sets { , , , } A 5 6 7 8 = and B ={11, 12, 13}.
We haveA B + Q = . So A and B are disjoint sets.
Two disjoint sets A and B are represented in
Venn diagram as shown in Fig.1.8
(i) The union of two disjoint sets A and B are
represented in Venn diagram as shown in
Fig.1.9
(ii) If A B + Q ! , then the two sets A and B are
said to be overlapping sets
Example 1.14
Given the sets A ={4, 5, 6, 7} and B ={1, 3, 8, 9}. Find A B + .
Solution A ={4, 5, 6, 7} and B ={1, 3, 8, 9}. So A B + Q = .
Hence A and B are disjoint sets.
1.5.7 Difference of Two Sets
Key Concept Difference of two Sets
The difference of the two sets A and B is the set of all elements belonging
to A but not to B. The difference of the two sets is denoted by A B - or \ A B.
Reading Notation
A B - or \ A B A difference B (or) A minus B
In symbol, we write : { : } A B x x A x B and g ! - =
Similarly, we write : { : } B A x x B x A and g ! - =
For example,
Consider the sets A ={2, 3, 5, 7, 11} and B ={5, 7, 9, 11, 13}
To fnd A B - , we remove the elements of B from A.
{2, 3} A B ` - =
A
U
B
Disjoint sets
Fig.1.8
B
U
A
A B , (shaded portion)
Fig.1.9
Note
Chapter 1
22
(i) Generally, A B B A ! - - . (ii) A B B A A B + - = - =
(iii) U A A - =
l (iv) U A A - =
l
The difference of two sets A and B can be represented by Venn diagram as shown in
Fig.1.10 and in Fig.1.11. The shaded portion represents the difference of the two sets
Example 1.15
If { , , , , }, { , , } A B 2 1 0 3 4 1 3 5 = - - = - , fnd (i) A B - (ii) B A -
Solution A ={ , , , , } 2 1 0 3 4 - - and B ={ , , } 1 3 5 - .
(i) A B - ={ , , } 2 0 4 - (ii) B A - ={ 5 }
1.5.8 Symmetric Difference of Sets
Key Concept Symmetric Difference of Sets
The symmetric difference of two sets A and B is the union of their differences
and is denoted by A B D .
Reading Notation
A B D
A symmetric B
Thus, ( ) ( ) A B A B B A , D = - -
For example,
Consider the sets A ={ , , , } a b c d and B ={ , , , } b d e f . We have
A B - ={a, c} and B A - ={e, f }
( ) ( ) A B A B B A ` , D = - - ={ , , , } a c e f
The symmetric difference of two sets A and B can
be represented by Venn diagram as shown in Fig.1.12 The
shaded portion represents the symmetric difference of the
two sets A and B.
Note
B
U
A
A -B
Fig.1.10
B
U
A
B -A
Fig.1.11
B
U
A
A-B
B-A
( ) ( ) A B A B B A 3 , = - -
Fig.1.12
Theory of Sets
23
(i) A A 3 Q = (ii) A B B A 3 3 =
(iii) From the Venn diagram, we can write
: A B x x A B 3 + g = " ,
So, we can fnd the elements of A B 3 , by listing the elements
which are not common to both A and B.
Example 1.16
If A = {2, 3, 5, 7, 11} and B = {5, 7, 9, 11, 13} , fnd A B 3 .
Solution Given A ={2, 3, 5, 7, 11} and B ={5, 7, 9, 11, 13}. So
A B - ={2, 3} and B A - ={9, 13}. Hence
A B 3 =( ) ( ) A B B A , - - ={2, 3, 9, 13}
Exercise 1.2
1. Find A B , and A B + for the following sets.
(i) A ={0, 1, 2, 4, 6} and B ={ , , , , , } 3 1 0 2 4 5 - -
(ii) A ={2, 4, 6, 8} and B =Q
(iii) A ={ : , 5} x x x N ! # and B ={ x : x is a prime number less than 11}
(iv) A ={ : , 2 7} { : , 0 6} x x x B x x x and N W 1 ! # ! # # =
2. If A ={x : x is a multiple of 5 , x # 30 and x N ! }
B ={1, 3, 7, 10, 12, 15, 18, 25},
Find (i) A B , (ii) A B +
3. If { : 2 , 20 } X x x n x n and N # ! = = and { : 4 , 20 } Y x x n x and n W # ! = =
Find (i) X Y , (ii) X Y +
4. U ={1, 2, 3, 6, 7, 12, 17, 21, 35, 52, 56},
P ={numbers divisible by 7}, Q ={prime numbers},
List the elements of the set { : } x x P Q + !
5. State which of the following sets are disjoint
(i) A ={2, 4, 6, 8} ; B ={x : x is an even number <10, x N ! }
(ii) X ={1, 3, 5, 7, 9}, Y ={0, 2, 4, 6, 8, 10}
(iii) P ={x : x is a prime <15}
Q ={x : x is a multiple of 2 and x <16}
(iv) R ={ , , , , }, { , , , , } a b c d e S d e a b c =
Note
Chapter 1
24
6. (i) If U ={ : 0 10, } x x x W # # ! and { : A x x = is a multiple of 3}, fnd Al
(ii) If U is the set of natural numbers and Al is the set of all composite numbers, then
what is A ?
7. If { , , , , , , , } U a b c d e f g h = , { , , , } { , , , } A a b c d B b d f g and = = ,
fnd (i) A B , (ii) ( ) A B , l (iii) A B + (iv) ( ) A B + l
8. If { : 1 10, } U x x x N # # ! = , A ={1, 3, 5, 7, 9} and B ={2, 3, 5, 9, 10},
fnd (i) Al (ii) Bl (iii) A B , l l (iv) A B + l l
9. Given that U ={3, 7, 9, 11, 15, 17, 18}, M ={3, 7, 9, 11} and N ={7, 11, 15, 17},
fnd (i) M N - (ii) N M - (iii) N M -
l (iv) M N -
l
(v) ( ) M M N + - (vi) ( ) N N M , - (vii) ( ) n M N -
10. If A ={3, 6, 9, 12, 15, 18}, B ={4, 8, 12, 16, 20}, C ={2, 4, 6, 8, 10, 12} and
D = {5, 10, 15, 20, 25}, fnd
(i) A B - (ii) B C - (iii) C D - (iv) D A - (v) ( ) n A C - (vi) ( ) n B A -
11. Let U ={x : x is a positive integer less than 50}, A ={x : x is divisible by 4} and
B ={x : x leaves a remainder 2 when divided by 14}.
(i) List the elements of U, A and B
(ii) Find A B , , A B + , ( ) n A B , , ( ) n A B +
12. Find the symmetric difference between the following sets.
(i) { , , , , }, { , , , , } X a d f g h Y b e g h k = =
(ii) { : 3 9, }, { : 5, } P x x x Q x x x N W 1 1 1 ! ! = =
(iii) { , , , , , }, { , , , , , } A B 3 2 0 2 3 5 4 3 1 0 2 3 = - - = - - -
13. Use the Venn diagram to answer the following questions
(i) List the elements of
U, E, F, E F , and E F +
(ii) Find ( ) n U , ( ) n E F , and ( ) n E F +
14. Use the Venn diagram to answer the following questions
(i) List U, G and H
(ii) Find Gl, Hl, G H + l l, ( ) n G H , l and ( ) n G H + l
3
E
U
F
10
1
2
4
7
9
11
5
G
U
H
9
1
4
8
2
6
10
3
Fig. 1.13
Fig. 1.14
Theory of Sets
25
1.6 Representation of Set Operations Using Venn Diagram
We shall now give a few more representations of set operations in Venn diagrams
(a) A B , (b) A B ,
l ^ h
(c)
A B , l l
Similarly the shaded regions represent each of the following set operations.
B
U
A
U
A
B
A B , l l (shaded portion)
Fig. 1.17
U
B
A
U
B A
Step 1 : Shade the regionAl
Step 2 : Shade the region Bl
U
B
A
B
U
A
B
U
A
A B + (shaded portion)
Fig. 1.18
A B +
l ^ h (shaded portion)
Fig. 1.19
A B + l (shaded portion)
Fig. 1.21
B
U
A B
U
A
A B + l (shaded portion)
Fig 1.20.
Fig. 1.15 Fig. 1.16
Chapter 1
26
We can also make use of the following idea
to represent sets and set operations in Venn
diagram.
In Fig. 1.22 the sets A and B divide the universal
set into four regions. These four regions are numbered for
reference. This numbering is arbitrary.
Region 1 Contains the elements outside of both the sets A and B
Region 2 Contains the elements of the set A but not in B
Region 3 Contains the elements common to both the sets A and B.
Region 4 Contains the elements of the set B but not in A
Example 1.17
Draw a Venn diagram similar to one at the side and shade the
regions representing the following sets
(i) Al (ii) Bl (iii) A B , l l (iv) ( ) A B , l (v) A B + l l
Solution
(i) Al
(ii) Bl
(iii) A B , l l
A
U
B
2 3 4
1
B
U
A
U
B A
Al (shaded portion)
Fig. 1.24
Tip to shade
Set Shaded Region
Al 1 and 4
B
U
A
Bl (shaded portion)
Fig. 1.25
Tip to shade
Set Shaded Region
Bl 1 and 2
B
U
A
A B , l l (shaded portion)
Fig. 1.26
Tip to shade
Set Shaded Region
Al 1 and 4
Bl 1 and 2
A B , l l 1, 2 and 4
Fig. 1.22
Fig. 1.23
R
e
m
a
r
k
Theory of Sets
27
(iv) ( ) A B , l
(v) A B + l l
I mportant Results
For any two fnite sets A and B, we have the following useful results(vi) ( ) ( ) ( ) n A n A n U + =
l
Example 1.18
From the given Venn diagram, fnd the following2, 3, 4, 5, 6, 7, 8, 9}, (ii) B ={3, 6, 9,},
(iii) A B , ={2, 3, 4, 5, 6, 7, 8, 9} and (iv) A B + ={3, 6, 9}
U
A B
( ) A B , l (shaded portion)
Fig. 1.27
Tip to shade
Set Shaded Region
A B , 2, 3 and 4
( ) A B , l 1
U
A B
A B + l l (shaded portion)
Fig. 1.28
Tip to shade
Set Shaded Region
Al 1 and 4
Bl 1 and 2
A B + l l 1
B
U
A
A-B
B-A
Fig. 1.29
Fig. 1.30
A B +
Chapter 1
28
We have ( ) , ( ) , ( ) , ( ) n A n B n A B n A B 8 3 8 3 , + = = = = . Now
( ) ( ) ( ) n A n B n A B + + - =8 3 3 + - =8
Hence, ( ) ( ) ( ) n A n B n A B + + - = ( ) n A B ,
Example 1.19
From the given Venn diagram fnd , , , , , } a b d e g h , (ii) B ={ , , , , , , } b c e f h i j ,
(iii) A B , ={ , , , , , , , , , a b c d e f g h i j } and (iv) A B + ={ , , } b e h
So, ( ) 6, ( ) 7, ( ) 10, ( ) 3 n A n B n A B n A B , + = = = = . Now
( ) ( ) ( ) n A n B n A B + + - =6 7 3 + - =10
Hence, ( ) ( ) ( ) n A n B n A B + + - = ( ) n A B ,
Example 1.20
If ( ) 12, ( ) 17 ( ) 21 n A n B n A B and , = = = , fnd ( ) n A B +
Solution Given that ( ) 12, ( ) 17 ( ) 21 n A n B n A B and , = = =
By using the formula ( ) ( ) ( ) ( ) n A B n A n B n A B , + = + -
( ) n A B + =12 17 21 + - =8
Example 1.21
In a city 65% of the people view Tamil movies and 40% view English movies, 20% of
the people view both Tamil and English movies. Find the percentage of people do not view
any of these two movies.
Solution Let the population of the city be 100. Let T denote the set of people who
view Tamil movies and E denote the set of people who view English movies. Then
( ) , ( ) , ( ) n T n E n T E 65 40 20 + = = = . So, the number of people who view either of these
movies is
( ) n T E , = ( ) ( ) ( ) n T n E n T E + + -
=65 40 20 + - =85
Hence the number of people who do not view any of these movies is 100 85 - =15
Hence the percentage of people who do not view any of these movies is 15
A
U
B
a
d
g
b
e
h
c
f
i
j
Fig. 1.31
A
B
Theory of Sets
29
Aliter
From the Venn diagram the percentage of people
who view at least one of these two movies is
45 20 20 85 + + =
Hence, the percentage of people who do not view any
of these movies =100 - 85 =15
Example 1.22
In a survey of 1000 families, it is found that 484 families use electric stoves, 552
families use gas stoves. If all the families use atleast one of these two types of stoves, fnd
how many families use both the stoves?
Solution Let E denote the set of families using electric stove and G denote the set of families
using gas stove. Then ( ) , ( ) , ( ) n E n G n E G 484 552 1000 , = = = . Let x be the number of
families using both the stoves . Then ( ) n E G x + = .
Using the results
( ) ( ) ( ) ( ) n E G n E n G n E G , + = + -
1000 = x 484 552 + -
x & =1036 1000 - = 36
Hence, 36 families use both the stoves.
Aliter
From the Venn diagram,
x x x 484 552 - + + - =1000
x 1036 & - =1000
& x - = 36 -
x =36
Hence, 36 families use both the stoves.
Example 1.23
In a class of 50 students, each of the student passed either in mathematics or in science
or in both. 10 students passed in both and 28 passed in science. Find how many students
passed in mathematics?
Solution Let M =The set of students passed in Mathematics
S =The set of students passed in Science
E
G
484- x x 552- x
Fig. 1.33
E G
T
E
45 20 20
Fig. 1.32
T
E
Chapter 1
30
Then, ( ) 28, ( ) 10 n S n M S + = = , ( ) n M S 50 , =
We have n M S , ^ h =n M n S n M S + + - ^ ^ ^ h h h
50 =n M 28 10 + - ^ h
& n M ^ h =32
Aliter
From the Venn diagram
x +10 +18 =50
x = 50 28 - =22
Number of students passed in Mathematics =x +10 =22 +10 =32
Exercise 1.3
1. Place the elements of the following sets in the proper location on the given Venn
diagram.
U ={5, 6, 7, 8, 9, 10, 11, 12, 13}
M ={5, 8, 10, 11}, N ={5, 6, 7, 9, 10}
2. If A and B are two sets such that A has 50 elements, B has 65 elements and A B , has
100 elements, how many elements does A B + have?
3. If A and B are two sets containing 13 and 16 elements respectively, then fnd the
minimum and maximum number of elements in A B , ?
4. If ( ) , ( ) , ( ) n A B n A B n A 5 35 13 + , = = = , fnd ( ) n B .
5. If ( ) , ( ) , ( ) , ( ) n A n B n A B n A 26 10 30 17 , = = = =
l , fnd ( ) n A B + and ( ) n , .
6. If ( ) , ( ) , ( ) , ( ) n n A n A B n B 38 16 12 20 , + = = = =
l , fnd ( ) n A B , .
7. Let A and B be two fnite sets such that , , . n A B n A B n A B 30 180 60 , + - = = = ^ ^ ^ h h h
Find n B ^ h
8. The population of a town is 10000. Out of these 5400 persons read newspaper A and
4700 read newspaper B. 1500 persons read both the newspapers. Find the number of
persons who do not read either of the two papers.
9. In a school, all the students play either Foot ball or Volley ball or both. 300 students
play Foot ball, 270 students play Volley ball and 120 students play both games. Find
(i) the number of students who play Foot ball only
(ii) the number of students who play Volley ball only
(iii) the total number of students in the school
M
S
x 10 18
Fig. 1.34
M
S
N
U
M
Fig. 1.35
M
N
Theory of Sets
31
10. In an examination 150 students secured frst class in English or Mathematics. Out of
these 50 students obtained frst class in both English and Mathematics. 115 students
secured frst class in Mathematics. How many students secured frst class in English
only?
11. In a group of 30 persons, 10 take tea but not coffee. 18 take tea. Find how many take
coffee but not tea, if each person takes atleast one of the drinks.
12. In a village there are 60 families. Out of these 28 families speak only Tamil and 20
families speak only Urdu. How many families speak both Tamil and Urdu.
13. In a School 150 students passed X Standard Examination. 95 students applied for
Group I and 82 students applied for Group II in the Higher Secondary course. If 20
students applied neither of the two, how many students applied for both groups?
14. Pradeep is a Section Chief for an electric utility company. The employees in his
section cut down tall trees or climb poles. Pradeep recently reported the following
information to the management of the utility.
Out of 100 employees in my section, 55 can cut tall trees, 50 can climb poles, 11
can do both, 6 can't do any of the two. Is this information correct?
15. A and B are two sets such that ( ) , ( ) n A B x n B A x 32 5 - = + - = and ( ) n A B x + =
Illustrate the information by means of a Venn diagram. Given that ( ) ( ) n A n B = .
Calculate (i) the value of x (ii) ( ) n A B , .
16. The following table shows the percentage of the students of a school who participated
in Elocution and Drawing competitions.
Competition Elocution Drawing Both
Percentage of Students 55 45 20
Draw a Venn diagram to represent this information and use it to fnd the percentage of
the students who
(i) participated in Elocution only
(ii) participated in Drawing only
(iii) do not participated in any one of the competitions.
17. A village has total population 2500. Out of which 1300 use brand A soap and 1050
use brand B soap and 250 use both brands. Find the percentage of population
who use neither of these soaps.
Chapter 1
32
Points to Remember
› A set is a well-defned collection of distinct objects
› Set is represented in three forms (i) Descriptive Form (ii) Set-builder Form
(iii) Roster Form
› The number of elements in a set is said to be the cardinal number of the set.
› A set containing no element is called the empty set
› If the number of elements in a set is zero or fnite, the set is called a fnite set.
Otherwise, the set is an infnite set.
› Two sets A and B are said to be equal if they contain exactly the same elements.
› A set A is a subset of a set B if every element of A is also an element of B.
› A set A is a proper subset of set B if A B 3 and A B !
› The power set of the set A is the set of all subsets of A. It is denoted by P(A).
› The number of subsets of a set with melements is 2
m
.
› The number of proper subsets of a set with melements is 2 1
m
-
› The set of all elements of the universal set that are not elements of a set A is called
the complement of A. It is denoted by Al.
› The union of two sets A and B is the set of elements which are in A or in B or in
both A and B.
› The intersection of two sets A and B is the set of all elements common to both A and B.
› If A and B are disjoint sets, then A B + Q =
› The difference of two sets A and B is the set of all elements belonging to A but not to B.
› Symmetric difference of two sets A and B is defned as ( ) ( ) A B A B B A 3 , = - -
› For any two fnite sets A and B, we haveReal Number System
33
2.1 I ntroduction
All the numbers that we use in normal day-to-day
activities to represent quantities such as distance, time, speed,
area, profit, loss, temperature, etc., are called Real Numbers.
The system of real numbers has evolved as a result of a process
of successive extensions of the system of natural numbers. The
extensions became inevitable as the science of Mathematics
developed in the process of solving problems from other fields.
Natural numbers came into existence when man first learnt
counting. The Egyptians had used fractions around 1700 BC;
around 500 BC, the Greek mathematicians led by Pythagoras
realized the need for irrational numbers. Negative numbers
began to be accepted around 1600 A.D. The development of
calculus around 1700 A.D. used the entire set of real numbers
without having defined them clearly. George Cantor can be
considered the first to suggest a rigorous definition of real
numbers in 1871 A.D.
Main Targets
RI CHARD DEDEK I ND
(1831-1916)
Richard Dedekind
(1831-1916) belonged to an
elite group of mathematicians
who had been students of the
legendary mathematician Carl
Friedrich Gauss.
He did important work in
abstract algebra, algebraic
number theory and laid the
foundations for the concept of
the real numbers. He was one
of the few mathematicians who
understood the importance
of set theory developed by
Cantor. While teaching
calculus for the first time at
Polytechnic, Dedekind came
up with the notion now called
a Dedekind cut, a standard
definition of the real numbers.
Life is good for only two things, discovering mathematics and
teaching mathematics
- SIMEON POISSON
● To recall Natural numbers, Whole numbers, Integers.
● To classify rational numbers as recurring / terminating
decimals.
● To understand the existence of non terminating and non
recurring decimals.
● To represent terminating / non terminating decimals on
the number line.
● To understand the four basic operations in irrational
numbers.
● To rationalise the denominator of the given irrational
numbers.
REAL NUMBER SYSTEM
33
Chapter 2
34
-5 -4 -3 -2 -1 0 1 2 3 4 5
Remark
The line extends
endlessly only to the
right side of 0.
0 1 2 3 4 5 6 7 8 9
Remark
Z is derived from
the German word
'Zahlen',
means 'to count'
1 2 3 4 5 6 7 8
The line extends
endlessly only to
the right side of 1.
The line extends
endlessly on both
sides of 0.
In this chapter we discuss some properties of real numbers. First, let us recall various
types of numbers that you have learnt in earlier classes.
2.1.1 Natural Numbers
The counting numbers 1, 2, 3, g are called natural numbers.
The set of all natural numbers is denoted by N.
i.e., N = {1, 2, 3, g}
The smallest natural number is 1, but there is no largest number as it goes
up continuously.
2.1.2 Whole Numbers
The set of natural numbers together with zero forms the
set of whole numbers.
The set of whole numbers is denoted by W.
W = { 0, 1, 2, 3, g }
The smallest whole number is 0
1) Every natural number is a whole number.
2) Every whole number need not be a natural number.
For, 0 W d , but 0 N g
3) N W 1
2.1.3 I ntegers
The natural numbers, their negative numbers together with zero are called integers.
The set of all integers is denoted by Z
Z ={ g -3, -2, -1, 0, 1, 2, 3, g}
N
W
Real Number System
35
1, 2, 3 g are called positive integers.
, , 1 2 3 - - - g are called negative integers.
1) Every natural number is an integer.
2) Every whole number is an integer.
3) N W Z 1 1
2.1.4 Rational Numbers
A number of the form
q
p
, where p and q are both integers and q 0 ! is called a
rational number.
For example, 3
1
3
= ,
6
5
- ,
8
7
are rational numbers.
The set of all rational numbers is denoted by Q.
Q = : , ,
q
p
p q q 0 and Z Z ! ! ! ' 1
1) A rational number may be positive, negative or zero.
2) Every integer n is also a rational number,
since we can write n as
n
1
.
3) N W Z Q 1 1 1
I mportant Results
1) If a and b are two distinct rational numbers, then
a b
2
+
is a rational number
between a and b such that a
a b
b
2
< <
+
.
2) There are infinitely many rational numbers
between any two given rational numbers.
Example 2.1
Find any two rational numbers between
4
1
and
4
3
.
Remark
Think and Answer !
Is zero a positive integer
or a negative integer?
-2 -1 0 1 2
4
3
-
2
1
-
1
4
-
4
1
2
1
4
3
Remark
Think and Answer !
Can you correlate the word ratio
with rational numbers ?
We find rational numbers in
between integers
Z
N
Q
W
N
W
Z
Chapter 2
36
Solution A rational number between
4
1
and
4
3
=
2
1
4
1
4
3
+ ` j
=
2
1
(1) =
2
1
Another rational number between
2
1
and
4
3
=
2
1
2
1
4
3
+ ` j
=
2
1
4
5
# =
8
5
The rational numbers
2
1
and
8
5
lie between
4
1
and
4
3
There are infnite number of rationals between
4
1
and
4
3
. The rationals
2
1
and
8
5
that we have obtained in Example 2.1 are two among them
Exercise 2.1
1. State whether the following statements are true or false.
(i) Every natural number is a whole number.
(ii) Every whole number is a natural number.
(iii) Every integer is a rational number.
(iv) Every rational number is a whole number.
(v) Every rational number is an integer.
(vi) Every integer is a whole number.
2. Is zero a rational number ? Give reasons for your answer.
3. Find any two rational numbers between
7
5
- and
7
2
- .
2.2 Decimal Representation of Rational Numbers
If we have a rational number written as a fraction
q
p
, we get the decimal representation
by long division.
When we divide p by q using long division method either the remainder becomes zero
or the remainder never becomes zero and we get a repeating string of remainders.
Case (i) The remainder becomes zero
Let us express
16
7
in decimal form. Then
16
7
= 0.4375
In this example, we observe that the remainder becomes zero after a few steps.
Note
Real Number System
37
Also the decimal expansion of
16
7
terminates.
Similarly, using long division method we can express the following
rational numbers in decimal form as
7
22
= 3.142857 142857 g =3.142857
The following table shows decimal representation of the reciprocals of the frst ten
natural numbers. We know that the reciprocal of a number n is
n
1
. Obviously, the reciprocals
of natural numbers are rational numbers.
Number Reciprocal Type of Decimal
1 1.0 Terminating
2 0.5 Terminating
3
. 0 3
Non-terminating
and recurring
4 0.25 Terminating
5 0.2 Terminating
6
. 0 16
Non-terminating
and recurring
7
. 0 142857
Non-terminating
and recurring
8 0.125 Terminating
9
. 0 1
Non-terminating
and recurring
10 0.1 Terminating
Thus we see that,
A rational number can be expressed by either a terminating or
a non-terminating and recurring decimal expansion.
Real Number System
39
The converse of this statement is also true.
That is, if the decimal expansion of a number is terminating or non-terminating and
recurring, then the number is a rational number.
We shall illustrate this with examples.
2.2.1 Representing a Terminating Decimal Expansion in the form
p
q
Terminating decimal expansion can easily be expressed in the form
q
p
(p, q Z ! and q ! 0).
This method is explained in the following example.
Example 2.2
Express the following decimal expansion in the form
q
p
, where p and q are integers
and q ! 0.
(i) 0.75 (ii) 0.625 (iii) 0.5625 (iv) 0.28
Solution (i) 0.75 =
100
75
=
4
3
(ii) 0.625 =
1000
625
=
8
5
150
13
=
2 3 5
13
2
# #
Since it is not in the form
2 5
p
m n
#
,
150
13
has a non-terminating and recurring decimal
expansion.
(iii)
75
11 -
=
3 5
11
2
#
-
Since it is not in the form
2 5
p
m n
#
,
75
11 -
has a non-terminating and recurring decimal
expansion.
(iv)
200
17
=
8 25
17
#
=
2 5
17
3 2
#
. So
200
17
has a terminating decimal expansion.
Chapter 2
42
Example 2.5
Convert 0.9 into a rational number.
Solution Let x =0.9. Then x = 0.99999g
Multiplying by 10 on both sides, we get
10x = 9.99999g = 9 + 0.9999g = 9 + x
( 9x = 9
( x = 1. That is, 0.9 = 1 (a 1 is rational number)
We have proved 0.9 = 1. Isn't it surprising?
Most of us think that 0.9999gis less than 1. But this is not the
case. It is clear from the above argument that 0.9 = 1. Also this result is
consistent with the fact that 3 0.333 . 0 999 # g g = , while 3
3
1
1 # = .
Similarly, it can be shown that any terminating decimal can
be represented as a non-terminating and recurri ng deci mal
expansi on wi t h an endl ess bl ocks of 9s.
For exampl e 6 = 5. 9999g, 2. 5 = 2. 4999g.
Exercise 2.2
1. Convert the following rational numbers into decimals and state the kind of decimal
expansion.
(i)
100
42
(ii) 8
7
2
(iii)
55
13
(iv)
500
459
(v)
11
1
(vi)
13
3
- (vii)
3
19
(viii)
32
7
-
2. Without actual division, fnd which of the following rational numbers have terminating
decimal expansion.
(i)
64
5
(ii)
12
11
(iii)
40
27
(iv)
35
8
3. Express the following decimal expansions into rational numbers.
(i) . 0 18 (ii) . 0 427 (iii) . 0 0001
(iv) . 1 45 (v) . 7 3 (vi) 0.416
4. Express
13
1
in decimal form. Find the number of digits in the repeating block.
5. Find the decimal expansions of
7
1
and
7
2
by division method. Without using the long
division method, deduce the decimal expressions of , , ,
7
3
7
4
7
5
7
6
from the decimal
expansion of
7
1
.
For your
Thought
Real Number System
43
2.3 I rrational Numbers
Let us have a look at the number line again. We have represented rational numbers on
the number line. We have also seen that there are infnitely many rational numbers between
any two given rational numbers. In fact there are
infnitely many more numbers left on the number line,
which are not rationals.
In other words there are numbers whose decimal
expansions are non-terminating and non-recurring.
Thus, there is a need to extend the system of rational
numbers. Consider the following decimal expansion
0.808008000800008g (1) .
This is non-terminating. Is it recurring?
It is true that there is a pattern in this decimal
expansion, but no block of digits repeats endlessly and so it is not recurring.
Thus, this decimal expansion is non-terminating and non-repeating (non-recurring).
So it cannot represent a rational number. Numbers of this type are called irrational numbers.
Key concept Irrational Number
A number having a non-terminating and non-recurring decimal
expansion is called an irrational number. So, it cannot be written in
the form
q
p
, where p and q are integers and q 0 ! .
For example,
, , , , , e 2 3 5 17 r , 0.2020020002g are a few examples of irrational numbers.
In fact, we can generate infinitely many non-terminating and non-recurring
decimal expansions by replacing the digit 8 in (1) by any natural number as we like.
Know about r: In the late 18
th
centurary Lambert and Legendre proved that r
is irrational.
We usually take
7
22
( a rational number) as an approximate value for r (an irrational
number).
Around 400 BC, the pythagorians,
followers of the famous Greek
mathematician Pythagoras, were the
first to discover the numbers which
cannot be written in the form of a
fraction. These numbers are called
irrational numbers.
Pythagoras
569BC - 479 BC
Note
Chapter 2
44
Classifcation of Decimal Expansions
2.4 Real Numbers
Key Concept Real Numbers
The union of the set of all rational numbers and the set of all irrational numbers
forms the set of all real numbers.
Thus, every real number is either a rational number or an irrational number.
In other words, if a real number is not a rational number, then it must be an
irrational number.
The set of all real numbers is denoted by R.
German mathematicians, George Cantor and R. Dedekind proved independently that
corresponding to every real number, there is a unique point on the real number line and
corresponding to every point on the number line there exists a unique real number.
Thus, on the number line, each point corresponds to a unique real number. And every
real number can be represented by a unique point on the number line.
The following diagram illustrates the relationships among the sets that make up the real
numbers
Rational Numbers
Z
Q
Integers
Whole Numbers
REal NumbERs R
W
Irrational
Numbers
Natural
Numbers
N
Decimal Expansions
Non-Repeating
(I rrational)
Repeating
(Rational)
Non-Terminating Terminating
(Rational)
Real Number System
45
Let us fnd the square root of 2 by long division method.
2 ` = 1.4142135g
If we continue this process, we observe that the decimal expansion has non-terminating
and non-recurring digits and hence 2 is an irrational number.
(i) The decimal expansions of 3 , 5 , 6 ,g are non-terminating
and non-recurring and hence they are irrational numbers.
(ii) The square root of every positive integer is not always irrational.
For example, 4 = 2, 9 = 3, 25 =5, g. Thus 4 , 9 , 25 , g
are rational numbers.
(iii) The square root of every positive but a not a perfect square number is an
irrational number
2.4.1 Representation of I rrational Numbers on the Number line
Let us now locate the irrational numbers 2 and 3 on the number line.
(i) Locating 2 on the number line.
Draw a number line. Mark points O and A such that O represents the number zero and
A represents the number 1. i.e., OA = 1 unit Draw AB=OA such that AB = 1unit. Join OB.
Note
2.00 00 00 00 00
1
100
96
400
281
11900
11296
60400
56564
383600
282841
10075900
8485269
159063100
141421325
17641775
1
24
281
2824
28282
282841
2828423
28284265
1.4142135g
h
Chapter 2
46
In right triangle OAB, by Pythagorean theorem,
OB
2
=OA
2
+AB
2
=1 1
2 2
+
OB
2
=2
OB = 2
With O as centre and radius OB, draw an arc to intersect the number line at C on the
right side of O. Clearly OC =OB = 2 . Thus, C corresponds to 2 on the number line.
(ii) Locating 3 on the number line.
Draw a number line. Mark points O and C on the number line such that O represents
the number zero and C represents the number 2 as we have seen just above.
` OC = 2 unit. Draw CD=OC such that CD =1 unit. J oin OD
In right triangle OCD, by Pythagorean theorem,
OD
2
=OC CD
2 2
+
= 1 2
2
2
+
^ h
=3
` OD = 3
With O as centre and radius OD, draw an arc to intersect the number line at E on the
right side of O. Clearly OE =OD = 3 . Thus, E represents 3 on the number line.
Example 2.6
Classify the following numbers as rational or irrational.
(i) 11 (ii) 81 (iii) 0.0625 (iv) 0.83 (v) 1.505500555g
Solution
(i) 11 is an irrational number. (11is not a perfect square number)
(ii) 81 =9 =
1
9
, a rational number.
(iii) 0.0625 is a terminating decimal.
` 0.0625 is a rational number.
(iv) 0.83 =0.8333g
The decimal expansion is non-terminating and recurring.
` 0.83 is a rational number.
(v) The decimal number is non-terminating and non-recurring.
` 1.505500555g is an irrational number.
-2 -3 -1 0
A
2 1
1
3
B
1
C
O
Fig. 2.6
2
2
-2 -3 -1 0 2 1
1
3
A
B
1
E
D
1
C
O
Fig. 2.7
2 3
3
2
Real Number System
47
Example 2.7
Find any three irrational numbers between
7
5
and
11
9
.
Solution
(8 is a perfect cube)
& x = 2 , a rational number.
(ii) x
2
= 81 = 9
2
(81 is a perfect square)
& x = 9, a rational number.
(iii) y
2
= 3 & y = 3 , an irrational number.
(iv) z
2
= 0.09 =
100
9
=
10
3
2
` j
& z =
10
3
, a rational number.
5.000000
4 9
10
7
30
28
20
14
7
0.714285g
60
56
40
35
50
j
9.0000
8 8
20
11
90
88
20
11
0.8181g
j
Chapter 2
48
Exercise 2.3
1. Locate 5 on the number line.
2. Find any three irrational numbers between 3 and 5 .
3. Find any two irrational numbers between 3 and 3.5.
4. Find any two irrational numbers between 0.15 and 0.16.
5. Insert any two irrational numbers between
7
4
and
7
5
.
6. Find any two irrational numbers between 3 and 2.
7. Find a rational number and also an irrational number between 1.1011001110001g
and 2.1011001110001g
8. Find any two rational numbers between 0.12122122212222g and 0.2122122212222g
2.4.2 Representation of Real Numbers on the Number line
We have seen that any real number can be represented as a decimal expansion. This
will help us to represent a real number on the number line.
Let us locate 3.776 on the number line. We know that 3.776 lies between 3 and 4.
Let us look closely at the portion of the number line between 3 and 4.
Divide the portion between 3 and 4 into 10 equal parts and mark each point of division
as in Fig 2.8. Then the frst mark to the right of 3 will represent 3.1, the second 3.2, and so
on. To view this clearly take a magnifying glass and look at the portion between 3 and 4. It
will look like as shown in Fig. 2.8. Now 3.776 lies between 3.7 and 3.8. So, let us focus on
the portion between 3.7 and 3.8 (Fig. 2.9)
Fig. 2.8
-5 -4 -3 -2 -1 0 1 2 3 4 5
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Fig. 2.9
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
3.7 3.71 3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.8
Real Number System
49
Again divide the portion between 3.7 and 3.8 into 10 equal parts. The frst mark will
represent 3.71, the next 3.72, and so on. To view this portion clearly, we magnify the portion
between 3.7 and 3.8 as shown in Fig 2.9
Again, 3.776 lies between 3.77 and 3.78. So, let us divide this portion into 10 equal
parts. We magnify this portion, to see clearly as in Fig. 2.10.
The frst mark represents 3.771, the next mark 3.772, and so on. So 3.776 is the 6
th
mark in this sub division.
This process of visualisation of representation of numbers on the number line, through
a magnifying glass is known as the process of successive magnifcation.
So, we can visualise the position of a real number with a terminating decimal expansion
on the number line, by suffcient successive magnifcations.
Now, let us consider a real number with a non-terminating recurring decimal expansion
and try to visualise the position of it on the number line.
Example 2.9
Visualise . 4 26 on the number line, upto 4 decimal places, that is upto 4.2626
Solution We locate . 4 26 on the number line, by the process of successive magnifcation.
This has been illustrated in Fig. 2.11
Step 1: First we note that . 4 26 lies between 4 and 5
Step 2: Divide the portion between 4 and 5 into 10 equal parts and use a magnifying glass to
visualise that . 4 26 lies between 4.2 and 4.3
Step 3: Divide the portion between 4.2 and 4.3 into 10 equal parts and use a magnifying
glass to visualise that . 4 26 lies between 4.26 and 4.27
Step 4: Divide the portion between 4.26 and 4.27 into 10 equal parts and use a magnifying
glass to visualise that . 4 26 lies between 4.262 and 4.263
Step 5: Divide the portion between 4.262 and 4.263 into 10 equal parts and use a magnifying
glass to visualise that . 4 26 lies between 4.2625 and 4.2627
Fig. 2.10
3.7 3.71 3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.8
3.77 3.771 3.772 3.773 3.774 3.775 3.776 3.777 3.778 3.779 3.78
Chapter 2
50
We note that . 4 26 is visualized closer to 4.263 than to 4.262.
The same procedure can be used to visualize a real number with a non-terminating and
non-recurring decimal expansion on the number line to a required accuracy.
From the above discussions and visualizations we conclude again that every real
number is represented by a unique point on the number line. Further every point on the
number line represents one and only one real number.
Exercise 2.4
1. Using the process of successive magnifcation
(i) Visualize 3.456 on the number line.
(ii) Visualize 6.73 on the number line, upto 4 decimal places.
2.4.3 Properties of Real Numbers
¯ For any two real numbers a and b, a b = or a b > or a b <
¯ The sum, difference, product of two real numbers is also a real number.
¯ The division of a real number by a non-zero real number is also a real number.
¯ The real numbers obey closure, associative, commutative and distributive laws
under addition and under multiplication that the rational numbers obey.
¯ Every real number has its negative real number. The number zero is its own
negative and zero is considered to be neither negative nor positive.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5
4.262 4.2621 4.2622 4.2623 4.2624 4.2625 4.2626 4.2627 4.2628 4.2629 4.263
4.26 4.261 4.262 4.263 4.264 4.265 4.266 4.267 4.268 4.269 4.27
4.2 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.3
Fig. 2.11
Real Number System
51
Further the sum, difference, product and quotient (except division by zero) of two
rational numbers, will be rational number. However, the sum, difference, product and quotient
of two irrational numbers may sometimes turnout to be a rational number.
Let us state the following facts about rational numbers and irrational numbers.
Key Concept
1. The sum or difference of a rational number and an irrational number is
always an irrational number
2. The product or quotient of non-zero rational number and an irrational
number is also an irrational number.
3. Sum, difference, product or quotient of two irrational numbers need not
be irrational. The result may be rational or irrational.
If a is a rational number and b is an irrational number then
(i) a b + is irrational (ii) a b - is irrational
(iii) a b is irrational (iv)
b
a
is irrational (v)
a
b
is irrational
For example, (i) 2 3 + is irrational (ii) 2 3 - is irrational
(iii) 2 3 is irrational (iv)
3
2
is irrational
2.4.4 square Root of Real Numbers
Let 0 a > be a real number. Then a b = means b a
2
= and 0 b > .
2 is a square root of 4 because 2 2 4 # = , but 2 - is also a square root of 4 because
( ) ( ) 2 2 4 # - - = . To avoid confusion between these two we defne the symbol , to mean
the principal or positive square root.
Let us now mention some useful identities relating to square roots.
Let a and b be positive real numbers. Then
1 ab = a b
2
b
a
=
b
a
3
a b a b + -
^ ^ h h =a b -
4 a b a b + -
^ ^ h h =a b
2
-
5
a b c d + +
^ ^ h h = ac ad bc bd + + +
6 a b
2
+
^ h =a b ab 2 + +
Remark
Chapter 2
52
Example 2.10
Give two irrational numbers so that their
(i) sum is an irrational number.
(ii) sum is not an irrational number.
(iii) difference is an irrational number.
(iv) difference is not an irrational number.
(v) product is an irrational number.
(vi) product is not an irrational number.
(vii) quotient is an irrational number.
(viii) quotient is not an irrational number.
Solution
(i) Consider the two irrational numbers 2 3 + and 3 2 - .
Their sum =2 3 + + 3 2 - =2 3 is an irrational number.
(ii) Consider the two irrational numbers 2 and 2 - .
Their sum = 2 + ( 2 - ) = 0 is a rational number.
(iii) Consider the two irrational numbers 3 and 2 .
Their difference = 3 2 - is an irrational number.
(iv) Consider the two irrational numbers 5 3 + and 3 5 - .
Their difference =(5 3 + ) - ( 3 5 - ) = 10 is a rational number.
(v) Consider the irrational numbers 3 and 5 .
Their product = 3 5 # = 15 is an irrational number.
(vi) Consider the two irrational numbers 18 and 2 .
Their product = 18 # 2 = 36 = 6 is a rational number.
(vii) Consider the two irrational numbers 15 and 3 .
Their quotient =
3
15
=
3
15
= 5 is an irrational number.
(viii) Consider the two irrational numbers 75 and 3 .
Their quotient =
3
75
=
3
75
= 5 is a rational number.
Real Number System
53
2.5 surds
We know that 2 , 3 , 5 are irrational numbers. These are square roots of rational
numbers, which cannot be expressed as squares of any rational number. 2
3
, 3
3
, 7
3
etc.
are the cube roots of rational numbers, which cannot be expressed as cubes of any rational
number. This type of irrational numbers are called surds or radicals.
Key Concept Surds
If 'a' is a positive rational number and n is a positive integer such that
a
n
is an irrational number, then a
n
is called a 'surd' or a 'radical'.
Notation
2.5.1 I ndex Form of a surd
The index form of a surd a
n
is a
n
1
When the order of the surds are different, we convert them to the same order
and then multiplication or division is carried out.
Result a
n
= a
m
n
m
For example, (i) 5
3
= 5
12
3
12
= 5
12 4
(ii) 11
4
= 11
8
4
8
= 11
8 2
2.6.4 Comparison of surds
Irrational numbers of the same order can be compared. Among the irrational numbers
of same order, the greatest irrational number is the one with the largest radicand.
If the order of the irrational numbers are not the same, we frst convert them to the
same order. Then, we just compare the radicands.
Example 2.19
Convert the irrational numbers 3 , 4
3
, 5
4
to the same order.
Solution The orders of the given irrational numbers are 2, 3 and 4.
LCM of 2, 3 and 4 is 12
3 = 3
12 6
= 729
12
Chapter 2
60
5. Express the following as pure surds.
(i) 6 5 (ii) 5 4
3
(iii) 3 5
4
(iv)
4
3
8
6. Simplify the following.
(i) 5 18 # (ii) 7 8
3 3
# (iii) 8 12
4 4
# (iv) 3 5
3 6
#
(v) 3 35 2 7 ' (vi) 48 72
4 8
'
7. Which is greater ?
(i) 2 or 3
3
(ii) 3
3
or 4
4
(iii) 3 or 10
4
8. Arrange in descending and ascending order.
(i) , , 5 3 4
4 3
(ii) , , 2 4 4
3 3 4
(iii) , , 2 4 3
3 9 6
2.7 Rationalization of surds
Rationalization of Surds
When the denominator of an expression contains a term with a
square root or a number under radical sign, the process of converting
into an equivalent expression whose denominator is a rational number is
called rationalizing the denominator.
If the product of two irrational numbers is rational, then each one
is called the rationalizing factor of the other.
Let a and b be integers and x, y be positive integers. Then
(i) a x +
^ h and a x -
^ h are rationalizing factors of each other.
(ii) a b x +
^ h and a b x -
^ h are rationalizing factors of each other.
(iii) x y + and x y - are rationalizing factors of each other.
(iv) a b + is also called the conjugate of a b - and a b - is called
the conjugate of a b + .
(v) For rationalizing the denominator of a number, we multiply its
numerator and denominator by its rationalizing factor.
Example 2.22
Rationalize the denominator of
3
2
Solution Multiplying the numerator and denominator of the given number by 3 , we get
Thus, if a >0, we have been able to give suitable meaning to a
x
for all rational numbers x.
Also for a >0itispossibletoextendthedefnitionofa
x
to irrational exponents x so that the laws
of exponents remain valid. We will not show how a
x
maybedefnedforirrationalx because the
defnitionof a
x
requiressomeadvancedtopicsinmathematics.So,weacceptnowthat,forany
a 0 > , a
x
isdefnedforallrealnumbersxandsatifesthelawsofexponents.
3.3.2 LogarithmicNotation
If a > 0, b > 0 and 1 a ! , then the logarithm of b to the base a is the number to which
a to be raised to obtain b.
Key Concept Logarithmic Notation
Let a be a positive number other than 1 and let x be a real number
(positive, negative, or zero). If a b
x
= , we say that the exponent x is the
logarithm of b to the base a and we write x = log b
a
.
x = log b
a
is the logarithmic form of the exponential form b a
x
= . In both the forms,
the base is same.
Chapter 3
72
For example,
Exponential Form Logarithmic Form
2 16
4
=
8 2
3
1
=
4
8
1 2
3
=
-
16 4 log
2
=
2 log
3
1
8
=
8
1
log
2
3
4
=- ` j
is x 4 - , whereas the coeffcient of
y
xz
2
is –4. A coeffcient such as –4, which involves no
variables, is called a numerical coeffcient. Terms such as 5x y
2
and 12x y
2
- , which differ
only in their numerical coeffcients, are called like terms or similar terms.
An algebraic expression such as 4 r
2
r can be considered as an algebraic expression
consisting of just one term. Such a one-termed expression is called a monomial. An algebraic
expression with two terms is called a binomial, and an algebraic expression with three terms is called
a trinomial. For instance, the expression 3 2 x xy
2
+ is a binomial, whereas 2 3 4 xy x
1
- + -
-
is
a trinomial. An algebraic expression with two or more terms is called a multinomial.
4.3 Polynomials
A polynomial is an algebraic expression, in which no variables appear in denominators
or under radical signs, and all variables that do appear are powers of positive integers. For
instance, the trinomial 2 3 4 xy x
1
- + -
-
is not a polynomial; however, the trinomial
3x y xy 2
2
1
2 4
+ - is a polynomial in the variables x and y. A term such as
2
1
- which
contains no variables, is called a constant term of the polynomial. The numerical coeffcients
of the terms in a polynomial are called the coeffcients of the polynomial. The coeffcients of
the polynomial above are 3, 2 and
2
1
- .
The degree of a term in a polynomial is the sum of the exponents of all the variables in
that term. In adding exponents, one should regard a variable with no exponent as being power
one. For instance, in the polynomial 9 12 3 2 xy x yz x
7 3 2
- + - , the term xy 9
7
has degree 1 +7
=8, the term 12x yz
3 2
- has degree 3 +1 +2 =6, and the term x 3 has degree one. The constant
term is always regarded as having degree zero.
Algebra
91
The degree of the highest degree term that appears with nonzero coeffcients in a
polynomial is called the degree of the polynomial.
For instance, the polynomial considered above has degree 8. Although the constant
monomial 0 is regarded as a polynomial, this particular polynomial is not assigned a degree.
4.3.1 Polynomials in One Variable
In this section we consider only polynomials in one variable.
Key Concept Polynomial in One Variable
A
where , , , , , a a a a a
n n 0 1 2 1
g
-
are constants and n is a non negative integer.
Here n is the degree of the polynomial and , , , , a a a a
n n 1 2 1
g
-
are the coeffcients of
, , , x x x x
n n 2 1
g
-
respectively. a
0
is the constant term. , , , , , a x a x a x a x a
. n
n
n
n
1
1
2
2
1 0
g
-
-
are the
terms of the polynomial p x ^ h.
For example, in the polynomial 5 3 1 x x
2
+ - , the coeffcient of x
2
is 5, the coeffcient of x is
3 and –1 is the constant term. The three terms of the polynomial are
5 , 3 1 x x and
2
-
.
4.3.2 Types of Polynomials
Key Concept Types of Polynomials Based on Number of Terms
Monomial
Polynomials which have only one term are known as monomials.
Binomial
Polynomials which have only two terms are called binomials.
Trinomial
Polynomials which have only three terms are named as trinomials.
1. A binomial is the sum of two monomials of different degrees.
2. A trinomial is the sum of three monomials of different degrees.
3. A polynomial is a monomial or the sum of two or more monomials.
Note
Chapter 4
92
Key Concept Types of Polynomials Based on the Degree
Constant Polynomial
A polynomial of degree zero is called a constant polynomial.
General form : ( ) p x =c, where c is a real number.
Linear Polynomial
A polynomial of degree one is called a linear polynomial.
General form : ( ) p x = ax+b, where a and b are real numbers and a 0 ! .
Quadratic Polynomial
A polynomial of degree two is called a quadratic polynomial.
General form: ( ) p x =ax bx c
2
+ + where a, b and c are real numbers and 0. a !
Cubic Polynomial
A polynomial of degree three is called a cubic polynomial.
General form : ( ) p x = ax bx cx d
2 3
+ + + , where , , a b c and d are real numbers
and a 0 ! .
Example 4.1
Classify the following polynomials based on number of terms.
(i) x x
3 2
- (ii) 5x (iii)
4 2 1 x x
4 3
+ +
(iv)
4x
3
(v) 2 x +
(vi) 3x
2
(vii) 1 y
4
+ (viii) y y y
20 18 2
+ +
(ix) 6 (x)
u u 2 3
3 2
+ +
(xi) u u
23 4
-
(xii) y
Solution
5x, 3 , x
2
4x
3
, y and 6 are monomials because they have only one term.
, x x
3 2
-
2, x + 1 y
4
+ and u u
23 4
- are binomials as they contain only two terms.
, x x y y y u u 4 2 1 2 3 and
4 3 20 18 2 3 2
+ + + + + +
are trinomials as they contain only
three terms.
Example 4.2
Classify the following polynomials based on their degree.
(i) ( ) p x =3 (ii)
( ) 1 p y y
2
5
2
= +
(iii) p x ^ h =2 4 1 x x x
3 2
- + +
(iv)
( ) 3 p x x
2
=
(v) ( ) p x x 3 = + (vi) ( ) p x =–7
(vii) p x ^ h =x 1
3
+ (viii) ( ) 5 3 2 p x x x
2
= - + (ix) ( ) 4 p x x =
(x) ( ) p x =
2
3
(xi) ( ) 1 p x x 3 = + (xii) ( ) p y =y y 3
3
+
Algebra
93
Solution
( ) p x =3, ( ) p x =–7, ( ) p x =
2
3
are constant polynomials.
( ) p x x 3 = + , ( ) 4 p x x = , ( ) 1 p x x 3 = + are linear polynomials, since the highest
degree of the variable x is one.
24 =
Substituting y 24 = in (3) we get, x 7 24 31 = + = .
` The required two numbers are 31 and 24.
Example 4.45
A number consist of two digits whose sum is 11. The number formed by reversing the
digits is 9 less than the original number. Find the number.
Solution
Let the tens digit be x and the units digit be y. Then the number is x y 10 + .
Sum of the digits is x y 11 + = (1)
The number formed by reversing the digits is y x 10 + .
Algebra
121
Given data, (10 ) 9 x y + - y x 10 = +
( x y y x 10 10 9 + - - =
x y 9 9 9 - =
Dividing by 9 on both sides, x y 1 - = (2)
Equation (2) becomes x y 1 = + (3)
Substituting x in (1) we get, y y 1 11 + + =
( y 2 1 11 + =
y 2 11 1 10 = - =
` y
2
10
5 = =
Substituting y = 5 in (3) we get, 1 x 5 6 = + =
` The number is ( ) x y 10 10 6 5 65 + = + =
4.9 Linear I nequations in One Variable
We know that x 4 6 + = is a linear equation in one variable. Solving we get x 2 = .
There is only one such value for x in a linear equation in one variable.
Let us consider,
x 4 + >6
ie x >6 4 -
x >2
So any real number greater than 2 will satisfy this inequation. We represent those real
numbers in the number line.
Unshaded circle indicates that point is not included in the solution set.
Example 4.46
Solve x 4 1 8 # - ^ h
Solution
x 4 1 8 # - ^ h
Dividing by 4 on both sides,
x 1 # - 2
( x 2 1 # + ( x # 3
The real numbers less than or equal to 3 are solutions of given inequation.
Shaded circle indicates that point is included in the solution set.
0 1 2 -2 -1 3 4
0 1 2 -2 -1 3 4
Chapter 4
122
Example 4.47
Solve 3 6 x 5 > - ^ h
Solution We have, 3 x 5 - ^ h >6
Dividing by 3 on both sides, 5 2 x > -
( 2 5 x > - - ( 3 x > - -
` 3 x < (See remark given below)
The real numbers less than 3 are solutions of given inequation.
(i) a b a b > < ( - - (ii) a b
a b
1 1
< > ( where a 0 ! , b 0 !
(iii) a b ka kb < < ( for k 0 > (iv) a b ka kb < > ( for k 0 <
Example 4.48
Solve 3 5 9 x # -
Solution We have, x 3 5 - #9
( x 5 - #9 3 - ( x 5 - # 6
( x 5 $ 6 - (x $
5
6
- (x $ . 1 2 -
The real numbers greater than or equal to . 1 2 - are solutions of given inequation.
Exercise 4.8
1. Solve the following equations by substitution method.
(i) x y 3 10 + = ; x y 2 5 + = (ii) x y 2 1 + = ; x y 3 4 18 - =
(iii) x y 5 3 21 + = ; x y 2 4 - = (iv)
x y
1 2
9 + = ;
x y
2 1
12 + = ( , ) x y 0 0 ! !
(v) 7
x y
3 1
+ = ;
x y
5 4
6 - = ( , ) x y 0 0 ! !
2. Find two numbers whose sum is 24 and difference is 8.
3. A number consists of two digits whose sum is 9. The number formed by reversing the
digits exceeds twice the original number by 18. Find the original number.
4 Kavi and Kural each had a number of apples . Kavi said to Kural "If you give me 4 of
your apples, my number will be thrice yours". Kural replied "If you give me 26, my
number will be twice yours". How many did each have with them?.
5. Solve the following inequations.
(i) x 2 7 15 > + (ii) x 2 2 3 1 - ^ h (iii) 2 x 7 9 # + ^ h (iv) 3 8 x 14 $ +
0 1
-1.2
-2 -1 2
0 1 2 -2 -1 3 4
Remark
Algebra
123
Points to Remember
› A where
, , , , , a a a a a
n n 0 1 2 1
g
-
are constants and n is a non negative integer .
› Let p x ^ h be a polynomial. If p a ^ h =0 then we say that a is a zero of the polynomial
p x ^ h
› If x a = satisfes the polynomial equation p x ^ h =0 then x a = is called a root of
the polynomial equation 0 p x = ^ h .
› Remainder Theorem : Let p x ^ h be any polynomial and a be any real number. If
p x ^ h is divided by the linear polynomial x a - , then the remainder is p a ^ h.
› Factor Theorem : Let p x ^ h be a polynomial and a be any real number. If p (a) =0,
then ( x–a) is a factor of p x ^ h.
› ( ) 2 2 2 x y z x y z xy yz zx
2 2 2 2
/ + + + + + + +
› ( ) 3 ( ) x y x y xy x y
3 3 3
/ + + + + ( )( ) x y x y x xy y
3 3 2 2
/ + + - +
› ( ) 3 ( ) x y x y xy x y
3 3 3
/ - - - - ( )( ) x y x y x xy y
3 3 2 2
/ - - + +
› 3 ( )( ) x y z xyz x y z x y z xy yz zx
3 3 3 2 2 2
/ + + - + + + + - - -
› x a x b x c x a b c x ab bc ca x abc
3 2
/ + + + + + + + + + + ^ ^ ^ ^ ^ h h h h h
Chapter 5
124
COORDINATE GEOMETRY
5.1 I ntroduction
Coordinate Geometry or Analytical Geometry is
a system of geometry where the position of points on the
plane is described using an ordered pair of numbers called
coordinates. This method of describing the location of points
was introduced by the French mathematician René Descartes
(Pronounced "day CART"). He proposed further that curves
and lines could be described by equations using this technique,
thus being the first to link algebra and geometry. In honour
of his work, the coordinates of a point are often referred
to as its Cartesian coordinates, and the coordinate plane as
the Cartesian Coordinate Plane. The invention of analytical
geometry was the beginning of modern mathematics.
In this chapter we learn how to represent points using
cartesian coordinate system and derive formula to find distance
between two points in terms of their coordinates.
I hope that posterity will judge me kindly, not only
as to the things which I have explained, but also as to
those which I have intentionally omitted so as to leave
to others the pleasure of discovery
- RENE DESCARTES
DESCARTES
(1596-1650)
D e s c a r t e s
(1596-1650) has been
cal l ed the father of
modern phi l osophy,
perhaps because he
attempted to bui l d a new
system of thought from the
ground up, emphasi zed
the use of l ogi c and
sci enti fi c method, and
was "profoundl y affected
i n hi s outl ook by
the new physi cs and
astronomy." Descartes
went far past Fermat i n
the use of symbol s, i n
' Ari thmeti zi ng' anal yti c
geometry, i n extendi ng
i t to equati ons of hi gher
degree. The fi xi ng of
a poi nt posi ti on i n the
pl ane by assi gni ng two
numbers - coordi nates -
gi vi ng i ts di stance from
two l i nes perpendi cul ar
to each other, was enti rel y
Descartes' i nventi on.
● To understand Cartesian coordinate system
● To identify abscissa, ordinate and coordinates of a point
● To plot the points on the plane
● To find the distance between two points
Main Targets
124
Coordinate Geometry
125
5.2 Cartesian Coordinate System
In the chapter on Real Number System,
you have learnt` how to represent real numbers
on the number line. Every real number,
whether rational, or irrational, has a unique
location on the number line. Conversely, a
point P on a number line can be specifed by a
real number x called its coordinate. Similarly,
by using a Cartesian coordinate system we
can specify a point P in the plane with two
real numbers, called its coordinates.
A Cartesian coordinate system or
rectangle coordinate system consists of two
perpendicular number lines, called coordinate
axes. The two number lines intersect at the
zero point of each as shown in the Fig. 5.1 and
this point is called origin 'O'. Generally the
horizontal number line is called the x-axis and
the vertical number line is called the y-axis. The
x coordinate of a point to the right of the y-axis is positive and to the left of y-axis is negative.
Similarly, the y coordinate of a point above the x-axis is positive and below the x-axis is negative.
We use the same scale (that is, the same unit
distance) on both the axes.
5.2.1 Coordinates of a Point
In Cartesian system, any point P in the plane
is associated with an ordered pair of real numbers.
To obtain these number, we draw two lines through
the point P parallel (and hence perpendicular) to
the axes. We are interested in the coordinates of the
points of intersection of the two lines with the axes.
There are two coordinates: x-coordinate on the x-axis
and y-coordinate on the y-axis. The x-coordinate is
called the abscissa and the y-coordinate is called the
ordinate of the point at hand. These two numbers
associated with the point P are called coordinates
of P. They are usually written as (x, y), the
abscissa coming frst, the ordinate second.
Fig. 5.1Fig. 5.2( , ) x y
y
x
P
Chapter 5
126
1. In an ordered pair (a, b), the two elements a and b are listed in a
specifc order. So the ordered pairs (a, b) and (b, a) are not equal, i.e.,
( , ) ( , ) a b b a ! .
2. Also ( , ) a b
1 1
=( , ) a b
2 2
is equivalent to a a
1 2
= and b b
1 2
=
3. The terms point and coordinates of a point are used interchangeably.
5.2.2 I dentifying the x-coordinate
The x-coordinate or abscissa, of a point
is the value which indicates the distance and
direction of the point to the right or left of the
y-axis. To fnd the x-coordinate of a point P:
(i) Drop a perpendicular from the point P
to the x-axis.
(ii) The number where the line meets the
x-axis is the value of the x-coordinate.
In Fig. 5.3., the x-coordinate of P is 1 and the
x-coordinate of Q is 5.
5.2.3 I dentifying the y-coordinate
The y-coordinate, or ordinate, of a
point is the value which indicates the distance
and direction of the point above or below the
x-axis. To fnd the y-coordinate of a point P:
(i) Drop a perpendicular from the point P
to the y-axis.
(ii) The number where the line meets the
y-axis is the value of the y-coordinate.
In Fig. 5.4 ., the y-coordinate of P is 6 and the
y-coordinate of Q is 2.
Remarks
Fig. 5.4
Fig. 5.3Coordinate Geometry
127
(i) For any point on the x-axis, the value of y-coordinate (ordinate) is zero.
(ii) For any point on the y-axis, the value of x-coordinate (abscissa) is zero.
5.2.4 Quadrants
A plane with the rectangular coordinate system
is called the cartesian plane. The coordinate axes divide
the plane into four parts called quadrants, numbered
counter-clockwise for reference as shown in Fig. 5.5.
The x coordinate is positive in the I and IV quadrants
and negative in II and III quadrants. The y coordinate
is positive in I and II quadrants and negative in III and
IV quadrants. The signs of the coordinates are shown
in parentheses in Fig. 5.5.
Region Quadrant Nature of x, y
Signs of the
coordinates
XOY I
, 0 x y 0 > >
+, +
X OY l
II
, x y 0 0 < > , - +
X OY l l
III
, x y 0 0 < < , - -
XOYl
IV
, x y 0 0 > < , + -
5.2.5 Plotting Points in Cartesian Coordinate
System
Let us now illustrate through an example
how to plot a point in Cartesian coordinate
system. To plot the point (5, 6) in cartesian
coordinate system we follow the x-axis until we
reach 5 and draw a vertical line at x =5. Similarly,
we follow the y-axis until we reach 6 and draw a
horizontal line at y =6. The intersection of these
two lines is the position of (5, 6) in the cartesian
plane.
That is we count from the origin 5 units along
the positive direction of x-axis and move along
the positive direction of y-axis through 6 units and
mark the corresponding point. This point is at a
distance of 5 units from the y-axis and 6 units from
the x-axis. Thus the position of (5, 6) is located in
the cartesian plane.
Note
Fig. 5.6(5, 6)
O
I Quadrant
(+, +)
IV Quadrant
(+, ) −
II Quadrant
( , +) −
III Quadrant
( , ) − −
X
Y
Xl
Yl
Fig. 5.5
Chapter 5
128
Example 5.1
Plot the following points in the rectangular coordinate system.
(i) A (5, 4) (ii) B (-4, 3) (iii) C (-2, -3) (iv) D (3, -2)
Solution (i) To plot (5,4), draw a vertical line at x =5 and draw a horizontal line at y = 4.
The intersection of these two lines is the position of (5, 4) in the Cartesian plane.
Thus, the point A (5, 4) is located in the Cartesian plane.
S
i
d
e
Vertex
O A
B
Trigonometry
145
When trigonometry was frst developed it was based on similar right triangles. All
right triangles that have a common acute angle are similar. So, for a given acute angle i, we
have many right triangles.
For each triangle above, the ratios of the corresponding sides are equal.
For example,
OA
AD
=
OB
BE
=
OC
CF
;
OA
OD
=
OB
OE
=
OC
OF
That is, the ratios depend only on the size of i and not on the particular right triangle
used to compute the ratios. We can form six ratios with the sides of a right triangle. Long ago
these ratios were given names.
The ratio
Hypotenuse
Opposite side
is called sine of angle i and is denoted by sini
The ratio
Hypotenuse
Adjacent side
is called cosine of angle i and is denoted by cos i
The ratio
Adjacent side
Opposite side
is called tangent of angle i and is denoted by tani
The ratio
Opposite side
Hypotenuse
is called cosecant of angle i and is denoted by coseci
The ratio
Adjacent side
Hypotenuse
is called secant of angle i and is denoted by seci
The ratio
Opposite side
Adjacent side
is called cotangent of angle i and is denoted by cot i
Key Concept Trigonometric Ratios
Let i be an acute angle of a right triangle. Then the six trigonometric
ratios of i are as followsO
D
i
A
B
E
O
i
F
C
i
O
Chapter 6
146
Reciprocal Relations
The trigonometric ratios coseci, seci and cot i are receprocals of sini, cos i
and tani respectively.
sin i =
cosec
1
i
cos i =
sec
1
i
tan i =
cot
1
i
at
the centre.
11. Find the length of the side of regular polygon of 12 sides inscribed in a circle of radius 6cm
12. Find the radius of the incircle of a regular hexagon of side 24cm.
Points to Remember
� Pythagoras Theorem:
The square of the length of the hypotenuse of a right triangle is equal to the sum of
the squares of the other two sides.
� Trigonometric Ratios:
Let i be an acute angle of a right triangle. Then the six trigonometric ratios of i
are as follows.� Reciprocal Relations:
sini =
cosec
1
i
cos i =
sec
1
i
tani =
cot
1
i
coseci =
sin
1
i
seci =
cos
1
i
cot i =
tan
1
i
� Trigonometric Ratios of Complementary Angles:
Let i be an acute angle of a right triangle. Then we have the following identities
for trigonometric ratios of complementary angles.
sini = (90 ) cos i - c coseci = (90 ) sec i - c
cos i = (90 ) sin i - c seci = (90 ) cosec i - c
tani = (90 ) cot i - c cot i = (90 ) tan i - c
Chapter 7
164
GEOMETRY
Truth can never be told so as to be understood,
and not to be believed
- William Blake
Main Targets
● To recall the basic concepts of geometry.
● To understand theorems on parallelograms.
● To understand theorems on circles.
7.1 I ntroduction
The very name Geometry is derived from two greek
words meaning measurement of earth. Over time geometry
has evolved into a beautifully arranged and logically
organized body of knowledge. It is concerned with the
properties of and relationships between points, lines, planes
and figures. The earliest records of geometry can be traced
to ancient Egypt and the Indus Valley from around 3000
B.C. Geometry begins with undefined terms, definitions,
and assumptions; these lead to theorems and constructions.
It is an abstract subject, but easy to visualize, and it has
many concrete practical applications. Geometry has long
been important for its role in the surveying of land and more
recently, our knowledge of geometry has been applied to
help build structurally sound bridges, experimental space
stations, and large athletic and entertainment arenas, just to
mention a few examples. The geometrical theorem of which
a particular case involved in the method just described in
the first book of Euclid's Elements.
THALES
(640 - 546 BC)
Thales (pronounced THAY-
lees) was born in the Greek city
of Miletus. He was known for
his theoretical and practical
understanding of geometry,
especially triangles. He
established what has become
known as Thales' Theorem,
whereby if a triangle is drawn
within a circle with the long
side as a diameter of the
circle then the opposite angle
will always be a right angle.
Thales used geometry to solve
problems such as calculating
the height of pyramids and
the distance of ships from
the shore. He is credited
with the first use of deductive
reasoning applied to geometry,
by deriving four corollaries to
Thales' Theorem. As a result,
he has been hailed as the first
true mathematician and is the
first known individual to whom
a mathematical discovery has
been attributed. He was one
of the so-called Seven Sages
or Seven Wise Men of Greece,
and many regard himas the
first philosopher in the Western
tradition.
7
164
Geometry
165
l
1
l
2
A C B
7.2 Geometry Basics
The purpose of this section is to recall some of the ideas that you have learnt in the
earlier classes.
Term Diagram Description
Parallel
lines
Lines in the same plane that do not intersect are
called parallel lines.
The distance between two parallel lines always
remains the same.
Intersecting
lines
Two lines having a common point are called
intersecting lines. The point common to the two
given lines is called their point of intersection. In
the fgure, the lines AB and CD intersect at a point
O.
Concurrent
lines
Three or more lines passing through the same point
are said to be concurrent. In the fgure, lines , , l l l
1 2 3
pass through the same point O and therefore they
are concurrent.
Collinear
points
If three or more points lie on the same straight
line, then the points are called collinear points.
Otherwise they are called non-collinear points.
7.2.1 Kinds of Angle
Angles are classifed and named with reference to their degree of measure.
Name Acute Angle Right Angle Obtuse Angle Refex Angle
Diagram
Measure
90 AOB + 1 c 90 AOB + = c 90 180 AOB 1 + 1 c c180 360 AOB 1 + 1 c c
Complementary Angles
Two angles are said to be complementary to each other if sum of their measures is 90c
For example, if A 52 + = c and 38 B + = c, then angles A + and B + are complementary to
each other.
C
A
B
D
O
l
1
l
2
l
3
O
A
B
O
A
B
O
A
B
O
B
O
A
Chapter 7
166
Supplementary Angles
Two angles are said to be supplementary to each other if sum of their measures is 180c.
For example, the angles whose measures are 112 68 and c c are supplementary to each other.
7.2.2 Transversal
A straight line that intersects two or more straight lines at distinct points is called a
transversal.
Suppose a transversal intersects two parallel lines. Then:
Name Angle Diagram
Vertically opposite angles are
equal
+1 =+3, +2 =+4,
+5 =+7, +6 =+8
Corresponding angles are
equal.
+1 =+5, +2 = +6,
+3 =+7, +4 = +8
Alternate interior angles are
equal.
+3 =+5, +4 = +6
Alternate exterior angles are
equal.
+1 =+7, +2 = +8
Consecutive interior angles
are supplementary.
+3 ++6 =180c;
+4 ++5 =180c
7.2.3 Triangles
The sum of the angles of a triangle is 180c.
In the Fig. 7.1., +A ++B ++C =180c
(i) If a side of a triangle is produced , then the exterior angle so
formed is equal to the sum of its interior opposite angles.
+ACD = +BAC + +ABC
(ii) An exterior angle of a triangle is greater than either of the interior
opposite angles.
(iii) In any triangle, the angle opposite to the largest side has the
greatest angle.
Congruent Triangles
Two triangles are congruent if and only if one of them can be made to superpose on
the other, so as to cover it exactly.
For congruence, we use the symbol '/'
l
2
l
1
m
1
2
3
4
5
6
7
8
Remarks
A
B C
Fig. 7.1
B
A
D C
Fig. 7.2
Geometry
167
Description Diagram
SSS
If the three sides of a triangle are
equal to three sides of another
triangle, then the two triangles are
congruent.
SAS
If two sides and the included angle
of a triangle are equal to two sides
and the included angle of another
triangle, then the two triangles are
congruent.
ASA
If two angles and the included side
of a triangle are equal to two angles
and the included side of another
triangle, then the two triangles are
congruent.
AAS
If two angles and any side of a
triangle are equal to two angles and
a side of another triangle, then the
two triangles are congruent.
RHS
If one side and the hypotenuse of
a right triangle are equal to a side
and the hypotenuse of another right
triangle, then the two triangles are
congruent
Exercise 7.1
1. Find the complement of each of the following angles.
(i) 63c (ii) 24c (iii) 48c (iv) 35c (v) 20c
2. Find the supplement of each of the following angles.
(i) 58c (ii) 148c (iii) 120c (iv) 40c (v) 100c
A
B
C
P
Q
R
ABC PQR T T /
A
B C
P
Q
R
ABC PQR T T /
A
B C
P
R
ABC PQR T T /
A
B C
P
Q R
ABC PQR T T /
A
B
C
P
Q R
ABC PQR T T /
Chapter 7
168
3. Find the value of x in the following fgures.
(i) (ii)
4. Find the angles in each of the following.
i) The angle which is two times its complement.
ii) The angle which is four times its supplement.
iii) The angles whose supplement is four times its complement.
iv) The angle whose complement is one sixth of its supplement.
v) Supplementary angles are in the ratio 4:5
vi) Two complementary angles are in the ratio3:2
5. Find the values of x, y in the following fgures.
(i) (ii) (iii)
6. Let l1 || l2 and m1 is a transversal . If F + =65c , fnd the measure
of each of the remaining angles.
7. For what value of x will l1 and l2 be parallel lines.
(i) (ii)
8. The angles of a triangle are in the ratio of 1:2:3. Find the measure of each angle of the
triangle.
9. In 3ABC, +A++B =70c and +B ++C =135c. Find the measure of each angle of the
triangle.
10. In the given fgure at right, side BC of 3ABC is produced to D.
Find +A and +C.
A
D
B
C
3x
2x
A
D
B
C
(3x+5)c
(2x–25)c
B A
D
C
E
F
H
G
l
2
l
1
m
1
t
t
l
1
(2x+20)c
l
2
(3x–10)c
(3x+20)c
2xc
l
1
l
2
A
B C
D
120c
40c
A B
D
C
( 30) x + c
xc
( ) x 115 - c
A B
xc
D
C
( ) x 20 - c
40c
A
D
B
C
E
O
3xc
60c
xc
y
c
90c
Geometry
169
7.3 Quadrilateral
A closed geometric fgure with four sides and four vertices is called a quadrilateral.
The sum of all the four angles of a quadrilateral is 360c.
7.3.1 Properties of Parallelogram, Rhombus and Trapezium
Parallelogram
Sides Opposite sides are parallel and equal.
Angles
Opposite angles are equal and sum of any two adjacent
angles is 180c.
Diagonals Diagonals bisect each other.
Rhombus
Sides All sides are equal and opposite sides are parallel.
Angles
Opposite angles are equal and sum of any two adjacent
angles is 180c.
Diagonals Diagonals bisect each other at right angles.
Trapezium
Sides One pair of opposite sides is parallel
Angles
The angles at the ends of each non-parallel side are
supplementary
Diagonals Diagonals need not be equal.
Isosceles
Trapezium
Sides
One pair of opposite sides is parallel, the other pair of sides
is equal in length.
Angles The angles at the ends of each parallel side are equal.
Diagonals Diagonals are equal in length.
Parallelogram
Rectangle
Rhombus
Square
Trapezium
Isosceles
Trapezium
Quadrilateral
Chapter 7
170
(i) A rectangle is an equiangular parallelogram
(ii) A rhombus is an equilateral parallelogram
(iii) A square is an equilateral and equiangular parallelogram.
(iv) Thus a square is a rectangle, a rhombus and a parallelogram.
7.4 Parallelogram
A quadrilateral in which the opposites sides are parallel is called a parallelogram.
7.4.1 Properties of Parallelogram
Property 1 : In a parallelogram, the opposite sides are equal.
Given : ABCD is a parallelogram. So, AB || DC and AD || BC
To prove : AB =CD and AD =BC
Construction : J oin BD
Proof :
Consider the TABD and the TBCD.
(i) +ABD = +BDC (AB || DC and BD is a transversal.
So, alternate interior angles are equal.)
(ii) +BDA =+DBC (AD || BC and BD is a transversal.
So, alternate interior angles are equal.)
(iii) BD is common side
` TABD / TBCD (By ASA property)
Thus, AB =DC and AD =BC (Corresponding sides are equal)
Converse of Property 1: If the opposite sides of a quadrilateral are equal, then the quadrilateral
is a parallelogram.
Property 2 : In a parallelogram, the opposite angles are equal.
Given : ABCD is a parallelogram,
where AB || DC, AD || BC
To prove : +ABC =+ADC and +DAB =+BCD
Construction : J oin BD
Note
A
D C
B
Fig. 7.3
A
D C
B
Fig. 7.4
Geometry
171
Proof :
(i) +ABD = +BDC (AB || DC and BD is a transversal.
So, alternate interior angles are equal.)
(ii) +DBC =+BDA (AD || BC and BD is a transversal.
So, alternate interior angles are equal.)
(iii) + ABD ++ DBC =+ BDC + + BDA
` +ABC = +ADC
Similarly, +BAD =+BCD
Converse of Property 2: If the opposite angles in a quadrilateral are equal, then the
quadrilateral is a parallelogram.
Property 3 : The diagonals of a parallelogram bisect each other.
Given : ABCD is a parallelogram, in which AB || DC and AD || BC
To prove : M is the midpoint of diagonals AC and BD .
Proof :
Consider the TAMB and TCMD
(i) AB =DC Opposite sides of the parallelogram are equal
(ii) +MAB =+MCD Alternate interior angles (a AB || DC)
+ABM =+CDM Alternate interior angles (a AB || DC)
(iii) TAMB / TCMD (By ASA property)
` AM =CM and BM =DM
i.e., M is the mid point of AC and BD
` The diagonals bisect each other
Converse of Property 3: If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
(i) A diagonal of a parallelogram divides it into two triangles
of equal area.
(ii) A parallelogram is a rhombus if its diagonals are perpendicular.
(iii) Parallelograms on the same base and between the same parallels are
equal in area.
A
D
C
B
M
Fig. 7.5
Note
Chapter 7
172
Example 7.1
If the measures of three angles of a quadrilateral are 100c, 84c and 76c then, fnd the
meausre of fourth angle.
Solution Let the measure of the fourth angle be xc.
The sum of the angles of a quadrilateral is 360c. So,
100c+84c+76c+xc = 360c
260c+xc =360c
i.e., xc =100c
Hence, the measure of the fourth angle is 100c.
Example 7.2
In the parallelogram ABCD if +A =65c, fnd +B, +C and +D.
Solution Let ABCD be a parallelogram in which +A =65c.
Since AD || BC we can treat AB as a transversal. So,
+A++B =180c
65c+B =180c
+B =180 65 - c c
+B =115c
Since the opposite angles of a parallelogram are equal, we have
+C =+A =65c and +D =+B =115c
Hence, +B =115c, +C =65c and +D =115c
Example 7.3
Suppose ABCD is a rectangle whose diagonals AC and BD intersect at O.
If +OAB =62c, fnd +OBC.
Solution The diagonals of a rectangle are equal and bisect each other. So,
OA =OB and +OBA =+OAB =62c
Since the measure of each angle of rectangle is 90c
+ABC =90c
+ABO ++OBC = 90c
62c ++OBC =90c
+OBC =90c - 62c
=28c
>
>
A
D
C
B
>
>
>
>
65
o
Fig. 7.6
D C
A B
O
62c
Fig. 7.7
Geometry
173
Example 7.4
If ABCD is a rhombus and if +A =76c, fnd +CDB.
Solution +A =+C =76c ( Opposite angles of a rhombus )
Let +CDB =xc. In 3 CDB, CD =CB
+CDB =+CBD =xc
+CDB++CBD ++DCB =180c (Angles of a triangle)
2 76 x + c c =180c ( x 2 =104c
xc =52c
`+CDB =52c
Exercise 7.2
1. In a quadrilateral ABCD, the angles +A, +B, +C and +D are in the ratio 2:3:4:6.
Find the measure of each angle of the quadrilateral.
2. Suppose ABCD is a parallelogram in which +A =108c. Calculate +B, +C and +D.
3. In the fgure at right, ABCD is a parallelogram
+BAO =30c, +DAO =45c and +COD=105c.
Calculate
(i) +ABO (ii) +ODC (iii) +ACB (iv) +CBD
4. Find the measure of each angle of a parallelogram, if larger angle is 30c less than
twice the smaller angle.
5. Suppose ABCD is a parallelogram in which AB =9 cm and its perimeter is 30 cm. Find
the length of each side of the parallelogram.
6. The length of the diagonals of a rhombus are 24 cm and 18 cm. Find the length of each
side of the rhombus.
7. In the following fgures, ABCD is a rhombus. Find the values of x and y.
(i) (ii) (iii)
A B
D C
76c
Fig. 7.8
A
D C
B
45
o
30
o
105
o
O
A
D
C
B
yc
xc
120c
A
D
C
B
yc
xc
40c
A
D
C
B
yc
xc
62c
Chapter 7
174
8. The side of a rhombus is 10 cm and the length of one of the diagonals is 12 cm. Find
the length of the other diagonal.
9. In the fgure at the right, ABCD is a parallelogram in which
the bisectors of +A and +B intersect at the point P. Prove
that +APB =90c.
7.5 Circles
Locus
Locus is a path traced out by a moving point which
satisfes certain geometrical conditions.
For example, the locus of a point equidistant from two
fxed points is the perpendicular bisector of the line segment
joining the two points.
Circles
The locus of a point which moves in such that the distance
from a fxed point is always a constant is a circle.
The fxed point is called its centre and the constant
distance is called its radius.
The boundary of a circle is called its circumference.
Chord
A chord of a circle is a line segment joining any two
points on its circumference.
Diameter
A diameter is a chord of the circle passing through the
centre of the circle.
Diameter is the longest chord of the circle.
Secant
A line which intersects a circle in two distinct points
is called a secant of the circle.
Tangent
A line that touches the circle at only one point is called
a tangent to the circle.
The point at which the tangent meets the circle is its point of contact.
Arc of a Circle
A continuous piece of a circle is called an arc of the circle.
The whole circle has been divided into two pieces, namely, major arc, minor arc.
A B
D C
E
P
P Q
Fig. 7.9
M
i
n
o
r
a
r
c
O
Segment
Sector
M
a
j
o
r
a
r
c
Fig. 7.10
S
e
c
a
n
t
R
a
d
i
u
s
Diameter
O
C
h
o
r
d
Tangent
Centre
Geometry
175
Concentric Circles
Circles which have the same centre but different radii are called
concentric circles.
In the given fgure, the two circles are concentric circles having
the same centre O but different radii r and R respectively.
Congruent Arcs
Two arcs AB
!
and CD
!
of a circle are said to be congruent
if they substend same angle at the centre and we write
AB
!
/ CD
!
. So,
AB
!
/ CD
!
, mAB
!
=mCD
!
,+AOB =+COD
7.5.1 Properties of Chords of a Circle
Result
Equal chords of a circle subtend equal angles at the centre.
In the Fig. 7.13., chord AB =chord CD ( +AOB = +COD
Converse of the result
If the angles subtended by two chords at the centre of a circle
are equal, then the chords are equal.
+AOB = +COD (chord AB =chord CD
Theorem 1
Perpendicular from the centre of a circle to a chord bisects
the chord.
Given : A circle with centre O and AB is a chord of the circle other than the diameter
and OC= AB
To prove: AC = BC
Construction: J oin OA and OB
Proof:
In Ts OAC and OBC
(i) OA = OB (Radii of the same circle.)
(ii) OC is common
(iii) +OCA = + OCB (Each 90c ,since OC = AB.)
(iv) TOAC / TOBC (RHS congruency.)
` AC = BC
O
C
D
A
B
Fig. 7.13
O
C D
A
B
Fig. 7.12
A
C
B
O
Fig. 7.14
O r
R
B
Fig. 7.11
A
Chapter 7
176
Converse of Theorem 1 : The line joining the centre of the circle and the midpoint of a chord
is perpendicular to the chord.
Theorem 2
Equal chords of a circle are equidistant from the centre.
Given: A cirle with centre O and radius r such that
chord AB =chord CD.
To prove: OL = OM
Construction: Draw OL = AB and OM = CD. J oin OA and OC
Proof:
(i) AL =
2
1
AB and CM =
2
1
CD (Perpendicular from the centre of a circle
to the chord bisects the chord.)
AB = CD (
2
1
AB =
2
1
CD ( AL = CM
(ii) OA = OC (radii)
(iii) +OMC= + OLA (Each 90c )
(iii) TOLA /
TOMC (RHS congruence.)
` OL = OM
Hence AB and CD are equidistant from O .
Converse of Theorem 2 : The chords of a circle which are equidistant from the centre
are equal.
Example 7.5
A chord of length 16 cm is drawn in a circle of radius 10 cm. Find the distance of the
chord from the centre of the circle.
Solution AB is a chord of length 16 cm
C is the midpoint of AB.
OA is the radius of length 10 cm
AB = 16 cm
AC =
2
1
16 # =8 cm
OA = 10 cm
In a right triangle OAC.
OC
2
=OA
2
- AC
2
Given : O is the centre of the circle. AXB is the arc. +AOB is the angle subtended by
the arc AXB
!
at the centre. +ACB is the angle subtended by the arc AXB
!
at
a point on the remaining part of the circle.
A
B
O
C M D
Fig. 7.17
(iv) OA = OB = OC ( radius )
+OCA = +OAC =25c
+OBC = +OCB =20c
+ACB = +OCA + +OCB
=25c +20c =45c
AOB = 2 +ACB
x =2 5 4 # c=90c
Example 7.8
In the Fig.7.27, O is the centre of a circle and
+ADC =120c. Find the value of x.
Solution ABCD is a cyclic quadrilateral.
we have
+ABC + +ADC =180c
+ABC =180c - 120c =60c
Also +ACB =90c ( angle on a semi circle )
In 3ABC we have
+BAC + +ACB + +ABC =180c
90 60 BAC + + + c c =180c
+BAC =180c - 150c =30c
C
A
B
O
xc
100c
C
A B
O
2
5
c
xc
2
0
c
C
A B
O
xc
56c
C
A B
O
D
xc
120c
Fig. 7.27
Geometry
181
Exercise 7.3
1. The radius of a circle is 15 cm and the length of one of its chord is 18 cm. Find the
distance of the chord from the centre.
2. The radius of a circles 17 cm and the length of one of its chord is 16 cm. Find the
distance of the chord from the centre.
3. A chord of length 20 cm is drawn at a distance of 24 cm from the centre of a circle.
Find the radius of the circle.
4. A chord is 8 cm away from the centre of a circle of radius 17 cm. Find the length of
the chord.
5. Find the length of a chord which is at a distance of 15 cm from the centre of a circle
of radius 25 cm.
6. In the fgure at right, AB and CD are two parallel chords of a
circle with centre O and radius 5 cm such that AB =6 cm and
CD =8 cm. If OP = AB and CD = OQ determine the length
of PQ.
7. AB and CD are two parallel chords of a circle which are on either sides of the centre.
Such that AB =10 cm and CD =24 cm. Find the radius if the distance between AB and
CD is 17 cm.
8. In the fgure at right, AB and CD are two parallel chords of a
circle with centre O and radius 5 cm. Such that AB =8 cm and
CD =6 cm. If OP = AB and OQ = CD
determine the length PQ.
9. Find the value of x in the following fgures.
(i) (ii) (iii)
A B
O
C
Q
D
P
C
A
D
B
O
P
Q
A
B
C
O
90c
xc
1
2
0
c
B
C
O
2
5
c
xc
A
3
0
c
O
x
35c
A B
C
Chapter 7
182
(iv) (v) (vi)
10. In the fgure at right, AB and CD are straight lines
through the centre O of a circle. If AOC + =98c
and
+
CDE =35c
fnd (i) +DCE (ii) +ABC
11. In the fgure at left, PQ is a diameter of a circle with
centre O. If +PQR =55c, +SPR =25c and +PQM =50c.
Find (i) +QPR, (ii) +QPM and (iii) +PRS.
12. In the fgure at right, ABCD is a cyclic quadrilateral
whose diagonals intersect at P such that
+
DBC =30c and
+BAC =50c.
Find (i) +BCD (ii) +CAD
13. In the fgure at left , ABCD is a cyclic quadrilateral in which
AB || DC. If +BAD =100c
fnd (i) +BCD (ii) +ADC (iii) +ABC.
P
25c
Q
S
M
50c
55c
R
O
D
C
B
A
P
50c
30c
D
C
B
A
1
0
0
c
A
B
C
O
130c
xc
A B
C
D
50c
O
A
B
C
48c
xc
O
xc
A
98c
B
E
C
D
35c
O
Geometry
183
D
C
B
A
100c
50c
100c
A
B
C
D
O
14. In the fgure at right, ABCD is a cyclic quadrilateral in
which
+
BCD =100c and
+
ABD =50c fnd +ADB
15. In the fgure at left, O is the centre of
the circle, +AOC = 100c and side AB is
produced to D.
Find (i) +CBD (ii) +ABC
Points to remember
� In a parallelogram the opposite sides are equal.
� In a parallelogram, the opposite angles are equal.
� The diagonals of a parallelogram bisect each other.
� A rectangle is an equiangular parallelogram
� A rhombus is an equilateral parallelogram
� A square is an equilateral and equiangular parallelogram. Thus a square is a
rectangle, a rhombus and a parallelogram.
� Each diagonal divides the parallelogram into two congruent triangles.
� A parallelogram is rhombus if its diagonals are perpendicular.
� Parallelograms on the same base and between the same parallels are equal in area.
� A diagonal of a parallelogram divides it into two triangles of equal area.
� Equal chords of a circle subtend equal angles at the centre.
� If two arcs of a circle are congruent then the corresponding chords are equal.
� Perpendicular from the centre of a circle to a chord bisects the chord.
� Equal chords of a circle are equidistant from the centre.
� The angle substended by an are of a circle at the centre is double the angel subtended
by it at any point on the remaining part of the circle.
� The angle in a semi circle is a right angle.
� Angle in the same segment of a circle are equal.
� The sum of either pair of opposite angle of opposite angles of a cyclic
quadrilateral is 180c
� If one side of a cyclic quadrilateral is produced then the exterior angle is equal to
the interior opposite angle.
Chapter 8
184
MENSURATION
8.1 I ntroduction
Every day, we see various shapes like triangles,
rectangles, squares, circles, spheres and so on all around us,
and we are already familiar with some of their properties:
like area and perimeter. The part of Mathematics that deals
with measurements of geometrical shapes is known as
Mensuration. It is considered very important because there
are various fields of life where geometry is considered as an
important field of study.
Perimeter, Area and Volume plays a vital role in
architecture and carpentry. Perimeter, Area and volume can
be used to analyze real-world situations. It is necessary for
everyone to learn formulas used to find the perimeter ,areas of
two-dimensional figures and the surface areas and volumes of
three dimensional figures for day- to-day life. In this chapter
we deal with arc length and area of sectors of circles and
area and volume of cubes and cuboids.
● To find the length of arc, area and perimeter of sectors
of circles.
● To find the surface area and volume of cubes.
● To find the surface area and volume of cuboids.
Main Targets
One of the very great
mathematicians of all time
was Archimedes, a native of
the Greek city of Syracus on
the island of Sicily. He was
born about 287 B.C. It was
Archimedes who inaugurated
the classical method of
computing r by the use of
regular polygons inscribed
in and circumscribed about a
circle. He is responsible for
the correct formulas for the
area and volume of a sphere.
He calculated a number of
interesting curvilinear areas,
such as that of a parabolic
segment and of a sector of the
now so called Archimedean
spiral. In a number of his
works he laid foundations of
mathematical physics.
Archimedes
287 - 212 B.C.
The most beautiful plane figure is – the circle and the most
beautiful solid figure – the sphere
- PYTHAGORAS
184
Mensuration
185
8.2 Sectors
Two points P and Q on a circle
with centre O determine an arc PQ
denoted by PQ
!
, an angle +POQ and a
sector POQ. The arc starts at P and goes
counterclockwise to Q along the circle.
The sector POQ is the region bounded by
the arc PQ
!
and the radii OP and OQ. As
Fig.8.1 shows, the arcs PQ
!
and QP
!
are
different.
Key Concept Sector
A sector is the part of a circle enclosed by any two radii of the circle and
their intercepted arc.
8.2.1 Central Angle or Sector Angle of a Sector
Key Concept Central Angle
Central Angle is the angle subtended by the arc of the sector at the centre
of the circle in which the sector forms a part.
In fg.8.2, the angle subtended by the arc PQ
!
at the centre is i.
So the central angle of the sector POQ is i.
For example,
1. A semi- circle is a sector whose central angle is 180c.
2. A quadrant of a circle is a sector whose central angle is
90c .
8.2.2 Length of Arc (Arc Length) of a Sector
In fg.8.3, arc length of a sector POQ is the length of the portion
on the circumference of the circle intercepted between the bounding
radii (OP and OQ) and is denoted by l.
P Q
O
r
q
Fig. 8.2
P Q
O
l
q
Fig. 8.3
Fig. 8.1
Arc PQ
!
Sector POQ
Arc QP
!
Sector QOP
Chapter 8
186
For example,
1. Length of arc of a circle is its circumference. i.e., l=2rr units, where r is the
radius.
2. Length of arc of a semicircle is l =2
360
r
180
# r
c
c
= r r units, where r is the radius
and the central angle is 180c.
3. Length of arc of a quadrant of a circle is l 2
360
90
r # r =
c
c
=
r
2
r
units, where r is
the radius and the central angle is 90c.
Key Concept Length of Arc
If i is the central angle and r is the radius of a sector, then its arc length
is given by l =
360
2 r #
i
r
c
units
8.2.3 Area of a Sector
Area of a sector is the region bounded by the bounding radii and the arc of the sector.
For Example,
1. Area of a circle is r
2
r square units.
2. Area of a semicircle is
r
2
2
r
square units.
3. Area of a quadrant of a circle is
r
4
2
r
square units.
Key Concept Area of a Sector
If i is the central angle and r is the radius of a sector, then the area of the
sector is
360
r
2
#
i
r
c
square units.
Let us fnd the relationship between area of a sector, its arc length l and radius r.
Area = r
360
2
#
i
r
c
=
360
r
r
2
2
# #
i r
c
=
360
r r
2
1
2 # # #
i
r
c
c m
= lr
2
1
#
Area of sector =
lr
2
square units.
P Q
O
q
Mensuration
187
8.2.4 Perimeter of a Sector
The perimeter of a sector is the sum of the lengths of all its boundaries. Thus, perimeter
of a sector is l r 2 + units.
Key Concept Perimeter of a Sector
If l is the arc length and r is the radius of a sector, then its perimeter P is
given by the formula P l r 2 = + units.
For example,
1. Perimeter of a semicircle is 2 r r + ^ h units.
2. Perimeter of a quadrant of a circle is
2
2 r
r
+ ` j
units.
1. Length of an arc and area of a sector are proportional to the central angle.
2. As r is an irrational number, we use its approximate val1ue
7
22
or 3.14 in
our calculations.
Example 8.1
The radius of a sector is 42 cm and its sector angle is 60c. Find its arc length, area and
perimeter.
Solution Given that radius r = 42 cm and i = 60c. Therefore,
length of arc l =
360
2 r #
i
r
c
(ii) Find the radius of the sector of area 225 cm
2
and having an arc length of 15 cm
(iii) Find the radius of the sector whose central angle is 140c and area 44 cm
2
.
5. (i) The perimeter of a sector of a circle is 58 cm. Find the area if its diameter is 9 cm.
(ii) Find the area of a sector whose radius and perimeter are 20 cm and 110 cm
respectively.
6. Find the central angle of a sector of a circle having
(i) area 352 cm
2
and radius 12 cm
(ii) area 462 cm
2
and radius 21 cm
7. (i) Calculate the perimeter and area of the semicircle whose radius is 14 cm.
(ii) Calculate the perimeter and area of a quadrant circle of radius 7 cm.
8. (i) Calculate the arc length of a sector whose perimeter and radius are 35 cm and
8 cm respectively.
(ii) Find the radius of a sector whose perimeter and arc length are 24 cm and 7 cm
respectively.
9. Time spent by a student in a day is shown in the fgure. Find how much
time is spent in
(i) school (ii) play ground (iii) other activities
10. Three coins each 2 cm in diameter are placed touching one another.
Find the area enclosed by them.
11. Four horses are tethered with ropes measuring 7 m each to the four corners of a
rectangular grass land 21 m # 24 m in dimension. Find
(i) the maximum area that can be grazed by the horses and
(ii) the area that remains ungrazed.
12. Find the area of card board wasted if a sector of maximum possible size is cut out from
a square card board of size 24 cm. . 3 14 r = 6 @.
A
C
B
O
School
P
l
a
y
G
r
o
u
n
d
OtherActivities
225
o
105
o
Mensuration
193
13. Find the area of the shaded portion in the adjoining fgure
14. Find the radius, central angle and perimeter of a sector whose length
of arc and area are 4.4 m and 9.24 m
2
respectively.
8.3 Cubes
You have learnt that a cube is a solid having six square faces. Example: Die.
In this section you will learn about surface area and volume of a cube.
8.3.1 Surface Area of a Cube
The sum of the areas of all the six equal faces is called the Total Surface Area
(T.S.A) of the cube.
In the adjoining figure, let the side of the cube measure a
units. Then the area of each face of the cube is a
2
square units.
Hence, the total surface area is a 6
2
square units.
In a cube, if we don't consider the top and bottom faces, the
remaining area is called the Lateral Surface Area (L.S.A). Hence,
the lateral surface area of the cube is a 4
2
square units.
Key Concept Surface Area of Cube
Let the side of a cube be a units. Then:
(i) The Total Surface Area (T.S.A) = a 6
2
square units.
(ii) The Lateral Surface Area (L.S.A) = a 4
2
square units.
8.3.2 Volume of a Cube
Key Concept Volume of Cube
If the side of a cube is a units, then its volume V is given by the formula
V =a
3
cubic units
Volume can also be defned as the number of unit cubes required to fll the
entire cube.
a
a
a
Note
C D
4
2
c
m
A B
2
1
c
m
O
Chapter 8
194
Example 8.16
Find the L.S.A, T.S.A and volume of a cube of side 5 cm.
Solution L.S.A = a 4
2
=4(5
2
) =100 sq. cm
T.S.A = a 6
2
= 6 (5
2
) = 150 sq. cm
Volume =a
3
=5
3
= 125 cm
3
Example: 8.17
Find the length of the side of a cube whose total surface area is 216 square cm.
Solution Let a be the side of the cube. Given that T.S.A = 216 sq. cm
i. e., a 6
2
= 216 ( a
2
=
6
216
36 =
` a = 36 = 6 cm
Example 8.18
A cube has a total surface area of 384 sq. cm. Find its volume.
Solution Let a be the side of the cube. Given that T.S.A = 384 sq. cm
a 6
2
= 384 ( a
2
=
6
384
64 =
` a = 64 =8 cm
Hence, Volume =a
3
=8
3
= 512 cm
3
Example 8.19
A cubical tank can hold 27,000 litres of water. Find the dimension of its side.
Solution Let a be the side of the cubical tank. Volume of the tank is 27,000 litres. So,
V = a
3
=
,
,
m
1 000
27 000 3
= m 27
3
` a = 27
3
= 3 m
Exercise 8.2
1. Find the Lateral Surface Area (LSA), Total Surface Area (TSA) and volume of the
cubes having their sides as
(i) 5.6 cm (ii) 6 dm (iii) 2.5 m (iv) 24 cm (v) 31 cm
2. (i) If the Lateral Surface Area of a cube is 900 cm
2
, fnd the length of its side.
(ii) If the Total Surface Area of a cube is 1014 cm
2
, fnd the length of its side.
(iii) The volume of the cube is 125 dm
3
. Find its side.
5cm
5
c
m
5
c
m
a
a
a
a
a
a
Mensuration
195
3. A container is in the shape of a cube of side 20 cm. How much sugar can it hold?
4. A cubical tank can hold 64,000 litres of water. Find the length of the side of the tank.
5. Three metallic cubes of side 3 cm, 4 cm and 5 cm respectively are melted and are
recast into a single cube. Find the total surface area of the new cube.
6. How many cubes of side 3 cm are required to build a cube of side 15 cm?
7. Find the area of card board required to make an open cubical box of side 40 cm. Also
fnd the volume of the box.
8. What is the total cost of oil in a cubical container of side 2 m if it is measured and sold
using a cubical vessel of height 10 cm and the cost is ` 50 per measure.
9. A container of side 3.5m is to be painted both inside and outside. Find the area to be
painted and the total cost of painting it at the rate of ` 75 per square meter.
8.4 Cuboids
A cuboid is a three dimensional solid having six rectangular faces.
Example: Bricks, Books etc.,
8.4.1 Surface Area of a Cuboid
Let l, b and h be the length, breadth and height of a cuboid respectively. To fnd the
total surface area, we split the faces into three pairs.
(i ) The total area of the front and back faces is
lh + lh = lh 2 square units.
(ii ) The total area of the side faces is
bh + bh =2bh square units.
(iii ) The total area of the top and bottom faces
is lb + lb =2lb square units.
The Lateral Surface Area (L.S.A) = 2( l + b)h square units.
The Total Surface Area (T.S.A) = 2( lb + bh + lh ) square units.
Key Concept Surface Area of a Cuboid
Let l, b and h be the length, breadth and height of a cuboid respectively.
Then
(i) The Lateral Surface Area (L.S.A) = 2( l + b)h square units
(ii) The Total Surface Area (T.S.A) = 2( lb + bh + lh ) sq. units
L.S.A. is also equal to the product of the perimeter of the base and the height.
l
h
b
Note
Chapter 8
196
8.4.2 Volume of a Cuboid
Key Concept Volume of a Cuboid
If the length, breadth and height of a cuboid are l, b and h respectively,
then the volume V of the cuboid is given by the formula
V =l b h # # cu. units
Example: 8.20
Find the total surface area of a cuboid whose length, breadth and height are 20 cm,
12 cm and 9 cm respectively.
Solution
Given that l = 20 cm, b = 12 cm, h =9 cm
` T.S.A = 2 (lb + bh + lh)
= 2[(20 12 # ) + (12 9 # ) + (20 9 # )]
= 2(240 + 108 + 180)
= 2#528
= 1056 cm
2
Example: 8.21
Find the L.S.A of a cuboid whose dimensions are given by 3 5 4 m m m # # .
Solution Given that l = 3 m, b = 5 m, h = 4 m
L.S.A = 2(l +b)h
= 2 # (3+5) # 4
=2 8 4 # #
= 64 sq. m
Example: 8.22
Find the volume of a cuboid whose dimensions are given by 11 m, 10 m and 7 m.
Solution Given that l = 11 m, b = 10 m, h = 7 m
volume =lbh
=11 10 7 # #
=770 cu.m.
20cm
1
2
c
m
9
c
m
11m
7
m
1
0
m
3m
5
m
4
m
Mensuration
197
Example: 8.23
Two cubes each of volume 216 cm
3
are joined to form a cuboid as shown in the fgure.
Find the T.S.A of the resulting cuboid.
Solution Let the side of each cube be a. Then a 216
3
=
` a = 216
3
= 6 cm
Now the two cubes of side 6 cm are joined to form a
cuboid. So,
` l = 6 + 6 = 12 cm, b = 6 cm, h = 6 cm
` T.S.A = 2 (lb + bh + lh)
= 2 [(12 6 # ) + (6 6 # ) + (12 6 # )]
= 2 [72 + 36 + 72]
= 2 # 180 = 360 cm
2
Example 8.24
J ohny wants to stitch a cover for his C.P.U whose length, breadth and height are
20 cm, 45 cm and 50 cm respectively. Find the amount he has to pay if it costs ` 50 per sq. m
Solution The cover is in the shape of a one face open cuboidal box.
l = 20 cm = 0.2 m, b = 45 cm =0.45 m, h = 50 cm = 0.5 m
` Area of cloth required = L.S.A + area of the top
= 2 (l + b) h + lb
= 2 (0.2 +0.45) 0.5 + (0.2# 0.45)
= . . . 2 0 65 0 5 0 09 # # +
= 0.65 + 0.09
= 0.74 sq.m
Given that cost of 1 sq. m of cloth is ` 50
` cost of 0.74 sq.m of cloth is 50 # 0.74 = ` 37.
20cm
4
5
c
m
5
0
c
m
Chapter 8
198
Example: 8.25
Find the cost for flling a pit of dimensions 5 m # 2 m # 1m with soil if the rate of
flling is ` 270 per cu. m
Solution The pit is in the shape of a cuboid having l = 5m, b = 2m and h = 1m.
`volume of the pit =volume of the cuboid
=lbh
= 5 # 2 # 1
=10 cu.m
Given that cost for flling 1 cu. m is ` 270
` cost for flling 10 cu. m is
270 # 10 =` 2700
Exercise 8.3
1. Find the L.S.A, T.S.A and volume of the cuboids having the length, breadth and height
respectively as
(i) 5 cm, 2 cm , 11cm (ii) 15 dm, 10 dm, 8 dm
(iii) 2 m, 3 m, 7 m (iv) 20 m, 12 m, 8 m
2. Find the height of the cuboid whose length, breadth and volume are 35 cm, 15 cm and
14175 cm
3
respectively.
3. Two cubes each of volume 64 cm
3
are joined to form a cuboid. Find the L.S.A and
T.S.A of the resulting solid.
4. Raju planned to stitch a cover for his two speaker boxes whose length, breadth and
height are 35 cm, 30 cm and 55 cm respectively. Find the cost of the cloth he has to
buy if it costs ` 75 per sq.m.
5. Mohan wanted to paint the walls and ceiling of a hall. The dimensions of the hall is
20 15 m m m 6 # # . Find the area of surface to be painted and the cost of painting it at
` 78 per sq. m.
6. How many hollow blocks of size 0cm cm cm 3 15 20 # # are needed to construct a wall
60m in length, 0.3m in breadth and 2m in height.
7. Find the cost of renovating the walls and the foor of a hall that measures 10m#45m#6m
if the cost is ` 48 per square meter.
Mensuration
199
Points to Remember
� A sector is the part of a circle enclosed by any two radii of the circle and their
intercepted arc.
� Central Angle is the angle subtended by the arc of the sector at the centre of the
circle in which the sector forms a part.
� If i is the central angle and r is the radius of a sector, then its arc length is given
by l= 2 r
360
#
i
r units
� If i is the central angle and r is the radius of a sector, then the area of the sector
is r
360
2
#
i
r square units.
� If l is the arc length and r is the radius of a sector, then its perimeter P is given by
the formula 2 P l r = + units.
� Let the side of a cube be a units. Then:
(i) The Total Surface Area (T.S.A) =6a
2
square units.
(ii) The Lateral Surface Area (L.S.A) =4a
2
square units.
� If the side of a cube is a units, then its volume V is given by the formula,
V =a
3
cubic units
� Let l, b and h be the length, breadth and height of a cuboid respectively. Then:
(i) The Lateral Surface Area (L.S.A) = 2( l + b)h squre units
(ii) The Total Surface Area (T.S.A) = 2( lb + bh + lh ) sq. units
� If the length, breadth and height of a cuboid are l, b and h respectively, then the
volume V of the cuboid is given by the formula V =l b h # # cu. units
Chapter 9
200
9.1 I ntroduction
The fundamental principles of geometry deal with
the properties of points, lines, and other figures. Practical
Geometry is the method of applying the rules of geometry
to construct geometric figures. "Construction" in Geometry
means to draw shapes, angles or lines accurately. The
geometric constructions have been discussed in detail in
Euclid's book 'Elements'. Hence these constructions are also
known as Euclidean constructions. These constructions use
only a compass and a straightedge (i.e. ruler). The compass
establishes equidistance and the straightedge establishes
collinearity. All geometric constructions are based on those
two concepts.
It is possible t
o construct rational and irrational
numbers using a straightedge and a compass as seen in chapter
II. In 1913 the Indian Mathematical Genius, Ramanujan
gave a geometrical construction for
355
113
r = . Today
with all our accumulated skill in exact measurements, it is
a noteworthy feature when lines driven through a mountain
meet and make a tunnel. How much more wonderful is it
that lines, starting at the corner of a perfect square, should be
raised at a certain angle and successfully brought to a point,
hundreds of feet aloft! For this, and more, is what is meant
by the building of a pyramid:
● To construct the Circumcentre
● To construct the Orthocentre
● To construct the Incentre
● To construct the Centroid
Main Targets
The Swiss mathematician
Leonhard Euler lived during
the 18th century. Euler wrote
more scientific papers than any
mathematician before or after
him. For Euler, mathematics was
a tool to decipher God's design
of our world. With every new
discovery, he felt a step nearer to
understanding nature and by this
understanding God.
Euler even found a new
theoremin Euclidean geometry,
a field which had been looked
at as completed. Here's a short
explanation of this theorem:
The three altitudes of a
triangle meet in point H, and the
three perpendicular bisectors
in point M. Point E in the
middle of the line between H
and M is the center of a circle
on which are all the inter-
sections of the altitudes and the
perpendicular bisectors with the
triangle. This circle known as 9
points circle.
LEONHARD EULER
1707 - 1783
A
B C
M
H E
M
c
M
a
M
b
H
c
H
b
H
a
PRACTICAL GEOMETRY
200
Practical Geometry
201
In class VIII we have learnt the construction of triangles with the given measurements.
In this chapter we learn to construct centroid, ortho-centre, in-centre and circum-centre
of a triangle.
9.2 Special line segments within Triangles
First let us learn to identify and to construct
(i) Perpendicular bisector to a given line segment
(ii) Perpendicular from an external point to a given line
(iii) Bisector of a given angle and
(iv) Line joining a given external point and the midpoint of a given line segment.
9.2.1 Construction of the Perpendicular Bisector of a given line segment
Step1 : Draw the given line segment AB.
Step 2 : With the two end points A and B of the line segment as
centres and more than half the length of the line segment
as radius draw arcs to intersect on both sides of the line
segment at C and D
Step 3 : J oin C and D to get the perpendicular bisector
of the given line segment AB.
Key Concept Perpendicular Bisector
The line drawn perpendicular through the midpoint of a given line
segment is called the perpendicular bisector of the line segment.
A B
A B
C
D
A B
C
D
M
Chapter 9
202
9.2.2 Construction of Perpendicular from an External Point to a given line
Step 1 : Draw the given line AB and mark the given
external point C.
Step 2 : With C as centre and any convenient radius
draw arcs to cut the given line at two points
P and Q.
Step 3 : With P and Q as centres and more than half the
distance between these points as radius draw
two arcs to intersect each other at E.
Step 4 : J oin C and E to get the required perpendicular
line.
Key Concept Altitude
In a triangle, an altitude is the line segment
drawn from a vertex of the triangle
perpendicular to its opposite side.
A B
C
Q P
E
A
C
Q P
E
B
D
A
C
B
C
A B P Q
A
C
Q P
E
B
D
Altitude
Practical Geometry
203
9.2.3 Construction of Angle Bisector
Step 1 : Draw the given angle +CAB with the given
measurement.
Step 2 : With A as centre and a convenient radius draw arcs to cut
the two arms of the angle at D and E.
Step 3 : With D and E as centres and a suitable radius draw arcs
to intersect each other at F.
Step 4 : J oin A and F to get the angle bisector AF of +CAB.
Key Concept Angle Bisector
The line which divides a given angle into two
equal angles is called the angle bisector of
the given angle.
B A
C
B A
C
E
D
B A
C
E
D
F
F
B A
C
E
D
B A
C
E
D
F
Angle
Bisector
Chapter 9
204
9.2.4 Construction of Line J oining a External Point and the Midpoint of a Line Segment
Step 1 : Draw a line segment AB with the given
measurement and mark the given point C (external
point).
Step 2 : Draw the perpendicular bisector of AB and
mark the point of intersection M which is the
mid point of line segment.
Step 3 : J oin C and M to get the required line.
Key Concept Median
In a triangle, a median is the line segment that joins
a vertex of the triangle and the midpoint
of its opposite side.
A
B M
C
A B
M
C
A B
C
A B
M
C
Median
Practical Geometry
205
9.3 The Points of Concurrency of a Triangle
As we have already learnt how to draw the Perpendicular Bisector, Altitude, Angle
Bisector and Median, now let us learn to locate the Circumcentre, Othocentre, Incentre and
Centroid of a given triangle.
9.3.1 Construction of the Circumcentre of a Triangle
Key Concept Circumcentre
The point of concurrency of the perpendicular
bisectors of the sides of a triangle is called
the circumcentre and is usually denoted by S.
Circumcircle
The circle drawn with S (circumcentre) as centre and passing through all the three
vertices of the triangle is called the circumcircle.
Circumradius
The radius of a circumcircle is called circumradius of the triangle. In other words,
the distance between the circumcentre S and any vertex of the triangle is the circumradius.
A
C
B
S
Circumradius
S
C
i
r
c
u
m
c
i
r
c
l
e
circumcentre
Chapter 9
206
Example 9.1
Construct the circumcentre of the ABC D with AB =5cm, 70 A + = c and 60 B + = c.
Also draw the circumcircle and fnd the circumradius of the ABC D .
Solution
Step 1 : Draw the ABC D with
the given measurements.
Step2 : Construct the perpendicular
bisectors of any two sides
(AC and BC) and let them
meet at S which is the
circumcentre.
Step 3 : With S as centre and
SA = SB = SC as radius
draw the circumcircle to
pass through A, B and C.
Circumradius = 3. 2cm
C
5cm
70c 0 6 c
A B
Rough
Diagram
A
70
o
60
o
C
B
5cm
A
70
o
60
o
C
B
S
5cm
A
70
o
60
o
C
B
S
3
.
2
c
m
5cm
Practical Geometry
207
1. The circumcentre of an acute angled triangle lies inside the triangle.
2. The circumcentre of a right triangle is at the midpoint of its hypotenuse.
3. The circumcentre of an obtuse angled triangle lies outside the triangle.
Exercise 9.1
1. Construct PQR D with PQ =5cm, 100 P + = c and PR = 5cm and draw its circumcircle.
2. Draw the circumcircle for
(i) an equilateral triangle of side 6cm
(ii) an isosceles right triangle having 5cm as the length of the equal sides.
3. Draw ABC D , where AB =7cm, BC =8cm and 60 B + = c and locate its circumcentre.
4. Construct the right triangle whose sides are 4.5cm, 6cm and 7.5cm. Also locate its
circumcentre.
9.3.2 Construction of the Orthocentre of a Triangle
Key Concept Orthocentre
The point of concurrency of the altitudes
of a triangle is called the orthocentre of the
triangle and is usually denoted by H.
Example 9.2
Construct ABC D whose sides are AB = 6cm, BC =4cm and AC = 5.5cm and locate
its orthocentre.
Solution
Step 1 : Draw the ABC D with
the given measurements.
Remark
C
6cm
A B
4
c
m
5
.
5
The point of intersection of the altitudes H is
the orthocentre of the given ABC D
1. Three altitudes can be drawn in a triangle.
2. The orthocentre of an acute angled triangle lies inside the triangle.
3. The orthocentre of a right triangle is the vertex of the right angle.
4. The orthocentre of an obtuse angled triangle lies outside the triangle.
Exercise 9.2
1. Draw ABC D with sides AB =8cm, BC =7cm and AC =5cm and construct its
orthocentre.
2. Construct the orthocentre of , LMN D where LM =7cm, 130 M + = c and MN = 6cm
3. Construct an equilateral triangle of sides 6cm and locate its orthocentre.
4. Draw and locate the orthocentre of a right triangle PQR right angle at Q, with
PQ = 4.5cm, RS = 6cm
5. Construct an isosceles triangle ABC with AB = BC and 0 B 8 + = c of sides 6cm and
locate its orthocentre.
9.3.3 Construction of the I ncentre of a Triangle
Key Concept Incentre
The point of concurrency of the internal angle
bisectors of a triangle is called the incentre
of the triangle and is denoted by I.
Remark
I
C
A B
Practical Geometry
209
I ncircle The circle drawn with the incentre (I)
as centre and touching all the three
sides of a triangle is the incircle of the
given triangle.
I nradius The radius of the incircle is called the
inradius of the triangle.
(or)
It is the shortest distance of any side of the triangle
from the incentre I.
Example 9.3
Construct the incentre of ABC D with AB =7cm, 50 B
o
+ = and BC = 6cm. Also draw
the incircle and measure its inradius.
Solution
Step 1 : Draw the ABC D with the
given measurments.
Step 2 : Construct the angle bisectors
of any two angles (A and B)
and let them meet at I. Then I
is the incentre of ABC D
I
50
o
6
c
m
7cm
A B
C
C
7cm
A
B
6
c
m
0 5 c
Rough Diagram
I
N
C
I
R
C
LE
I
N
R
A
D
I
U
S
I
A
B C
50
o
6
c
m
7cm
A B
C
Chapter 9
210
Step 3 : With I as an external point
drop a perpendicular to any
one of the sides to meet
at D.
Step 4 : With I as centre and ID as
radius draw the circle. This
circle touches all the sides of the
triangle.
Inradius =1.8 cm
The incentre of any triangle always lies inside the triangle.
Exercise 9.3
1. Draw the incircle of ABC D , where AB =9 cm, BC =7cm, and AC =6cm.
2. Draw the incircle of ABC D in which AB =6 cm, AC =7 cm and 40 A + = c. Also
fnd its inradius.
3. Construct an equilateral triangle of side 6cm and draw its incircle.
4. Construct ABC D in which AB =6 cm, AC =5 cm and 0 A 11 + = c. Locate its incentre
and draw the incircle.
I
50
o
6
c
m
7cm
1
.
8
c
m
A B
C
D
I
50
o
6
c
m
7cm
1
.
8
c
m
A B
C
D
Remark
Practical Geometry
211
9.3.4 Construction of the Centroid of a Triangle.
Key Concept Centroid
The point of concurrency of the medians of a
triangle is called the Centroid of the triangle and
is usually denoted by G.
Example 9.4
Construct the centroid of ABC D whose sides are AB =6cm, BC =7cm,
and AC =5cm.
Solution
=
100
3970
= 39.7
Hence the correct mean is 39.7.
Exercise 11.2
1. Obtain the mean number of bags sold by a shopkeeper on 6 consecutive days from the
following table
Days Monday Tuesday Wednesday Thursday Friday Saturday
No. of bags sold 55 32 30 25 10 20
2. The number of children in 10 families in a locality are 2, 4, 3, 4, 1, 6, 4, 5, x, 5. Find x
if the mean number of children in a family is 4
Chapter 11
236
3. The mean of 20 numbers is 59. If 3 is added to each number what will be the new mean?
4. The mean of 15 numbers is 44. If 7 is subtracted from each number what will be the
new mean?
5. The mean of 12 numbers is 48. If each numbers is multiplied by 4 what will be the new
mean?
6. The mean of 16 numbers is 54. If each number is divided by 9 what will be the new
mean?
7. The mean weight of 6 boys in a group is 48 kg. The individual weights of 5 of them
are 50kg, 45kg, 50kg, 42kg and 40kg. Find the weight of the sixth boy.
8. Using assumed mean method fnd the mean weight of 40 students using the data given
below.
weights in kg. 50 52 53 55 57
No. of students 10 15 5 6 4
9. The arithmetic mean of a group of 75 observations was calculated as 27. It was later
found that one observation was wrongly read as 43 instead of the correct value 53.
Obtain the correct arithmetic mean of the data.
10. Mean of 100 observations is found to be 40. At the time of computation two items
were wrongly taken as 30 and 27 instead of 3 and 72. Find the correct mean.
11. The data on number of patients attending a hospital in a month are given below. Find
the average number of patients attending the hospital in a day.
No. of patients 0-10 10-20 20-30 30-40 40-50 50-60
No. of days attending hospital 2 6 9 7 4 2
12. Calculate the arithmetic mean for the following data using step deviation method.
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of students 8 15 22 20 10 5
13. In a study on patients, the following data were obtained. Find the arithmetic mean.
Age (in yrs) 10-19 20-29 30-39 40-49 50-59
No. of patients 1 0 1 10 13
14. The total marks obtained by 40 students in the Annual examination are given below
Marks 150 - 200 200 - 250 250 - 300 300 - 350 350 - 400 400 - 450 450
- 500
Students 2 3 12 10 4 6 3
Using step deviation method to fnd the mean of the above data.
Statistics
237
15. Compute the arithmetic mean of the following distribution.
Class Interval 0 - 19 20 - 39 40 - 59 60 - 79 80 - 99
Frequency 3 4 15 14 4
11.4 Median
Median is defned as the middle item of the given observations arranged in order.
11.4.1 Median - Raw Data
Steps:
(i) Arrange the n given numbers in ascending or descending order of magnitude.
(ii) When n is odd,
n
2
1
th
+
` j
observation is the median.
(iii) When n is even the median is the arithmetic mean of the two middle values.
i.e., when n is even,
Median = Mean of
n
2
th
` j
and
n
2
1
th
+ ` j
observations.
Example 11.10
Find the median of the following numbers
(i) 24, 22, 23, 14, 15, 7, 21 (ii) 17, 15, 9, 13, 21, 32, 42, 7, 12, 10.
Solution
(i) Let us arrange the numbers in ascending order as below.
7, 14, 15, 21, 22, 23, 24
Number of items n = 7
Median =
n
2
1
th
+
` j
observation (a n is odd)
=
2
7 1
th
+
` j
observation
= 4
th
observation = 21
(ii) Let us arrange the numbers in ascending order
7, 9, 10, 12, 13, 15, 17, 21, 32, 42.
Number of items n = 10
Median is the mean of
n
2
th
` j
and
n
2
1
th
+ ` j
observations. (a n is even)
item is 25.
` Median = 25.
Statistics
239
Example 11.12
Find the median for the following distribution.
Value 1 2 3 4 5 6
f 1 3 2 4 8 2
Solution
Value f cf
1 1 1
2 3 4
3 2 6
4 4 10
5 8 18
6 2 20
n = 20
n = 20 (even)
Position of the median =
2
20 1
th
+
` j
th observation
=
2
21
th
` j
th observation = (10.5)
th
observation
The median then, is the average of the tenth and the eleventh items. The tenth item is
4, the eleventh item is 5.
Hence median =
2
4 5 +
=
2
9
= 4.5.
11.4.3 Median - Grouped Frequency Distribution
In a grouped frequency distribution, computation of median involves the following
steps.
(i) Construct the cumulative frequency distribution.
(ii) Find
N
2
th
term.
(iii) The class that contains the cumulative frequency
N
2
is called the median class.
(iv) Find the median by using the formula:
Median = l
f
N
m
c
2
# +
-
,
where l = Lower limit of the median class, f = Frequency of the median class
c = Width of the median class, N = The total frequency
m= cumulative frequency of the class preceeding the median class
Chapter 11
240
Example 11.13
Find the median for the following distribution.
Wages (Rupees
in hundreds)
0-10 10-20 20-30 30-40 40-50
No of workers 22 38 46 35 20
Solution
Wages f cf
0-10 22 22
10-20 38 60
20-30 46 106
30-40 35 141
40-50 20 161
N = 161
Here,
N
2
=
2
161
= 80.5. Median class is 20-30.
Lower limit of the median class l = 20
Frequency of the median class f = 46
Cumulative frequency of the class preceeding the median class m= 60
Width of the class c = 10
Median= l
f
N
m
c
2
# +
-
=
.
20
46
80 5 60
10 # +
-
= . 20
46
10
20 5 # +
= 20
46
205
+ = 20 + 4.46 = 24.46
` Median = 24.46
Example 11.14
Find the median for the following data.
Marks 11-15 16-20 21-25 26-30 31-35 36-40
Frequency 7 10 13 26 9 5
Solution
Since the table is given in terms of inclusive type we convert it into exclusive type.
Statistics
241
Marks f cf
10.5- 15.5 7 7
15.5-20.5 10 17
20.5-25.5 13 30
25.5-30.5 26 56
30.5-35.5 9 65
35.5-40.5 5 70
N = 70
N = 70,
N
2
=
2
70
= 35
Median class is 25.5-30.5
Lower limit of the median class l = 25.5
Frequency of the median class f = 26
Cumulative frequency of the preceding median class m = 30
Width of the median class c = 30.5 - 25.5 = 5
Median = l
f
N
m
c
2
# +
-
= . 25 5
26
35 30
5 # +
-
= . 25 5
26
25
+ = 26.46
Exercise 11.3
1. Find the median of the following data.
(i) 18,12,51,32,106,92,58
(ii) 28,7,15,3,14,18,46,59,1,2,9,21
2. Find the median for the following frequency table.
Value 12 13 15 19 22 23
Frequency 4 2 4 4 1 5
3. Find the median for the following data.
Height (ft) 5-10 10-15 15-20 20-25 25-30
No of trees 4 3 10 8 5
Chapter 11
242
4. Find the median for the following data.
Age group 0-9 10-19 20-29 30-39 40-49 50-59 60-69
No. of persons 4 6 10 11 12 6 1
5. Calculate the median for the following data
Class interval 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35
Frequency 1 18 25 26 7 2 1
6. The following table gives the distribution of the average weekly wages of 800 workers
in a factory. Calculate the median for the data given below.
Wages
(` in hundres)
20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 45 - 50 50 - 55 55 - 60
No. of persons 50 70 100 180 150 120 70 60
11.5 Mode
The Mode of a distribution is the value at the point around which the items tend to be
most heavily concentrated.
11.5.1 Mode - Raw Data
In a raw data, mode can be easily obtained by arranging the observations in an array
and then counting the number of times each observation occurs.
For example, consider a set of observations consisting of values 20,25,21,15,14,15.
Here, 15 occurs twice where as all other values occur only once. Hence mode of this
data = 15.
Mode can be used to measure quantitative as well as qualitative data. If
a printing press turns out 5 impressions which we rate very sharp, sharp,
sharp, sharp and Blurred, then the model value is sharp.
Example 11.15
The marks of ten students in a mathematics talent examination are 75,72,59,62,
72,75,71,70,70,70. Obtain the mode.
Solution Here the mode is 70, since this score was obtained by more students than any
other.
A distribution having only one mode is called unimodal.
Remark
Note
Statistics
243
Example 11.16
Find the mode for the set of values 482,485,483,485,487,487,489.
Solution In this example both 485 and 487 occur twice. This list is said to have two modes
or to be bimodal.
(i) A distribution having two modes is called bimodal.
(ii) A distribution having three modes is called trimodal.
(iii) A distribution having more than three modes is called multimodal.
11.5.2 Mode - Ungrouped Frequency Distribution
In a ungrouped frequency distribution data the mode is the value of the variable having
maximum frequency.
Example 11.17
A shoe shop in Chennai sold hundred pairs of shoes of a particular brand in a certain
day with the following distribution.
Size of shoe 4 5 6 7 8 9 10
No of pairs sold 2 5 3 23 39 27 1
Find the mode of the following distribution.
Solution Since 8 has the maximum frequency with 39 pairs being sold the mode of the
distribution is 8.
11.5.3 Mode - Grouped Frequency Distribution
In case of a grouped frequency distribution, the exact values of the variables are
not known and as such it is very difficult to locate mode accurately. In such cases, if the
class intervals are of equal width an appropriate value of the mode may be determined by
using the formula
Mode = l
f f f
f f
c
2 1 2
1
# +
- -
-
c m ,
where l = lower limit of the modal class
f = frequency of modal class
c = class width of the modal class
f1 = frequency of the class just preceeding the modal class.
f2 = frequency of the class succeeding the modal class.
Note
Chapter 11
244
Example 11.18
Calculate the mode of the following data.
Size of item 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50
No of items 4 8 18 30 20 10 5 2
Solution
Size of the item f
10-15 4
15-20 8
20-25 18
25-30 30
30-35 20
35-40 10
40-45 5
45-50 2
Modal class is 25-30 since it has the maximum frequency.
Lower limit of the modal class l = 25
Frequency of the modal class f = 30
Frequency of the preceding the modal class f1 = 18
Frequency of the class reducing the modal class f2 = 20
Class width c = 5
Mode = l
f f f
f f
c
2 1 2
1
# +
- -
-
^ h
=25
60 18 20
30 18
5 # +
- -
-
` j
= 25
22
12 5 #
+
= 25
22
60
+ = . 25 2 73 + = 27.73
Mode = 27.73
Exercise 11.4
1. The marks obtained by 15 students of a class are given below. Find the modal marks.
42,45,47,49,52,65,65,71,71,72,75,82,72,47,72
2. Calculate the mode of the following data.
Size of shoe 4 5 6 7 8 9 10
No. of Pairs sold 15 17 13 21 18 16 11
Statistics
245
3. The age (in years) of 150 patients getting medical treatment in a hospital in a month
are given below. Obtain its mode.
Age (yrs) 10-20 20-30 30-40 40-50 50-60 60-70
No of patients 12 14 36 50 20 18
4. For the following data obtain the mode.
Weight (in kg) 21-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60
No of students 5 4 3 18 20 14 8 3
5. The ages of children in a scout camp are 13, 13, 14, 15, 13, 15, 14, 15, 13, 15 years.
Find the mean, median and mode of the data.
6. The following table gives the numbers of branches and number plants in a garden of
a school.
No. of branches 2 3 4 5 6
No. of plants 14 21 28 20 17
Calculate the mean, median and mode of the above data.
7. The following table shows the age distribution of cases of a certain disease reported
during a year in a particular city.
Age in year 5 - 14 15 - 24 25 - 34 35 - 44 45 - 54 55 - 64
No. of cases 6 11 12 10 7 4
Obtain the mean, median and mode of the above data.
8. Find the mean, mode and median of marks obtained by 20 students in an examination.
The marks are given below.
Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50
No. of students 1 4 5 8 2
Chapter 11
246
ACTI VI TY
1. Find the mean of 10,20,30,40 and 50.
* Add 10 to each value and fnd the mean.
* Subtract 10 from each value and fnd the mean.
* Multiply each value by 10 and fnd the mean.
* Divide each value by 10 and fnd the mean.
Make a general statement about each situation and compare it with the properties of mean.
2. Give specifc examples of your own in which,
(i) The median is preferred to arithmetic mean.
(ii) Mode is preferred to median.
(iii) Median is preferred to mode.
Points to Remember
The mean for grouped data
The direct method
x
f
fx
/
/
=
The assumed mean method
x A
f
fd
/
/
= +
The step deviation method
x A
f
fd
C #
/
/
= +
The cumulative frequency of a class is the frequency obtained by adding the frequencies
of all up to the classes preceeding the given class.
The median for grouped date can be found by using the formula
median l
f
N
m
c
2
# = +
-
The mode for the grouped data can be found by using the formula
l
f f f
f f
c
2
mode
1 2
1
# = +
- -
-
c m
Probability
247
PROBABILITY
12.1 I ntroduction
From dawn to dusk any individual makes decisions
regarding the possible events that are governed at least
in part by chance. Few examples are: "Should I carry an
umbrella to work today?", "Will my cellphone battery last
until tonight?", and "Should I buy a new brand of laptop?".
Probability provides a way to make decisions when the person
is uncertain about the things, quantities or actions involved
in the decision. Though probability started with gambling, it
has been used extensively, in the fields of Physical Sciences,
Commerce, Biological Sciences, Medical Sciences,
Insurance, Investments, Weather Forecasting and in various
other emerging areas.
Consider the statements:
Probably Kuzhalisai will stand first in the forth
coming annual examination.
Possibly Thamizhisai will catch the train today.
The prices of essential commodities are likely to be
stable.
There is a chance that Leela will win today's Tennis
match.
The words "Probably", "Possibly" , "Likely",
"Chance" , etc., will mean "the lack of certainty" about the
● To understand repeated experiments and observed
frequency approach of Probability
● To understand Empirical Probability
Main Targets
All business proceeds on beliefs, or judgments of
probabilities, and not on certainty - CHARLES ELIOT
The statistical, or empirical,
attitude toward probability
has been developed mainly
by R.F. Fisher and R.
Von Mises. The notion of
sample space comes from
R.Von Mises. This notion
made it possible to build
up a strictly mathematical
theory of probability based
on measure theory. Such
an approach emerged
gradually in the last century
under the influence of many
authors. An axiomatic
treatment representing the
modern development was
given by A. Kolmogorov.
Richard Von Mises
(1883-1953)
247
Chapter 12
248
events mentioned above. To measure "the lack of certainty or uncertainty", there is no perfect
yardstick, i.e., uncertainty is not perfectly quantifable one. But based on some assumptions
uncertainty can be measured mathematically. This numerical measure is referred to as
probability. It is a purposeful technique used in decision making depending on, and changing
with, experience. Probability would be effective and useful even if it is not a single numerical
value.
12.2BasicConceptsandDefnitions
Before we start the theory on Probability, let us defne some of the basic terms required
for it.
Experiment
Random Experiment
Trial
Sample Space
Sample Point
Events
Key Concept Experiment
An experiment is defned as a process whose result is well defned
Experiments are classifed broadly into two ways:
EXPERIMENT
RANDOM DETERMI NI STI C
Probability
249
1. Deterministic Experiment : It is an experiment whose outcomes can
be predicted with certainty, under identical conditions.
For example, in the cases-when we heat water it evaporates,
when we keep a tray of water into the refrigerator it freezes into ice and
while fipping an unusual coin with heads on both sides getting head
- the outcomes of the experiments can be predicted well in advance.
Hence these experiments are deterministic.
2. Random Experiment : It is an experiment whose all possible outcomes are known, but it
is not possible to predict the exact outcome in advance.
For example, consider the following
experiments:
(i) A coin is fipped (tossed)
(ii) A die is rolled.
These are random experiments, since we cannot predict
the outcome of these experiments.
Key Concept
Trial
A Trial is an action which results
in one or several outcomes.
For example,
" Flipping" a coin and "Rolling" a
die are trials
Sample
Space
A sample space S is the set of all
possible outcomes of a random
experiment.
For example,
While fipping a coin the sample space,
S ={ Head, Tail}
While rolling a die, sample space
S ={ 1, 2, 3, 4, 5, 6}
Sample
Point
Each outcome of an experiment is
called a sample point.
While fipping a coin each outcome
{Head}, {Tail} are the sample
points.
While rolling a die each outcome,
{1} {2} {3} {4} {5} and {6} are
are corresponding sample points
Event
Any subset of a sample space is
called an event.
For example,
When a die is rolled some of the
possible events are {1, 2, 3}, {1, 3},
{2, 3, 5, 6}
Chapter 12
250
12.3 ClassifcationofProbability
According to various concepts of probability, it can be classifed mainly in to three
types as given below:
(1) Subjective Probability
(2) Classical Probability
(3) Empirical Probability
12.3.1SubjectiveProbability
Subjective probabilities express the strength of one's belief with regard to the uncertainties.
It can be applied especially when there is a little or no direct evidence about the event desired,
there is no choice but to consider indirect evidence, educated guesses and perhaps intuition and
other subjective factors to calculate probability .
12.3.2 ClassicalProbability
Classical probability concept is originated in connection with games of chance. It applies
when all possible outcomes are equally likely. If there are n equally likely possibilities of which
one must occur and s of them are regarded as favorable or as a success then the probability of a
success is given by
s
n
^ h
.
12.3.3 EmpiricalProbability
It relies on actual experience to determine the likelihood of outcomes.
12.4 Probability-AnEmpiricalApproach
In this chapter, we shall discuss only about empirical probability. The remaining
two approaches would be discussed in higher classes. Empirical or experimental or Relative
frequency Probability relies on actual experience to determine the likelihood of outcomes.
EmpiricalProbability ClassicalProbability
Subjective Objective
PROBABILITY
Probability
251
Empirical approach can be used whenever the experiment can be repeated many times and
the results observed. Empirical probability is the most accurate scientifc 'guess' based on the
results of experiments about an event.
For example, the decision about people buying a certain brand of a soap, cannot be
calculated using classical probability since the outcomes are not equally likely. To fnd the
probability for such an event, we can perform an experiment such as you already have or
conduct a survey. This is called collecting experimental data. The more data we collect the
better the estimate is.
Key Concept Empirical Probability
Let mbe the number of trials in which the event E happened (number of observations
favourable to the event E) and n be the total number of trials (total number of
observations) of an experiment. The empirical probability of happening of an event E,
denoted by P(E), is given by
E P( ) =
Total number of trials
Number of trials in which the event happened
(or)
P(E) =
Total number of observations
Number of favourable observations
` P( ) E
n
m
=
Clearly 0 m n # # ( 0 1
n
m
# # , hence ( ) P E 0 1 # # .
0 1 ( ) P E # #
i.e.the probability of happening of an event always lies from 0 to 1.
Probability is its most general use is a measure of our degree of confdence that a thing
will happen. If the probability is 1.0, we know the thing happen certainly, and if probability
is high say 0.9, we feel that the event is likely to happen. A probability of 0.5 denotes that the
event is a equally likely to happen or not and one of 0 means that it certainly will not. This
interpretation applied to statistical probabilities calculated from frequencies is the only way
of expecting what we know of the individual from our knowledge of the populations.
If P(E) =1 then E is called Certainevent or Sureevent.
If P(E) =0 then E is known is an Impossibleevent.
Remark
Chapter 12
252
If ( ) P E is the probability of an event, then the probability of not happening of E is
denoted by ( ) ( ) P or P E E l
We know, P(E ) + ( ) P El =1; ( ) P E & l = ( ) P E 1 -
( ) P El = ( ) P E 1 -
We shall calculate a few typical probabilities, but it should be kept in mind that
numerical probabilities are not the principal object of the theory. Our aim is to learn
axioms, laws, concepts and to understand the theory of probability easily in higher
classes.
I llustration
A coin is fipped several times. The number of times head and tail appeared and
their ratios to the number of fips are noted below.
Number of
Tosses (n)
Number of
Heads (m
1
)
P(H) =
n
m1
Number of
Tails (m
2
)
P(T) =
n
m2
50 29
50
29
21
50
21
60 34
60
34
26
60
26
70 41
70
41
29
70
29
80 44
80
44
36
80
36
90 48
90
48
42
90
42
100 52
100
52
48
100
48
From the above table we observe that as we increase the number of fips more and
more, the probability of getting of heads and the probability of getting of tails come closer
and closer to each other.
Activity(1)Flippingacoin:
Each student is asked to fip a coin for 10 times and tabulate the number of heads and
tails obtained in the following table.
Outcome
Tally
Marks
Number of heads or
tails for 10 fips.
Head
Tail
Probability
253
Repeat the experiment for 20, 30, 40, 50 times and tabulate the results in the same
manner as shown in the above example. Write down the values of the following fractions.
Activity(2)Rollingadie:
Roll a die 20 times and calculate the probability of obtaining each of six outcomes.
Outcome
Tally
Marks
Number of
outcome for 20 rolls. .
.
Total no of times the die is rolled
No of times corresponding outcomes come up
1
2
3
4
5
6
Repeat the experiment for 50, 100 times and tabulate the results in the same manner.
Activity(3)Flippingtwocoins:
Flip two coins simultaneously 10 times and record your observations in the table.
Outcome Tally
Number of outcomes
for 10 times .
.
Total no of times the two coins are flipped
No of times corresponding outcomes comes up
Two Heads
One head and
one tail
No head
In Activity (1) each fip of a coin is called a trial. Similarly in Activity (2) each roll of a die
is called a trial and each simultaneous fip of two coins in Activity (3) is also a trial.
In Activity (1) the getting a head in a particular fip is an event with outcome "head".
Similarly, getting a tail is an event with outcome tail.
In Activity (2) the getting of a particular number say " 5" is an event with outcome 5.
Total number of times the coin is flipped
Number of times head turn up
d
d
=
Total number of times the coin is flipped
Number of times tail turn up
d
d
=
Chapter 12
254
The value
fliped Total number of times the coins
Number of heads comes up
is called an experimental or
empirical probability.
Example 12.1
A manufacturer tested 1000 cell phones at random and found that 25 of them were
defective. If a cell phone is selected at random, what is the probability that the selected
cellphone is a defective one.
Solution Total number of cell phones tested =1000 i.e., n =1000
Let E be the event of selecting a defective cell phone.
n(E) =25 i.e., m=25
P(E) =
Total number of cellphones tested
Number of defective cellphones
=
n
m
=
1000
25
=
40
1
Example 12.2
In T-20 cricket match, Raju hit a "Six" 10 times out of 50 balls he played. If a ball was
selected at random fnd the probability that he would not have hit a "Six".
Solution Total Number of balls Raju faced =50 i.e., n =50
Let E be the event of hit a "Six" by Raju
n(E) =10 i.e., m=10
P(E) =
" "
Total number of balls faced
Number of times Raju hits a Six
Chapter 12
256
(iv) Let E
4
be the event of a vehicle travelling the speed between 40-69 km/h.
n(E
4
) =28+35+52 =115 i.e. m
4
=115
P(E
4
) =
n
m
4
=
160
115
=
32
23
Example 12.5
A researcher would like to determine whether there is a relationship between a
student's interest in statistics and his or her ability in mathematics. A random sample of 200
students is selected and they are asked whether their ability in mathematics and interest in
statistics is low, average or high. The results were as follows:
Ability in mathematics
Interest in
statistics
Low Average High
Low 60 15 15
Average 15 45 10
High 5 10 25
If a student is selected at random, what is the probability that he / she
(i) has a high ability in mathematics (ii) has an average interest in statistics
(iii) has a high interest in statistics (iv) has high ability in mathematics and high
interest in statistics and (v) has average ability in mathematics and low interest in
statistics.
Solution
Total number of students =80+70+50=200. i.e. n =200
(i) Let E
1
be the event that he/she has a high ability in mathematics .
n(E
1
) =15+10+25=50 i.e. m
1
=50
P(E
1
) =
n
m
1
=
200
50
=
4
1
(ii) Let E
2
be the event that he/she has an average interest in statistics.
n(E
2
) =15+45+10 =70 i.e. m
2
=70
P(E
2
) =
n
m
2
=
200
70
=
20
7
(iii) Let E
3
be the event that he/she has a high interest in statistics.
n(E
3
) =5+10+25 =40 i.e. m
3
=40
P(E
3
) =
n
m
3
=
200
40
=
5
1
Probability
257
(iv) Let E
4
be the event has high ability in mathematics and high interest in
statistics.
n(E
4
) =25 i.e. m
4
=25
P(E
4
) =
n
m
4
=
200
25
=
8
1
(v) Let E
5
be the event has has average ability in mathematics and low interest in
statistics.
n(E
5
) =15 i.e. m
5
=15
P(E
5
) =
n
m
5
=
200
15
=
40
3
Example 12.6
A Hospital records indicated that maternity patients stayed in the hospital for the
number of days as shown in the following.
No. of days stayed 3 4 5 6 more than 6
No. of patients 15 32 56 19 5
If a patient was selected at random fnd the probability that the patient stayed
(i) exactly 5 days (ii) less than 6 days
(iii) at most 4 days (iv) at least 5 days
Solution
Total number of patients of observed =127 i.e., n =127
(i) Let E
1
be the event of patients stayed exactly 5 days.
n(E
1
) =56 i.e., m
1
=56
P(E
1
) =
n
m
1
=
127
56
(ii) Let E
2
be the event of patients stayed less than 6 days.
n(E
2
) =15 +32 +56 =103 i.e., m
2
=103
P(E
2
) =
n
m
2
=
127
103
(iii) Let E
3
be the event of patients stayed atmost 4 days (3 and 4 days only).
n(E
3
) =15 +32 =47 i.e., m
3
=47
P(
E
3
) =
n
m
3
=
127
47
(iv) Let E
4
be the event of patients stayed atleast 5 days (5, 6 and 7 days only).
n(E
4
) =56 +19 +5 =80 i.e., m
4
=80
P(E
4
) =
n
m
4
=
127
80
Chapter 12
258
Exercise 12.1
1. A probability experiment was conducted. Which of these cannot be considered as
a probability of an outcome?
i) 1/3 ii) -1/5 iii) 0.80 iv) -0.78 v) 0
vi) 1.45 vii) 1 viii) 33% ix) 112%
2. Defne: i) experiment ii) deterministic experiment iii) random experiment
iv) sample space v) event vi) trial
3. Defne empirical probability.
4. During the last 20 basket ball games, Sangeeth has made 65 and missed 35 free throws. What
is the empirical probability if a ball was selected at random that Sangeeth make a foul shot?
5. The record of a weather station shows that out of the past 300 consecutive days, its
weather was forecasted correctly 195 times. What is the probability that on a given
day selected at random, (i) it was correct (ii) it was not correct.
6. Gowri asked 25 people if they liked the taste of a new health drink. The responses
are,
Responses Like Dislike Undecided
No. of people 15 8 2
Find the probability that a person selected at random
(i) likes the taste (ii) dislikes the taste (iii) undecided about the taste
7. In the sample of 50 people, 21 has type "O" blood, 22 has type "A" blood, 5 has
type "B" blood and 2 has type "AB" blood. If a person is selected at random fnd the
probability that
(i) the person has type "O" blood (ii) the person does not have type "B" blood
(iii) the person has type "A" blood (iv) the person does not have type "AB" blood.
8. A die is rolled 500 times. The following table shows that the outcomes of the die.
Outcomes 1 2 3 4 5 6
Frequencies 80 75 90 75 85 95
Find the probability of getting an outcome (i) less than 4 (ii) less than 2
(iii) greater than 2 (iv) getting 6 (v) not getting 6.
Probability
259
9. 2000 families with 2 children were selected randomly, and the following data were
recorded.
Number of girls in a family 2 1 0
Number of families 624 900 476
Find the probability of a family, chosen at random, having (i) 2 girls (ii) 1 girl (iii) no girl
10. The follwing table gives the lifetime of 500 CFL lamps.
Life time (months) 9 10 11 12 13 14 more than 14
Number of Lamps 26 71 82 102 89 77 53
A bulb is selected at random. Find the probability that the life time of the selected bulb is
(i) less than 12 months (ii) more than 14 months
(iii) at most 12 months (iv) at least 13 months
11. On a busy road in a city the number of persons sitting in the cars passing by were
observed during a particular interval of time. Data of 60 such cars is given in the
following table.
No. of persons in the car 1 2 3 4 5
No. of Cars 22 16 12 6 4
Suppose another car passes by after this time interval. Find the probability that it has
(i) only 2 persons sitting in it (ii) less than 3 persons in it
(iii) more than 2 persons in it (iv) at least 4 persons in it
12. Marks obtained by Insuvai in Mathematics in ten unit tests are listed below.
Unit Test I II III IV V VI VII VIII IX X
Marks obtained (%) 89 93 98 99 98 97 96 90 98 99
Based on this data fnd the probability that in a unit test Insuvai get
(i) more than 95% (ii) less than 95% (iii) more than 98%
13. The table below shows the status of twenty residents in an apartment
Status
Gender
College Students Employees
Male 5 3
Female 4 8
If one of the residents is chosen at random, fnd the probability that the chosen resident
will be (i) a female (ii) a college student (iii) a female student (iv) a male employee
Chapter 12
260
14. The following table shows the results of a survey of thousand customers who bought
a new or used cars of a certain model
Satisfaction level
Type
Satisfed Not Satisfed
New 300 100
Used 450 150
If a customer is selected at random, what is the probability that the customer
(i) bought a new car (ii) was satisfed (iii) bought an used car but not satisfed
15. A randomly selected sample of 1,000 individuals were asked whether they were
planning to buy a new cellphone in the next 12 months. A year later the same persons
were interviewed again to fnd out whether they actually bought a new cellphone. The
response of both interviews is given below
Buyers Non-buyers
Plan to buy 200 50
No plan to buy 100 650
If a person was selected at random, what is the probability that he/she (i) had a plan to buy
(ii) had a plan to buy but a non-buyer (iii) had no plan to buy but a buyer.
16. The survey has been undertaken to determine whether there is a relationship
between the place of residence and ownership of an automobile. A random sample
of car owners, 200 from large cities, 150 from suburbs and 150 from rural areas
were selected and tabulated as follow
Type of Area
Car ownership
Large city Suburb Rural
Own a foreign car 90 60 25
Do not own a foreign car 110 90 125
If a car owner was selected at random, what is the probability that he/she
(i) owns a foreign car.
(ii) owns a foreign car and lives in a suburb.
(iii) lives in a large city and does not own a foreign car.
(iv) lives in large city and owns a foreign car.
(v) neither lives in a rural area nor owns a foreign car.
Probability
261
17. The educational qualifcations of 100 teachers of a Government higher secondary
school are tabulated below
Education
Age
M.Phil Master Degree
Only
Bachelor Degree
Only
below 30 5 10 10
30 - 40 15 20 15
above 40 5 5 15
If a teacher is selected at random what is the probability that the chosen teacher has (i)
master degree only (ii) M.Phil and age below 30 (iii) only a bachelor degree and age
above 40 (iv) only a master degree and in age 30-40 (v) M.Phil and age above 40
18. A random sample of 1,000 men was selected and each individual was asked to indicate
his age and his favorite sport. The results were as follows.
Sports
Age
Volleyball Basket ball Hockey Football
Below 20 26 47 41 36
20 - 29 38 84 80 48
30 - 39 72 68 38 22
40 - 49 96 48 30 26
50 and above 134 44 18 4
If a respondent is selected at random, what is the probability that
(i) he prefers Volleyball (ii) he is between 20 - 29 years old
(iii) he is between 20 and 29 years old and prefers Basketball
(iv) he doesn't prefer Hockey (v) he is at most 49 of age and prefers Football.
19. On one Sunday Muhil observed the vehicles at a Tollgate in the NH-45 for his science project
about air pollution from 7 am. to 7 pm. The number of vehicles crossed are tabulated below.
Time interval
Vehicles
7 a.m. to 11 a.m. 11 a.m. to 3 p.m. 3 p.m. to 7 p.m.
Bus 300 120 400
Car 200 130 250
Two Wheeler 500 250 350
A vehicle is selected at random. Find the probability that the vehicle chosen is a
(i) a bus at the time interval 7 a.m. to 11 a.m. (ii) a car at the time interval 11 a.m. to 7 p.m.
(iii) a bus at the time interval 7 a.m. to 3 p.m. (iv) a car at the time interval 7 a.m. to 7 p. m.
(v) not a two wheeler at the time interval 7 a.m. to 7 p.m.
Chapter 12
262
Pointstoremember
› Uncertainty or probability can be measured numerically.
› Experiment is defned as a process whose result is well defned.
› Deterministic Experiment : It is an experiment whose outcomes can be predicted
with certainty, under identical conditions.
› Random Experiment is an experiment whose all possible outcomes are known,
but it is not possible to predict the exact outcome in advance.
› A trial is an action which results in one or several outcomes.
› A sample space S is a set of all possible outcomes of a random experiment.
› Each outcome of an experiment is called a sample point.
› Any subset of a sample space is called an event.
› Classifcation of probability
(1) Subjective probability (2) Classical probability (3) Empirical probability
› The empirical probability of happening of an event E, denoted by P(E), is given
by
E P( ) =
Total number of trials
Number of trials in which the event happened | 677.169 | 1 |
Numerical Analysis with Algorithms and Programming
Book Description
Numerical Analysis with Algorithms and Programming is the first comprehensive textbook to provide detailed coverage of numerical methods, their algorithms, and corresponding computer programs. It presents many techniques for the efficient numerical solution of problems in science and engineering.
Along with numerous worked-out examples, end-of-chapter exercises, and Mathematica® programs, the book includes the standard algorithms for numerical computation:
Root finding for nonlinear equations
Interpolation and approximation of functions by simpler computational building blocks, such as polynomials and splines
The text develops students' understanding of the construction of numerical algorithms and the applicability of the methods. By thoroughly studying the algorithms, students will discover how various methods provide accuracy, efficiency, scalability, and stability for large-scale systems. | 677.169 | 1 |
Algebra Helps in Real World Applications
Using algebra in real-world applications happens on a daily basis. Calculating sales tax percentages, fuel efficiency in cars and determining travel times to destinations are all algebraic equations in some form. Without these, humans would be lost and late all the time. Algebra should be taught because children need it later to survive modern society.
Algebra is an essential course for all students.
The educational system generally fails in how they sell the idea of the importance of learning algebra. Learning algebra is not about learning to solve impractical math problems nor is it about becoming a mathematician. It is about training young minds to think critically. It helps to develop a flexible mind that's capable of resolving real-world problems.
Algebra Tools Are Useful
Algebra tools are what are useful. The idea that we need two years worth of these courses is ridiculous. The algebra course is erratic and inefficient. There would be a much lower drop-out rate in high school if we developed an algebra course that provided situations in which our students could apply their knowledge. | 677.169 | 1 |
Showing 1 to 30 of 40
Functional Business Systems
Some Information Systems are cross-functional
Example: A TPS can affect several different business areas: Accounting,
Human Resources, Production, etc.
Some Information Systems concentrate on one particular business
area (Ac
1
Functions, Limits and Dierentiation
1.1
Introduction
Calculus is the mathematical tool used to analyze changes in physical quantities.
It was developed in the 17th century to study four major classes of scientific
and mathematical problems of the time:
Handbook of Mathematical Functions
The Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables [1] was the
culmination of a quarter century of NBS work on core
mathematical tools. Evaluating commonly occurring
mathematical functi
Q. Suppose XX is a random variable which follows standard normal distribution
then how is KXKX (KK is constant) defined. Why does it follow a normal
distribution with mean 00 and variance K2K2.
A. For a random variable XX with finite first and second mome
Random Variable and Constants
CHM 621 Home Page
Statistics TOC
> Sums >
PDF for adding a constant. Define a new random
variable, y = x + a, where a is a constant. If f(x) is the pdf for the
independent random variable, what is the functional form of the
p
Problem Solving
Techniques &
Algorithm
(c) Khaeroni, S.Si, M.Si
General Problem-Solving Concepts
CHAPTER #1
OBJECTIVE
When you have finished this chapter, you should be able to:
1. Describe the difference between heuristic and algorithmic
solutions to pro
NAME LIYBARFE LOUIZA
COURSE: PROBLEM SOLVING
COURSE CODE: CSC 103
DATE: 9/II/2014
MAIN INSTRUCTOR: MR NFOR NELSON
1
Name three current problems in your life that could be solve through an algorithmic process. Explain
why this problem is algorithmic in nat
C Tutorial variables and constants
In this C programming language tutorial we take a look at variables and
constants.
Variables
If you declare a variable in C (later on we talk about how to do this), you ask
the operating system for a piece of memory. Thi
Practice: Topic Sentence and Supporting Details
I.
WRITING A TOPIC SENTENCE.
Write a topic sentence that exactly matches the details provided in each group.
1. Topic sentence:
_
_
a. When we brought a welcome to the neighborhood present, the family next d
Based on Developing Compositions Skills: Academic Writing and Grammar 3rd ed.
GEEN 1203 / Prof. Marylian Rivera
Following a writing process helps the writer to focus on one task at a
time.
We will be focusing on three main steps: (1) prewriting (2) draf
THESIS STATEMENT
Reference: pages 166-167
Thesis Statement
Is the main idea for the entire essay.
It gives the attitude, opinion, or idea that the writer is
going to develop.
Like a topic sentence, the thesis statement tells the
topic and the controlli
For a manufacturing company has a total monthly fixed costs of 100,000, variable
costs per units $10, income tax rate of 20% targeted net income 10,000. If sales in
units (quantities) increase fixed cost per unit.
*DECREASE
For a manufacturing company has | 677.169 | 1 |
Pre-Calculus: Trigonometry
Pre-Calculus: Trigonometry
Pre-Calculus: Trigonometry
University of California, Irvine
About this course: ThisIn this module, you will get an overview of the course and the foundations of trigonometry. We will also begin exploring angles and different systems for angle measure.
5 videos, 4 readings
Graded: Angles and their Measure
WEEK 2
The Unit Circle and Solving Right Triangles
In this module, we will explore circles and right triangles. We will see several special angles related to particular right triangles and we will learn how to find measurements of sides and angles in right triangles using trigonometric functions.
4 videos, 1 reading
Graded: Unit Circle and Solving Right Triangles
WEEK 3
Properties of Trigonometric Functions
In this module we will explore several properties of trigonometric functions and discover how to compute values of these functions given information about an angle or a unit circle point.
3 videos, 1 reading
Graded: Properties of Trigonometric Functions
WEEK 4
Inverse Trigonometric Functions
We will now explore the inverse trigonometric functions. These are useful to go backwards - we will seek to find the angles which produce a given value of a trigonometric function.
3 videos, 1 reading
Graded: Inverse Trigonometric Functions
WEEK 5
Basic Trigonometric Identities I
There are several useful trigonometric identities which allow us to simplify trigonometric expressions and find values for the trigonometric functions beyond the special angles. We will begin by exploring the sum and difference identities. Warning: Generally, ...
4 videos, 1 reading
Graded: Basic Trigonometric Identities I
WEEK 6
Basic Trigonometric Identities II
In this module, we continue our exploration of trigonometric function identities. We will begin by learning how to verify such identities. We will then talk about the double-angle and half-angle identities.
4 videos, 1 reading
Graded: Quiz: Basic Trigonometric Identities II
WEEK 7
Trigonometric Equations
In this module, we will focus on solving equations involving trigonometric functions. These are usually equations in which the variable appears inside of a trigonometric function and we must use a combination of algebra skills and trigonometry manipulation to ...
4 videos, 1 reading
Graded: Trigonometric Equations
WEEK 8
Law of Sines and Law of Cosines
The Law of Sines and the Law of Cosines give useful properties of the trigonometry functions that can help us solve for unknown angles and sides in oblique (non-right angle) triangles. We will focus on utilizing those laws in solving triangles, including those...
6 videos, 1 reading
Graded: Law of Sines and Law of Cosines
WEEK 9
Trigonometry Final Exam
We have completed the new content for the course. In this final module, you will review and practice the topics covered throughout the course. You will end by taking the comprehensive final exam.
Graded: Trigonometry7 out of 5 of 90 ratings
MR
Very good course. Lots of materials to study and the practice exercises are well designed and require good thinking. I learned the basics to continue studying calculus in a more advanced level. | 677.169 | 1 |
Schaum's Outline of Mathematics for Nurses by Larry J. Stephens
Book Description
"Schaum's Outline" gives you: practice problems with full explanations that reinforce knowledge; coverage of the most up-to-date developments in your course field; and, in-depth review of practices and applications. 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" means your problem solved.
Author Biography - Larry J. Stephens
Larry L. Stephens is Professor of Mathematics at the University of Nebraska at Omaha, and has been on the faculty for more than 25 years. Lana C. Stephens is a Nurse Practitioner for Alegent Health Care in Council Bluffs, Iowa. She was Program Director for the Medical Assistants program at Iowa Western Community College and a research coordinator at the University of Nebraska Medical Center. Eizo Nishiura is Associate Professor Emeritus of Mathematics at Queensborough Community College | 677.169 | 1 |
Analysis with Ultrasmall Numbers (Textbooks in Mathematics) download
Analysis with Ultrasmall Numbers presents an intuitive treatment of mathematics using ultrasmall numbers. With this modern approach to infinitesimals, proofs become simpler and more focused on the combinatorial heart of arguments, unlike traditional treatments that use epsilon-delta methods. Students can fully prove fundamental results, such as the Extreme Value Theorem, from the axioms immediately, without needing to master notions of supremum or compactness. The book is suitable for a calculus course at the undergraduate or high school level or for self-study with an emphasis on nonstandard methods. The first part of the text offers material for an elementary calculus course while the second part covers more advanced calculus topics. The text provides straightforward definitions of basic concepts, enabling students to form good intuition and actually prove things by themselves. It does not require any additional "black boxes" once the initial axioms have been presented. The text also includes numerous exercises throughout and at the end of each chapter. | 677.169 | 1 |
About this product
Description
Description
Based on the successful at a Glance approach, with integrated double page presentations explaining the mathematics required by undergraduate students of chemistry, set in context by detailed chemical examples, this book will be indispensable to all students of chemistry. By bringing the material together in this way the student is shown how to apply the maths and how it relates to familiar concepts in chemistry. By including problems (with answers) on each presentation, the student is encouraged to practice both the mathematical manipulations and the application to problems in chemistry. More detailed chemical problems at the end of each topic illustrate the range of chemistry to which the maths is relevant and help the student acquire sufficient confidence to apply it when necessary. | 677.169 | 1 |
Fundamentals of Transportation
Fundamentals of Transportation...
More requires a switch in thinking from simply solving given problems to formulating the problem mathematically before solving it, i.e. from straight-forward calculation often found in undergraduate Calculus to vaguer word problems more reflective of the real world Transportation to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Fundamentals of Transportation
Select this link to open drop down to add material Fundamentals of Transportation to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
This book helps learn geometry from an all-new angle! Now anyone with an interest in basic, practical geometry can master it - without formal training, unlimited time, or a genius IQ.In "Geometry Demystified", best-selling author Stan Gibilisco provides a fun, effective, and totally painless way to learn the fundamentals and general concepts of geometry. With "Geometry Demystified", you master the subject one simple step at a time - at your own speed. This unique self-teaching guide offers multiple-choice questions at the end of each chapter and section to pinpoint weaknesses, and a 100-question final exam to reinforce the entire book. Simple enough for beginners but challenging enough for advanced students, "Geometry Demystified" is your direct route to learning or brushing up on this essential math subject. Get ready to: learn all about points, lines, and angles; figure out perimeters, areas, and volumes; improve your spatial perception; envision warped space and hyperspace; and much more!
"synopsis" may belong to another edition of this title.
From the Back Cover:
LEARN GEOMETRY FROM AN ALL-NEW ANGLE!
Now anyone with an interest in basic, practical geometry can master it -- without formal training, unlimited time, or a genius IQ. In Geometry Demystified, best-selling author Stan Gibilisco provides a fun, effective, and totally painless way to learn the fundamentals and general concepts of geometry.
With Geometry Demystified you master the subject one simple step at a time -- at your own speed. This unique self-teaching guide offers multiple-choice questions at the end of each chapter and section to pinpoint weaknesses, and a 100-question final exam to reinforce the entire book.
Simple enough for beginners but challenging enough for advanced students, Geometry Demystified is your direct route to learning or brushing up on this essential math subject.
Get ready to: * Learn all about points, lines, and angles * Figure out perimeters, areas, and volumes * Improve your spatial perception * Envision warped space and hyperspace * And much more!
About the Author:
Stan Gibilisco is one of McGraw-Hill's most diverse and best-selling authors. His clear, friendly, easy-to-read writing style makes his electronics titles accessible to a wide audience and his background in mathematics and research make him an ideal handbook editor. He is the author of The TAB Encyclopedia of Electronics for Technicians and Hobbyists Teach Yourself Electricity and Electronics, and The Illustrated Dictionary of Electronics. Booklist named his book, The McGraw-Hill Encyclopedia of Personal Computing, one of the Best References of 1996. | 677.169 | 1 |
Circuit Training - Building Functions / Using Notation
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Be sure that you have an application to open this file type before downloading and/or purchasing.
0.21 MB | 2 pages
PRODUCT DESCRIPTION
Keep your Algebra I students engaged as they practice with writing simple functions and using function notation in the circuit format! Students will build and evaluate functions from verbal descriptions and from tables. Students start on an easy problem and then must hunt for their answer to advance in the circuit. 20 problems which build function fluency -- intended to be worked without a calculator. Suitable for Algebra 2, college math, and standardized test review. Inspired by Common Core State Standards (CCSS) test items | 677.169 | 1 |
Graphing Sines and Cosines By Dan Penner. My name is Daniel Penner. I teach at Enterprise High School in Redding California. My lesson is for a Trig-Pre-calculus.
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Presentation on theme: "Graphing Sines and Cosines By Dan Penner. My name is Daniel Penner. I teach at Enterprise High School in Redding California. My lesson is for a Trig-Pre-calculus."— Presentation transcript:
1
Graphing Sines and Cosines By Dan Penner
2
My name is Daniel Penner. I teach at Enterprise High School in Redding California. My lesson is for a Trig-Pre-calculus class. This class is predominantly a junior and senior level class. In my lesson the students will start to become familiar with the basic shape and properties of sine and cosine waves.
3
Graphing Sines and Cosines Students spent some time viewing applets getting familiar with the patterns of these basic trigonometry graphs.
4
Graphing Sines and Cosines In taking this class I was hoping to learn how to quickly create lessons that students can access from home. Specifically this would be a tool for me to become more engaging in my instruction, but also more organized. As a significant byproduct students that miss class would be able to catch up more easily. Expectations
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Graphing Sines and Cosines I actually learned a great deal in this class. The type of page we designed was actually quite easy. How to create links How to set targets How to find many interesting gifs. How much of a time vacuum this process is. Outcomes
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Graphing Sines and Cosines Overall I felt that the project was successful. Many students seem naturally drawn to anything that utilizes technology, so using the Internet as an instructional tool helps to make learning more engaging. I think that in the future to make this lesson work better I need to have students working in a lab environment and I also need to have many more lessons in a similar format so that they are used to this style of teaching. successes
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Graphing Sines and Cosines Graphing Sines and Cosines success ÙStudents experienced math in a new format. ÙStudents learned that the Internet is a valuable tool for learning mathematics. ÙStudents could use site to help study for their exam.
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Graphing Sines and Cosines Graphing Sines and Cosines Data The chart reflects each students growth in their understanding of the topics covered. Most students showed marked improvement. (Granted most had no previous experience with this topic.)
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Graphing Sines and Cosines Graphing Sines and Cosines Revisions ÙStudents did not have sufficient individual access to computers. ÙThe lesson took a significant amount of time. ÙI had to link to other sites. I would like my site to be self sufficient. ÙI plan to cover this lesson in the future in a lab. ÙI will be more patient. Sometimes good things take time. ÙI plan to learn how to impart Java applets into web pages. | 677.169 | 1 |
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Reviewer Comments:
This activity is set-up as an interactive problem solving worksheet (Excel). Students enter the intermediate steps to solving equations, and then the answer while receiving the feedback "correct", when matching the expected answer. The program is very particular about the numbers, variables, symbols, spaces and order of entries. A student entering "10 = x" would get the message "try again" even if the solution is "x = 10". This worksheet could be used as a handout to reinforce solving equations. It would not make a good learning tool, as it is difficult to match the answer exactly, | 677.169 | 1 |
Product Description:
Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject ... a highly welcome addition to the existing literature. --Zentralblatt MATH The interaction between number theory and algebraic geometry has been especially fruitful. In this volume, the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the one-dimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes, which will aid the reader who goes to the next level of this rich subject.
REVIEWS for An Invitation to Arithmetic | 677.169 | 1 |
Math Bridge
Three items are added to Geometry section.
Length (Pythagorean theorem)
Area (Triangle, rectangle and circle)
Geometry with Algebra (Triangle figured by three linear functions)
Description
In this Math App, there are no words to explain except at index. So let me introduce the concept of Math Bridge in English. You can try free sample version on web.
This App is not Math itself, but Math-related supplemental program. Math can only exist when you are doing it as if your thinking exists when you think.
1. Calculation
In this App, there are three levels and one challenge mode in following calculations.
i) Addition:
ii) Subtraction:
iii) Multiplication:
iv) Division:
After every five questions, quick review is set to check right and wrong. All four challenge modes for calculations have a five seconds timer to answer the question. They are difficult but concentration can enable us to get right answers.
5 Differentiation
In calculus, a branch of mathematics, the derivatives is a measure to know how a function changes as its input changes. (from wikipedia)
In this App, you can find points of extreme by a quadratic derivative in trinomials.
6 Integration
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. (from wikipedia)
In this App, you can learn the calculation of an area between quadratic function and linear function. At first, the cross-points of two functions are calculated by factoring and calculate the area by integration. Please touch screen to take steps.
An Orbit of thrown object as quadratic curve can be a bridge connected between the real world and math. Math is a tool as well as a world. The difference of the real world and a world is ambiguous, because we can't see the real world without a world, which is our own world. Math has a history over 3000 years as well as language, both of which are very powerful to explore the real world. Life may be a problem much more complex than math, but logic is useful for both of them. To handle logic or theory requires much energy in brain so that doing math is a good training for thinking about the world and life with your own logic. These can be reasons why we learn math. How about you?
Have fun!
ver 1.1
"Quadratic formula" and "Trinomial" are added in Equation.
In Quadratic formula, you can see the course from a general quadratic equation to the quadratic formula by 7 steps. The logic of four arithmetic operations and notion of equal is the basic skill.
In trinomial, you can solve cubic equations by factoring. Please note that plus and minus notations are needed to be cared for as well as calculating fraction. When f(a) = 0 in f(x)=0, it means that the "a" can be factored. | 677.169 | 1 |
Literal Equations Guided Cornell Notes
1,925 Downloads
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PRODUCT DESCRIPTION
These guided notes start with 3 equations students can solve for bellwork. The notes start with a short video (the link is in the footer) that explains to the students what a literal equation is. The students are able to fill in the blanks on their own. It gives examples that of what it means when an equation is solved for a variable and how to solve for a different variable | 677.169 | 1 |
Econometrics and Mathematical Economics
Course Introduction
The preliminary year is designed for students with high academic ability but lacking a sufficient background in economics, econometrics, statistics or mathematics. Its purpose is to enable students to develop their skills to the point where they are eligible for progression to the Master's in econometrics and mathematical economics | 677.169 | 1 |
...
Show More uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical | 677.169 | 1 |
Nghi Nguyen's Discussions
Factoring trinomials is a main subject of high school math. The AC Method has been so far the most popular one to factor trinomials. However, this method can be considerably improved if we apply the…Continue
So far, the factoring AC Method has been the most popular to solve quadratic equations in standard form that can be factored. However, this method can be considerably improved if we apply the Rule of…Continue
So far, the existing Factoring AC Method (YouTube)has been the most popular method to solve quadratic equations. However, it can be considerably improved if we add into its solving process the Rule…Continue
There are some differences between the US basic solving process for algebra equations and inequalities and the world wide one. We teach students to solve equations by observing the principle of…Continue
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"This article explains the advantages of the transposing method in solving algebraic equations and inequalities. It proposes to begin teaching students this method at pre-algebra and algebra I classes."
Factoring trinomials is a main subject of high school math. The AC Method has been so far the most popular one to factor trinomials. However, this method can be considerably improved if we apply the Rule of Sign into its solving approach.The improved method, called by the author of this article,"The New AC Method for factoring trinomials" proceeds faster and avoids the lengthy factoring by grouping.The strong points of this new method are: simple, fast, no guessing, systematic, no factoring by…See More
There are some differences between the US basic solving process for algebra equations and inequalities and the world wide one. We teach students to solve equations by observing the principle of balancing the two sides of an equation. The common saying is likely:"Do the same thing to the right side what you did to the left side". On the contrary, the world wide solving process performs by transposing terms from one side to another, while always keeping the equation balanced. The likely saying…See More
Conversion of a quadratic function from one form to another (standard form, intercept form, vertex form) is extra-learning that provides students with a better understanding of the various aspects of the quadratic functions and quadratic equations.This article explains general methods to convert quadratic equations from one form to another.See More
There are so far 8 common methods to solve quadratic equations. When the given quadratic equation can't be factored, the quadratic formula is the obvious choice to solve it. There is a new quadratic formula in intercept form that is simpler and easier to remember.See More
This article generally explains the concept and methods for solving trig equations. This concept has been popular in Europe and it provides a needed key to solve all trig equations and inequalities. Solving trig equations is a tricky work. Answers should be always carefully checked.This article will help students to excel in trig classes and to be ready for further college studies. This article was updated on Nov 05, 2014.See More
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Ihr Matheaufgabe Shop
Take the mystery out of basic math with the latest edition of BarCharts´ best-selling Math Review QuickStudy guide. With updated content and an additional 2 pages of information, Math Review includes hard-to-remember formulas and properties, along with numerous examples and illustrations to improve understanding. This comprehensive math guide will assist you way beyond your high school and college years.
Candy School Chapter Book Want to teach core concepts with a fun story about the thing kids love most? Math Candy is the first in a series of beginning level chapter books featuring two boys, Danny and Sam. They have a lot to learn but when they visit Mr. Candor´s Candy Store, math concepts suddenly seem fun and tasty. In this book, educator Katrina Streza focuses on major math concepts such as addition, units of measurement, estimation, money, sorting, multiplication and comparison. Kids see the real-life application of math and enjoy the fun and engaging stories. Math Candy features 7 short chapters with black and white illustrations and is a good bridge for beginning readers or as a read-aloud. Make sure you check out all the Candy School Chapter Books: 1. Math Candy 2. Grammar Candy 3. Map Candy 4. Science Candy 5. History Candy
Currently in its eighteenth printing in Japan, this best-selling novel is available in English at last. Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks. Math Girls has something for everyone, from advanced high school students to math majors and educators.
Having trouble figuring out the total cost of a contracting job or calculating the square yardage of a living room floor? Our comprehensive 6-page guide will be of incomparable assistance. Each page is full of charts, diagrams, formulas and equations covering elements of construction where math is neededall arranged in our attractive, easy-to-use format. It is the perfect resource for the professional contractor or the do-it-yourselfer. Topics include: Basic Arithmetic Measurement Basics Volume Surface Measurement Roofs Building Materials Excavations Concrete And more! | 677.169 | 1 |
This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given. Proofs using induction, recurrence relations and proofs by contradiction are covered. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are used. Basic topics in probability and statistics are presented. The student is introduced to elements of saving and investing that are of life-long practical use. These include savings and checking accounts, certificates of deposit, student loans, credit cards, mortgages, buying and selling bonds, and buying and selling stocks. | 677.169 | 1 |
Technology & Textbooks
Technology Information
Calculators - Students in MATH 101, MATH 105, MATH 150, MATH 151, and MATH 201 are expected to have a scientific calculator. Scientific calculators can be purchased for around $10 to $15 at discount retail stores. Your professor may not allow graphing or programmable calculators to be used on graded assessments. Please view your course syllabus for details with regard to your class.
Computers - Students are not required to own a computer. There are many computer labs on campus available for student use. If you are able to purchase a personal computer, either a PC or Mac will serve you well.
Mathematics majors who have completed or are enrolled in MATH 202 can print, fill out, and return the Bancroft 173 Access Form (pdf - 55KB) in order to obtain the access code for the Mathematics Department's student computer lab.
Software - Students who enroll in MATH 202 must enroll in MAED 200 during the same semester. In MAED 200, students learn to use Mathematica as a tool for exploring and visualizing mathematical concepts. Mathematica is installed on computers in the student computer labs. Mathematica workbooks may be used in lower-level math courses as well. Workbooks created by professors for students to use outside of class can be viewed on the free Mathematica Player.
Winthrop's site license for Mathematica now allows students to download a copy of Mathematica onto their own computers for free, as long as those computers are on campus for the download process. To download Mathematica: | 677.169 | 1 |
Concepts covered in this lecture : Complex numbers. Equations to curves in the plane in terms of z and z*. The Riemann sphere and stereographic projection. Analytic functions of z and the Cauchy-Riemann conditions. The real and imaginary parts of an analytic function.
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Summary and Info
When information is transmitted, errors are likely to occur. This problem has become increasingly important as tremendous amounts of information are transferred electronically every day. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected.The traditional tools of coding theory have come from combinatorics and group theory. Since the work of Goppa in the late 1970s, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed-Solomon codes as coming from evaluating functions associated to divisors on the projective line, one can see how to define new codes based on other divisors or on other algebraic curves. For instance, using modular curves over finite fields, Tsfasman, Vladut, and Zink showed that one can define a sequence of codes with asymptotically better parameters than any previously known codes.This book is based on a series of lectures the author gave as part of the IAS/Park City Mathematics Institute (Utah) program on arithmetic algebraic geometry. Here, the reader is introduced to the exciting field of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. The construction of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink result mentioned above is discussed.
More About the Author
Judy Leavitt Walker is an American mathematician. She is the Aaron Douglas Professor of Mathematics at the University of Nebraska–Lincoln, where she chaired the mathematics department from 2012 through 2016 and currently serves as the Interim Associate Vice Chancellor for Faculty Affairs. | 677.169 | 1 |
1Basic Concepts and NotationLinear algebra provides a way of compactly representing and operating on sets of linearequations. For example, consider the following system of equations:4x1-5x2=-13-2x1+3x2=9.This is two equations and two variables, so as you know from high school algebra, youcan ±nd a unique solution forx1andx2(unless the equations are somehow degenerate, forexample if the second equation is simply a multiple of the ±rst, but in the case above thereis in fact a unique solution). In matrix notation, we can write the system more compactlyasAx=bwithA=±4-5-23²,b=±-139².As we will see shortly, there are many advantages (including the obvious space savings)to analyzing linear equations in this form.1.1Basic NotationWe use the following notation:•ByA∈Rm×nwe denote a matrix withmrows andncolumns, where the entries ofAare real numbers.•Byx∈Rn, we denote a vector withnentries. By convention, ann-dimensional vectoris often thought of as a matrix withnrows and 1 column, known as acolumn vector.If we want to explicitly represent arow vector— a matrix with 1 row andncolumns— we typically writexT(herexTdenotes the transpose ofx, which we will de±neshortly).•Theith element of a vectorxis denotedxi:x=x1x2...xn.2
•We use the notationaij(orAij,Ai,j, etc) to denote the entry ofAin theith row andjth column:A=a11a12···a1na21a22a2n............am1am2amn.•We denote thejth column ofAbyajorA:,j:A=|||a1a2an|||.•We denote theith row ofAaTiorAi,::A=—aT1——aT2—...—aTm—.•Note that these deFnitions are ambiguous (for example, thea1andaT1in the previoustwo deFnitions arenotthe same vector). Usually the meaning of the notation shouldbe obvious from its use.
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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 1. Computer Arithmetic The purpose of computing is insight, not numbers. R.W. Hamming, [ 23 ] The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathematical quantities, including functions, integrals, solu- tions of algebraic equations, solutions of differential equations (both ordinary and partial), solutions of minimization problems, and so on. The objects of interest typically (but not exclusively) arise in applications, which seek not only their qualitative properties, but also quantitative numerical data. The goal of this course of lectures is to introduce some of the most important and basic numerical algorithms that are used in practical computations. Beyond merely learning the basic techniques, it is crucial that an informed practitioner develop a thorough understanding of how the algorithms are constructed, why they work, and what their limitations are. In any applied numerical computation, there are four key sources of error: ( i ) Inexactness of the mathematical model for the underlying physical phenomenon. ( ii ) Errors in measurements of parameters entering the model. ( iii ) Round-off errors in computer arithmetic. ( iv ) Approximations used to solve the full mathematical system. Of these, ( i ) is the domain of mathematical modeling, and will not concern us here. Neither will ( ii ), which is the domain of the experimentalists. ( iii ) arises due to the finite numerical precision imposed by the computer. ( iv ) is the true domain of numerical analysis , and refers to the fact that most systems of equations are too complicated to solve explicitly, or, even in cases when an analytic solution formula is known, directly obtaining the precise numerical values may be difficult. There are two principal ways of quantifying computational errors. Definition 1.1. Let x be a real number and let x ? be an approximation. The absolute error in the approximation x ? x is defined as | x ?- x | . The relative error is defined as the ratio of the absolute error to the size of x , i.e., | x ?- x | | x | , which assumes x 6 = 0; otherwise relative error is not defined. 3/15/06 1 c 2006 Peter J. Olver For example, 1000001 is an approximation to 1000000 with an absolute error of 1 and a relative error of 10- 6 , while 2 is an approximation to 1 with an absolute error of 1 and a relative error of 1. Typically, relative error is more intuitive and the preferred determiner of the size of the error. The present convention is that errors are always 0, and are = 0 if and only if the approximation is exact. We will say that an approximation x ? has k significant decimal digits if its relative error is < 5 10- k- 1 . This means that the first k digits of x ?...
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This note was uploaded on 02/10/2012 for the course MATH 5485 taught by Professor Olver during the Fall '09 term at University of Central Florida. | 677.169 | 1 |
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13. The passage as a whole is best characterized as
(A) a case study of an unusual type of
bioluminescence
(B) a survey of popular misconceptions about
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(C) a discussion of the evolutionary origins of
bioluminescence
(D) an overview of the various functions of
bioluminescence in sea creatures
(E) an examination of luminescent species that
use camouflage
-6-5-
19. In line 25, the author most likely mentions
that copepods "do their best to be transparent"
in order to
14. In line 2, "given" most nearly means
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(E)
inclined
transported
devoted
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...Calculus
is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integralcalculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamentaltheorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-definedlimit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot.
Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on.
BRANCHES OF CALCULUSCalculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do...
...The Simple Ledger: Ledger Accounts
You will now learn the system used to maintain an up to date financial position.
They use an account and ledger.
Account:
Page specially used to record financial changes
There is one account for each different item affecting the financial position. (Bank, equipment, automobile…)
Ledger:
All the accounts together are called the ledger
Group or file of accounts
Used to record business transaction and keep track of the balances in each specific account
If you wanted to know how much cash the business has to write a cheque, you would look in the ledger (Cash Account)
If you wanted to know how much the business owes on a bank loan, you would look in the ledger (Bank Loan Account)
A ledger can be prepared in different ways (cards, looseleaf ledger, computer system)
T-account:
Simple type of account. (A quick and easy way to track what is happening in each account)
Accounting form we use to keep track of the specific balance in an account
Shaped like a "T"
The formal account, the one actually used in business, will be introduced at a later time.
Important Features of Ledger Accounts
1. Each individual balance sheet item is given its own specially divided page with the name of item at the top (for now think of each "T" as a page)
Each of these pages is called an account
You must learn to call each one by name. i.e., cash account, bank loan account, and so on.
2. The dollar figure for each item is...
...1. Physical Properties of Water and Ice
1. Molecular Weight:
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Water, Molar mass
Triple Point
The temperature and pressure at which solid, liquid, and gaseous water coexist in equilibrium is called the triple point of water. This point is used to define the units of temperature (the kelvin, the SI unit of thermodynamic temperature and, indirectly, the degree Celsius and even the degree Fahrenheit).
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Density of water and ice
Density of ice and water as a function of temperature
Density of liquid water
Temp (°C)
Density (kg/m3)[20][21]
+100
958.4
+80
971.8
+60
983.2
+40
992.2
+30
995.6502
+25
997.0479
+22
997.7735
+20
998.2071
+15
999.1026
+10
999.7026
+4
999.9720
0
999.8395
−10
998.117
−20
993.547
−30
983.854
The values below 0 °C refer to supercooled water.
The density of water is approximately one gram per cubic centimeter. It is dependent on its temperature, but the relation is not...
...children in the age group 6-14 years a fundamental right? Your help will go a long way in providing educational opportunities for children who would otherwise be left behind.
Everyone has the right to go to public school. This right does not matter what your culture is, your religious affiliation, your abilities, your physical disabilities or learning disabilities. We have a country (one of the only a few countries) that services students with emotional, learning, physical disabilities as well as to cater to Gifted and Talented students.
It should be a fundamental right because everyone is entitled to an education to make their lives better. Everyone should be given the opportunity to improve themselves and their lives.
Having education as a fund mental right not only places the United States is a high status of other countries, it allows our citizens to make educated decisions on candidates, purchases (home/vehicles) and the ability to make budgets.
If our country did not have well established public schools our citizens would not be educated and our country would not be a world power with a population that was not educated.
If we think of education as a process by which individuals acquire skills and knowledge that help them to achieve their potential, then the right to be educated is as important to to right to freedom and freedom of speech. However, if we want to include it as a... | 677.169 | 1 |
Accessing the M in STEM
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
19.74 MB | 204 pages
PRODUCT DESCRIPTION
This book is a wrap up of information gathered over the first two years of a three year NSF Grant in Bio-Technology and 38 years of teaching mathematics. Before you read the book, I would like to state that the book contains what I perceive as reasons students dislike math, struggle with math, struggle with math in science class, changes I see needed to improve student attitude toward MATH and ways to improve the student's math skills. This book is meant to stir your emotions and thoughts about mathematics, math education, the future of mathematics, and give you tools and problems to help improve math education. Keep in mind, it is your option to agree, disagree, like or dislike what I say or do in this book | 677.169 | 1 |
Maths Quest by Lyn Elms
Book Description
Maths Quest Maths A Year 11 for QLD 2E is part of a complete Maths package which includes Teacher Editions, Fully Worked Solution Manuals, and now also supported with eBookPLUS and eGuidePLUS! The second editions of this highly successful maths series have been updated to meet the requirements of the revision of Maths Year 11 syllabus for implementation from 2009. Textbook Features * New technology - this new edition contains explanations and screen shots for the CASIO fx-9860G AU and TI-Nspire CAS model of calculators. It also includes the equivalent screen shots for the TI-89 in an appendix at the back of the book recognising that many schools will continue to use this model * Extra modeling and problem solving style questions * Additional suggested learning experiences to support each topic * Icons to indicate supporting material on the accompanying JacarandaPLUS website. (See the Weblinks tab Maths Quest Maths A Year 11 for QLD 2E eBookPLUS is an electronic version of the textbook and a complementary set of targeted digital resources. These flexible and engaging ICT activities are available to you online at the JacarandaPLUS website ( | 677.169 | 1 |
Students often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and J. Matt Switzer's students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood "if its mental representation is part of a network of representations" and that the "degree of understanding is determined by the number and strength of the connections" (p. 67). Therefore, Switzer developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities. The design of the activity also aligns with and supports the Common Core Standards for Mathematical Practice. These standards describe mathematically proficient students as being able to identify important quantities in a practical situation and map mathematical relationships using such tools as diagrams, two-way tables, graphs, flow charts, and formulas. In this article, Switzer describes the fragile understanding and lack of connections his students had when graphing functions and inequalities. He then provides an overview of how he drew on the trichotomy axiom and evaluation of given expressions to connect students' understanding of functions, inequalities, and their corresponding graphs. Next, he discusses how students incorporated mathematical practices to make sense of the graphing process and the relationships between models. | 677.169 | 1 |
Product Description:
We learn by doing. We learn mathematics by doing problems. This is the third volume of Problems in Mathematical Analysis. The topic here is integration for real functions of one real variable. The first chapter is devoted to the Riemann and the Riemann-Stieltjes integrals. Chapter 2 deals with Lebesgue measure and integration. The authors include some famous, and some not so famous, integral inequalities related to Riemann integration. Many of the problems for Lebesgue integration concern convergence theorems and the interchange of limits and integrals. The book closes with a section on Fourier series, with a concentration on Fourier coefficients of functions from particular classes and on basic theorems for convergence of Fourier series. The book is primarily geared toward students in analysis, as a study aid, for problem-solving seminars, or for tutorials. It is also an excellent resource for instructors who wish to incorporate problems into their lectures. Solutions for the problems are provided in the book.
REVIEWS for Problems in Mathematical Analysis | 677.169 | 1 |
College Math 2 Documents
Showing 1 to 30 of 42
Project Part 2
Shake, Rattle and Roll
J.R.Teeter
How Fast Is A Speeding Bullet?
The velocity of a bullet is given in feet per
second (fps) in the United States. The
0.22 rim fire cartridge, which has a very small
powder charge, sends its bullets
on their
Project Part 1:
Design a Home Theatre
Line of Sight
(Vertical Subtended Angle)
The Following pages will give the line of sight for the fives rows
These slides will show Angles of Elevation and depression for the Vertical LoS
1ft D
I
A
m
2 ft high
1ft
2 fe
Project
Project Part 1: Design a Home Theater
Purpose
You will apply trigonometry design a home theater. In this project you will use a line of sight in a theater
situation, sound waves, and the amount of space required for various sized groups.
Learning
Project Part 2: Shake, Rattle, and Roll
Purpose
Using an event, the March 2011 Japanese earthquake, you will use mathematics to provide a way to
represent and understand the size of the quake and forces involved. The mathematics provides a way to
represen
Project Part 3: Chaos and Fractals
Purpose
Out of chaos come unexpected events that are interrelated. Fractals are a form of chaos. The
mathematical formula for fractals was created prior to our ability to see the results and was created by a
blind mathem
Project Part 1 Design a Home Theater Done
A. You will have a screen that is 160 Wide x 120 in height
B. You will have 5 tiered rows of seating able to seat 25 people\
C. Every row will have line of sight view
D. Research the concept of subtended angles of
Project Part 3: Chaos and Fractals
FRACTALS
Mathematics, a geometrical or physical structure having an uneven or fragmented shape at all
scales of measurement between a greatest and smallest scale such that certain mathematical or
physical properties of t
Project Part 2: Shake, Rattle, and Roll
What is faster than a speeding bullet?
The velocity of a bullet is given in feet per second (fps). The 0.22 rim fire cartridge, which has a
very small powder charge, sends its bullets on their way at between 370 m/s
Project Part 1: Design a Home Theater
Requirements
5 tiered rows to seat 25 people
Every row must have a line of sight view
Seating distance from the screen measured
Measurements
Screen = 160inches long by 120inches high
Seating = 36inches long by 12inche
David Lariviere
Chris Arthur
ClassProjectMATH
DesignaHomeTheater
In this I will show what you have to do and equipment that you have to put into it. With
our home theater we started the first row seventy two inches from the screen. Then every row
after th | 677.169 | 1 |
reviews important concepts in calculus and provides solved problems and step-by-step solutions. Once students have mastered the basic approaches to solving calculus word problems, they will confidently apply these new mathematical principles to even the most challenging advanced problems. Each chapter features an introduction to a problem type, definitions, related theorems, and formulas. Topics range from vital pre-calculus review to traditional calculus first-course content. Sample problems with solutions and a 50-problem chapter are ideal for self-testing. Fully explained examples with step-by-step | 677.169 | 1 |
Summary and Info
From the Preface :Teach Yourself Trigonometry has been substantially revised and rewritten to take account of modern needs and recent developments in the subject.It is anticipated that every reader will have access to a scientific calculator which has sines, cosines and tangents, and their inverses. It is also important that the calculator has a memory, so that intermediate results can be stored accurately. No support has been given about how to use the calculator, except in the most general terms. Calculators vary considerably in the keystrokes which they use, and what is appropriate for one calculator may be inappropriate for another.There are many worked examples in the book, with complete, detailed answers to all the questions. At the end of each worked example, you will find the symbol I to indicate that the example has been completed, and what follows is text.Contents========ContentsPreface01 - Historical Background Introduction What Is Trigonometry The Origins of Trigonometry02 - The Tangent Introduction The Idea of the Tangent Ratio A Definition of Tangent Values of the Tangent Notation for angles and Sides Using Tangents Opposite and adjacent Sides03 - Sine and Cosine Introduction Definition of Sine and Cosine Using the Sine and Cosine Trigonometric Ratios of 45°, 30° and 60° Using the Calculator Accurately Slope and Gradient Projections Multistage Problems04 - In Three Dimensions Introduction Pyramid Problems Box Problems Wedge Problems05 - Angles of Any Magnitude Introduction Sine and Cosine for Any Angle Graphs of Sine and Cosine Functions The Tangent of any Angle Graph of the Tangent Function Sine, Cosine and Tangent06 - Solving Simple Equations Introduction Solving Equations Involving Sines Solving Equations Involving Cosines Solving Equations Involving Tangents07 - The Sine and Cosine Formulae Notation Area of a Triangle The Sine Formula for a Triangle The Ambiguous Case The Cosine Formula for a Triangle Introduction to Surveying Finding the Height of a Distant Object Distance of an Inaccessible Object Distance Between Two Inaccessible but Visible Objects Triangulation08 - Radians Introduction Radians Length of a Circular Arc Converting from Radians to Degrees Area of a Circular Sector09 - Relations Between the Ratios Introduction Secant, Cosecant and Cotangent10 - Ratios of Compound Angles Compound Angles Formulae for Sin(A + 8) and Sin(A - 8) Formulae for Cos(A + 8) and Cos(A - 8) Formulae for Tan(A + 8) and Tan(A - 8) Worked Examples Multiple angle Formulae Identities More Trigonometric Equations11 - The Form A Sin(X) + B Cos(X) Introduction The Form Y = A Sin(X) + B Cos(X) Using the Alternative Form12 - The Factor Formulae The First Set of Factor Formulae The Second Set of Factor Formulae13 - Circles Related to a Triangle The Circumcircle The Incircle The Ecircles Heron's Formula: The area of a Triangle14 - General Solution of Equations The Equation Sin θ = Sin α The Equation Cos θ = Cos α The Equation Tan θ = Tan αSummary of ResultsGlossarySummary of Trigonomeb1c FormulaeAnswersIndex
More About the Author
P. J. Abbott (born May 28, 1964, in Bloomington, Indiana) is an American race car driver. In 2004, he drove in two races in the Infiniti Pro Series for Michael Crawford Motorsports. | 677.169 | 1 |
Math 13X00 Arithmetic Skills Quiz Study GuideThere will be three Arithmetic Skills Quizzes given during each semester of the 13X00 courses. Each quiz has five problems. This is a selection of the type of problems to expect. No calculators can be used. Problems that are copied incorrectly will receive NO credit. All fraction answers should be simplified. All improper fraction answers should
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This note was uploaded on 01/18/2012 for the course MATH 152 taught by Professor staff during the Fall '09 term at Purdue University-West Lafayette. | 677.169 | 1 |
Math extended essays
We provide excellent essay writing service 24/7 Enjoy proficient essay writing and custom writing services provided by professional academic writers. Home: Fun & Learning About Bridges: Bridge Statistics: Bridges Project - Rice University - Awesome searchable database of bridge information. Teacher Login / Registration : Teachers: If your school or district has purchased print student editions, register now to access the full online version of the book. Free narrative papers, essays, and research papers These results are sorted by most relevant first (ranked search) You may also sort these by color rating or.
Welcome to Pearson SuccessNet! We have made some important updates to Pearson SuccessNet! Please see the Feature Summary for more details As always, please. ClassZone Book Finder Follow these simple steps to find online resources for your book. Eureka Math is America's #1 Math Curriculum Eureka Math, only three years old, is now the most widely used math curriculum in the United States, according to a new.
Eureka Math is America's #1 Math Curriculum Eureka Math, only three years old, is now the most widely used math curriculum in the United States, according to a new. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. More generally, the Fibonacci numbers can be extended to a real number via.
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Learn why the Common Core is important for your child What parents should know; Myths vs facts. No Fear Shakespeare No Fear Shakespeare puts Shakespeare's language side-by-side with a facing-page translation into modern English—the kind of English people. | 677.169 | 1 |
Math Exchange
Recent News
Search
Math Exchange
The Exchange is Open!
The Department of Mathematics and Applied Mathematics is beginning a new program called the Math Exchange, designed to deliver innovative Pre-Calculus (Math 151) instruction. The Exchange employs up-to-date software developed by ALEKS, born from a National Science Foundation grant on the study of knowledge spaces, an area within artificial intelligence.
This cutting-edge software breaks down the myriad of skills and techniques within a topic and scaffolds them according to their various relationships with one another. Then it simultaneously challenges and assesses students along these lines, helping students to master material by identifying and repairing weaknesses along the way. ALEKS continues to reinforce methods as well, in order to build long-term mastery.
Due to the success of its use at nearly 3,000 universities and colleges, VCU has invested in developing it here. With 2,000—2,500 students taking Math 151 each year as a prerequisite to studies in disciplines from STEM to Business, the Exchange has the potential to affect roughly one third of all undergraduates, impacting almost every major on campus.
With some lucky timing, sufficient space opened up on Grace St. just as the Department of Mathematics was in the midst of planning the Exchange. Campus police moved into their new environs between the Medical and Monroe Park campuses this past Fall, and reconstruction of 940 W. Grace St. (between N. Harrison St. and the Grace St. Theater) began immediately.
The new interior includes a variety of group-oriented study and seminar spaces, through which instructors and teaching assistants can move to assist individualized learning. Plans are also in the works to incorporate both recorded and live video for supplemental learning support.
To better develop, improve, and master our implementation of the Exchange, mathematics education expert Dr. Aimee Ellington will study the program over the course of three semesters. Half of our sections will be traditional and half will use ALEKS. Feedback from students, along with their results in Pre-Calculus and subsequent courses, will help to inform our direction as we grow and include more components. | 677.169 | 1 |
2Introduction No application of numerical methods in science and engineering is more exciting than the solution of partial differential equations (PDE) and systems of partial differential equations. Development of the computational power to solve the PDE that constitute the mathematical models of global climate change, blood flow near artificial valves, drug metabolism in the liver, evolution of severe storms, production of oil reservoirs, performance of financial markets, or atmospheric reentry of space vehicles, has given scientists and engineers an indispensable new tool that has transformed modern technology. While we have found Excel and Visual Basic to be a powerful combination for many numerical tasks, it is not an adequate tool for solving partial differential equations of interest to scientists and engineering. Comparing the numerical solution of an ordinary differential to the numerical solution of a partial differential equation is like comparing a straight line to a square, a cube, or even a hypercube. The addition of new dimensions to the equations greatly increases the demand for CPU speed, storage capacity, and the ability to represent the solution in an accessible manner. Consequently, this chapter will focus on providing only an introduction to the fundamental techniques used in the solution of PDE. The techniques we will study are quite simple, but they will serve to present the concepts and terminology essential for the study of more advanced techniques or the use of more powerful software packages. Excel with VB is more than adequate to introduce us to this central topic of modern science and engineering. For simplicity we will mostly limit our study to second order PDE. Extension of these techniques to first order or to higher order PDE is surprisingly straightforward, so the student should not be concerned about the adequacy of this introduction to prepare them for using even the most sophisticated simulations. The examples we will use are examples of linear PDE, but these examples will prepare the student for extension of the approach to nonlinear equations. The method of lines in particular is well suited for application to nonlinear PDE.
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This note was uploaded on 01/26/2011 for the course CH E 2112 taught by Professor Dr.harwell during the Spring '10 term at The University of Oklahoma. | 677.169 | 1 |
Welcome to My Parent Page!
My algebra and pre-algebra pages on this web site contain several sites to provide help and practice on concepts. There are websites on the math and prealgebra web pages under resources that also provide explanations of concepts in our textbooks. IXL can be used to practice concepts. Please refer to the parent letter for aides that are provided to assist students. Explore my web site to see what is available.
Algebra Class News
Students are working on chapter 8 which covers slope and equations of lines. We are working on coding in Basic.
Algebra 2 Class News
Students are currently working on chapter 6 which covers operations on matrices.
Pre-algebra Class News
Students are currently working on chapter 6 which covers linear geometry. We are also coding in BASIC.
Literature Class News
Students are currently studying their lines for the play. Practice is during literature class. The students are also working on the mythology unit. | 677.169 | 1 |
Did I understand correctly that
the Eclectic Education Series includes curriculum for K-12 grades all
together? Yes,
that's correct, it's a one-time purchase. The Ray's Arithmetic CD includes books for all 12 grades, and
all of the CD's provided in the Eclectic Education Series start with basic materials for the beginning
grades and work their way up, with the exception of the Science CD, which starts at a more advanced
level for obvious reasons.
Can the Eclectic Education Series be used with the
Robinson Curriculum? Yes, the combined use of the
EES and RC is a very effective formula, they work very well together. For instance the RC needs a math
program to be used with it, and the Ray's Arithmetic supplies that need. Also while the RC supplies
a great many wonderful autobiographies and close up looks at history from people who were leaders of
their time, it lacks books to provide a broader perspective and overview, which is a role the
Thalheimer's Histories in the EES fill admirably. One of the wonderful things about homeschooling is that children learn
to love reading. The RC and EES both provide wonderful books for these homeschooled children who
are eager to learn.
I have an old version of the EES, how can I
upgrade? The current version of the EES is 2.1.0, if you have a previous version you
may wish to upgrade. Upgrades cost $10 per disk, E-mail us with a request for an upgrade along with your
name and address, we'll send you a bill via paypal upon verifying that you are a past customer, and when
you have payed, we will send you the upgraded CD.
Do your math and science CD's include answer keys for
the problems/questions? Yes, the math CD provides problems along with the answer key's; the
Science CD focuses on teaching the subjects, it does have some example problems and there are answers for
these, but in theNature Study
book for instance most of the 'problems' are more essay
style questions to show what you have learned, and the answer key is the book itself.
I want to order, but I don't have a paypal account.
How do I order? You don't have to have a paypal
account to buy the CD, if you click on the paypal button you will be taken to a page with two choices on
it, to log into paypal on the right or to click "Continue" to use a credit or debit card on the left.
Paypal then will simply process your credit card like any other system. If you would rather order over
the phone, you can reach us at 1-517-304-4844. If you would like to mail in your order, make
out a check or money order to Aaron Jagt, and mail it to 3321 Sesame Dr. Howell MI
48843.
Does the Science and History have ANY evolution
content at all? No, the books don't have any
evolution in them. Rather, the books are from a Christian perspective- for instance, Thalheimer's
History books are special because of the fact that he wrote them from a firmly Christian perspective,
which allows him to talk about events in history which most historians skip over because the most
important record from that period is the Bible.
What is the cost is to print each book allowing for
paper and printer ink? That depends on the printer you use, a good duplex printer will allow you to
print for about a penny a page. I would recommend the Brother HL-5250DN, which costs about $250 dollars
brand new.
Is it possible to see a sample of the books on
CD? Certainly, there are a few pages availablehere: or if you e-mail us
at we'll be happy to send
you some sample pages.
My son is bored with math and doesn't focus well,
will he be able to use Ray's? Ray's Arithmetic can be helpful to children bored with traditional math,
due to Ray's interesting use of word problems, which help children apply math to life.
Are your Math CD's available forApple MAC
Computers? The books on the CD's are PDF files and they will work on any
computer; Mac, Vista, XP,Linux, you name it.
The Ray's books that I have found have the answers
next to the problems, are they like that on the CD? We spent quite a bit of time a couple of years ago removing the answers from next to
the problems. We still provide the books with the answers there as well, but we refer to these as
"Teachers Editions". There are also Answer Key's supplied for the books which spend time
explaining the problems
"
I am 16 and a happy graduate of the EES curriculum-- I absolutely LOVE it! It is
wonderful, a true God-send. When I went to take the GED and the SAT upon my graduation,
my scores were superb and well above average. The curriculum gave me a firm foundation
and helped me to clearly and realistically grasp the concepts of true education.
Nothing was lacking and I give it a full five stars! Thank you very much for all
you have done to bring back the beautiful texts and materials which were so unjustly
done away with." ~
Erica | 677.169 | 1 |
About this product
Description
Description
Why do 5000 girls a year t get credit for AP Calculus? How do our mindsets affect our learning? Can we change our own brains, get smarter, or improve our willpower? What happens in your brain when you concentrate on learning? What is the major factor th | 677.169 | 1 |
Three Skills For Algebra - Alan Selby
This book describes three skills key to the algebraic way of writing and thinking, offering a first image of mathematics beyond arithmetic. It also describes the first elements of logic or rule-based reason, needed in all disciplines for writing or a
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TI-83 / TI-84 Games and Programs - Bill Paetzke
Bill Paetzke offers TI-83/TI-84 Plus programs to help students and teachers be more efficient in using their calculators. The programs are organized under these categories: Algebra, Finance, Geometry, Physics, Programs, Statistics, Trigonometry, and Tutorials.
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Ti 84 Plus Calculator
Instructional videos include using the parametric function to construct a pentagram, hypothesis testing, sketching polynomial functions, finding critical points of a function, and using the TVM (Time Value of Money) Solver method. The site also offers
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TI89Prog - Mickaël Nicotera
A French database of programs for the TI-89, TI-92+ and V200 graphing calculators, organized into categories such as math, games, ebooks, pictures, and cours prépa. With an online discussion forum, links to other resources, and more. Available
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Tickling the Mind - Ivars Peterson (MathTrek)
... Different math teachers may use different styles and methods, and they may favor different topics. However, they all share the goal of building an appreciation of both the usefulness and the intellectual wonder of mathematics. Much of the current
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Tiesforteachers
A company in the UK selling neckties specifically with the school curriculum in mind. The growing collection includes ties for teachers of maths, science, English, music, ICT and other general-purpose school subjects.
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Tilton's Algebra - Ken Tilton
Guided, self-paced algebra discovery in the Socratic style. Tilton's tutorials provide several sequences of whiteboard videos, demonstrations, hints, and feedback at each step of a student's work on a problem. Formerly known as Theory Y Algebra and Algebra
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Time Engineers - Software Kids
A CD-ROM game of interactive, open-ended engineering problems to solve by using math and science skills, designed to help middle and high school students explore and apply some of the fundamental principles of engineering. Time Engineers encourages trial-and-error
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Time Math - Education 4 Kids, Inc.
Time, at different levels of complexity (clock and calendar). You are presented with two times (in one of the formats selected) and asked to determine the difference between the two.
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Time (SMILE) - Gracie Davenport, Spalding School
A lesson designed to teach physically Handicapped (EMH) students to tell time using a number chart; to demonstrate their ability to tell time by using a clock; and to use this skill to follow a daily schedule. From the Practical and Applied Math section
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Time to Set the Hook - John Stevens
A site dedicated to helping other teachers "set the hook with technology. It might be a video that demonstrates your standard being applied in a real life setting. It might be a screencast of a step-by-step solution to a problem. It might be a blog that
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Time Zone Converter - David Svenson
Enter multiple cities, countries, or time zones, and this JavaScript plots them on a timeline that reveals their temporal differences. Slide the circled "host" marker through business hours, evening, past midnight, night, and morning.
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Synopsis
Oxford GCSE Maths for AQA: Evaluation Pack by
This new Oxford GCSE Maths for AQA series offers a unique choice of four student books and stunning new OxBox software to match. Each book allows access to a grade C; you simply allocate a single targeted book to each student for the whole course, making learning simpler and more focused (particularly on students who fall between Foundation and Higher tiers) and therefore enabling students to achieve their highest possible grade. This evaluation pack contains all four student books, plus a homework book, plus extra sample material so you can see how it will work for you. | 677.169 | 1 |
Algebra 1 Variables and Expressions Lesson Plan
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
2.24 MB | 17 pages
PRODUCT DESCRIPTION
This lesson plan teaches variables and expressions for first year algebra (Algebra 1). This is lesson 1 for Jill Alumbaugh's Algebra 1 curriculum. It is promethean board and projector ready. There are 17 example and "try" | 677.169 | 1 |
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