MIAMI-DADE COLLEGE, HIALEAH CAMPUS

DEPARTMENT OF MATHEMATICS, LAS

***A STUDY GUIDE***

*OF* ****FREQUENTLY USED DEFINITIONS AND TERMS****

**IN**

*****ALGEBRA*****

**FOR**

MAT0002, MAT0020, MAT0024, MAT1033, MAC1105, MAC1114, MAC1140, MAC1147,

MAC2233, MAC2311/2312/MAC2313, MGF1106/MGF1107, MTG2204, STA2023, AND OTHER MATH STUDENTS

Compiled and Edited

by

Dr. Mohammad Shakil Department of Mathematics

and

Professor Ricardo A. Sanchez Academic Support Lab & Dept of Mathematics

Miami-Dade College, Hialeah Campus

2

****FREQUENTLY USED DEFINITIONS AND TERMS****

**IN**

***ALGEBRA***

**FOR**

MAT0002, MAT0020, MAT0024, MAT1033, MAC1105, MAC1114, MAC1140, MAC1147,

MAC2233, MAC2311/2312/MAC2313, MGF1106/MGF1107, MTG2204, STA 2023,

AND OTHER MATH STUDENTS

This hand-out of frequently used definitions and terms in algebra has been prepared for free distribution to the math students of Miami Dade College,

Hialeah Campus. It has been compiled to serve the MAT 0024, MAT 1033, MAC 1105, and other mathematics students of the Hialeah Campus. The hand-out has been updated by including by including some notes on Set Theory, Properties of

Real Numbers, Calculus, Geometry, & Probability, provided in the Appendices. It is hoped that this hand-out will meet the needs of our math students, and save

their time in cross checking frequently used important definitions, terms, and principles of algebra in their courses.

We acknowledge our sincere indebtedness to the works of various authors and resources on the subject which we have freely and liberally consulted .

MDC, Hialeah Campus Dr. M. Shakil & Prof. R. A. Sanchez

Spring Term, 2009 Compilers

3

***FREQUENTLY USED DEFINITIONS AND TERMS IN ALGEBRA***

A

Absolute value: (i) Absolute value of a number is its distance from zero on the

number line, i.e., aaa . (ii) It‟s the distance between two numbers on the

number line, i.e., abba (a and b are real numbers). Example:

5335 , i.e., 22 , i.e., 22 .

Absolute value of a complex number: The absolute value of a complex

number a + bi is the square root of a2 plus b2, i.e., 22 babia . Example:

525916343422i

Addition: It is the process to find the sum of two or more numbers. Example: 3

and 4 add up to give 7.

Addition Property of Equality: If the same number is added to both sides of

an equation, the two sides remain equal. That is, if x = y, then x + z = y + z.

Example: 5 = 5. Add the same number, say, 4 to both sides, i.e., 5+ 4 = 5 + 4, i.e., 9 = 9, it‟s true. Addition Property of Inequality: Adding the same number to both sides of an

inequality, does not affect the inequality. That is, if x > y, then x + z > y + z, and

if x < y, then x + z < y + z. Example: 2 < 3. Add the same number to both sides, i.e., 2 + 1 < 3 + 1, i.e., 3 < 4, it‟s true.

Additive inverses: Pairs of real number that have the sum 0, i.e., 0)( aa .

Example: 0)5(5 , i.e., 5 and - 5 are additive inverses.

Additive inverse of a matrix: The matrix obtained by taking the opposite of

every matrix element. The additive inverse of matrix A is written –A. Note: The

sum of a matrix and its additive inverse is the zero matrix. Example:

1101

452A

1101

452A

000

000)( AA

Algebraic expressions: Expressions that are made up of variables, numbers, grouping symbols, operation signs, and exponents. Examples: zyx 723 ;

13472 2 pzy ; 64

322

5

x

xx

4

Amplitude: Half the difference between the maximum and minimum values of a

periodic function.

Angle: is the figure formed by two rays sharing a common endpoint, called the

vertex of the angle. Example:

In the figure shown below, angle AOB is formed by the rays OA and OB with a common endpoint O.

Angle of depression: An angle that measures between 0 and 90 degrees

formed by a horizontal ray and a ray from the observer to the object observed

below, as shown in the following figure::

Angle of elevation: An angle that measures between 0 and 90 degrees formed

by a horizontal ray and a ray from the observer to the object observed above, as shown in the following figure:

Area: It is a physical quantity expressing the size of a part of a surface. Area is

defined as the number of square units that covers a closed figure.

5

In the example shown below, the area of the yellow square is 16 square units. That means, 16 square units are needed to cover the surface

enclosed by the square.

Formulas are defined to calculate the area of regular geometric figures like

square, rectangle, circle, etc.

Arithmetic mean (Average): It is the number that is found by dividing the sum of

data by the number of items in the data set. It is also called the average.

Example: The average height of students in Dr. Bestard's class is 142.5 cm. This means that this average was found by adding the heights of all students in his class and then dividing that sum by the total number of students.

Arithmetic sequence: It‟s a sequence (finite or infinite list of real numbers for

which each term is the previous term plus a constant ( called the common difference). If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence is

given by: dnaan 11 , and, in general, dmnaa mn .

For example, starting with 1 and using a common difference of 4, we get the finite arithmetic sequence: 1, 5, 9, 13, 17, 21; and also the infinite sequence 1, 5,

9, 13, 17, … , 4n+1, … . Arithmetic series: It‟s the indicated sum of terms of an arithmetic sequence.

Arithmetic series are commonly expressed using sigma notation. As an example,

the arithmetic series: ,12....2 11111 dnadnadadaa can

be written using sigma notation as: 1

0

n

i

ida . Likewise, an arithmetic series

mm aaaaa 1321 ....... , can be written as m

j

ja1

Ascending order: Numbers listed in ascending order are listed from smallest to

largest. If you listed the numbers 5,7,6,4 and 9 in ascending order, the list would be: 4, 5, 6, 7, 9.

Associative property of addition: It states that the way in which numbers being

added are grouped does not change the sum. Example: 5)23(235

6

Associative property of multiplication: It states that the way in which numbers

being multiplied are grouped does not change the product. Example:

5)23(235

Asymptote: is a straight line or curve A to which another curve B (the one being

studied) approaches closer and closer as one moves along it.

Axis of symmetry: A line in the plane of a graph such that the part of the graph

on one side of the line is a reflection of the part on the other side.

Axiom: Axiom is a rule or a statement that is accepted as true without proof. An

axiom is also called a postulate. Example: Here is an axiom of addition and

multiplication: Let x and y be real numbers. Then x + y is also a real number and xy is also a real number.

B

Bar Graph: A way of displaying data using horizontal or vertical bars so that the

height or length of the bars indicates its value. The following are characteristics

of a bar graph:

All bars have the same width (equal intervals). The equal intervals are shown on one of the axes.

The frequency of the data in each interval is represented by the height/length of the bar.

Example: A bar graph is shown in the following figure.

7

Bar Notation: The use of a horizontal bar over decimal digits to indicate that they

repeat indefinitely. Example: 333333333.1 . The bar above 3 indicates that the 3

repeats forever (i.e., indefinitely). Base: Base is the bottom of a plane or a solid. In case of trapezoids and prisms,

both the top and bottom are called bases as they are parallel to each other.

The base of a two-dimensional figure is a line.

The base of a three-dimensional figure is a plane.

Circle is the base of a cone.

Square is the base of a cube.

Base (of a Power): The number or expression used as a factor for repeated

multiplication. Example: 55553 . The number 5 is the base of a power 3 .

Binary System: The base 2 number system that uses the digits 0 and 1.

Example: The binary number 1011 can be represented as a decimal number by (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20). This simplifies to (1 x 8) + (0 x 4) + (1 x 2)

+ (1 x 1) = 8 + 0 + 2 + 1 = 11. This means that the binary number 1011 is equivalent to 11 in the decimal number system. Binomial: An algebraic expression with two unlike terms.

Examples of Binomials Non-Examples of Binomials

2 + 5p2 ; 4xy + 20 m4n2

42pr3 (A Monomial) 2x2 + 3x + 4 (A Trinomial)

Bisector of a line: In a plane, a bisector of a line divides a line segment into two

congruent segments.

8

C

Canceling Property: If a common factor is found in both parts of a fraction (in

the denominator and in the numerator), it can be "factored out" or "cancelled".

We only can cancel the same values in the numerator and in the denominator if they are multiplying. Example 1: First, the numerator and denominator are written as a product of

their factors. Then common factors are cancelled.

7

5

73

53

21

15

Example 2: When common factors exist on opposite sides of an equation, you

can also cancel as the following shows:

yx

yx 33

Example 3: We can also cancel using addition and subtraction.

yx

yx 55

In the above example, since there is a 5 added to both sides of the equation, we

can "cancel" both 5's. Non-Examples of Canceling:

In order to cancel a number, the operations being performed on the numbers

must be the same. Consider the following expression:

1

6

3

63 xx

23

6

3

3

3

63x

xx

Capacity: It is the amount of liquid a container can hold. In other words, capacity

is the volume of a container given in terms of liquid measurement. Center of a Circle: The point inside the circle that is the same distance from all

of the points on the circumference.

9

Central angle: Central angles are angles in which the vertex of the angle is the

center of a circle.

Example: In the circle below, there are four central angles. In each case the vertex of the angles is Point B, which is the center of the circle.

Classification of numbers:

Classification

Definition

Examples

Counting / Natural

They are the positive numbers we use

to count objects

1,2,3,4,5,6,7,8,9,10…

Whole Numbers

The whole numbers are the numbers

0, 1, 2, 3, 4, and so on (the natural numbers and zero).

0,1,2,3,4,5,6,7,8,9,

10…

Integers

The integers are all the whole

numbers and their opposites (the positive whole numbers, the negative whole numbers, and zero).

…-3,-2,-1, 0, 1, 2,3…

Rational Numbers

Fractions, repeating and terminating

decimals. All natural numbers, whole numbers, and integers are rationales, but not all rational numbers are

natural numbers, whole numbers, or integers.

25.0

,33333.1

,9

8,

4

3,

1

5

Irrationals

An irrational number is a number with a decimal that neither terminates nor repeats. An irrational number cannot

be written as a fraction b

awhere a and

2 1.41421...

= 3.14159... 3 2 1.25992...

10

b are integers

Real

All the rational numbers and all the irrational numbers together form the

real numbers. Every rational number is real, and every irrational number is

real.

25.0,33333.1,1

5

2 1.41421...

-1, 0, 1

Coefficient: A number multiplied by a variable in an algebraic expression.

Example: In xxx 1065 23 , 5, 6 and 10 are coefficients.

Column Matrix: A matrix with only one column. Example:

7

5

1

Common Denominator: A whole number that is the lowest common multiple of

the denominators of two or more fractions. Example: What is the common

denominator for the fractions below?

6

7

4

5;

3

2and

First, find the multiples of the denominators, 3, 4 and 6.

Denominator Multiples

3 3 6 9 12 15 18 21 24

4 4 8 12 16 20 24 28 32

6 6 12 18 24 30 36 42 48

The first common multiple among them is the Common Denominator. In this case

is 12.

11

Common Factor: A number (or an algebraic expression) that is a factor of two

or more numbers (or algebraic expressions).

Numbers Factors Common

factors GCF

36 36,18,12,9,6,4,3,2,1

6,3,2,1

6 24 24,12,8,6,4,3,2,1

18 18,9,6,3,2,1

Example of Relatively Prime Numbers:

Numbers Factors Common

factors GCF

32 32,16,8,4,2,1

1

1

65 65,13,5,1

GCF: Greatest Common Factor.

Note that neither 32 nor 65 are prime. Since they have the number 1 as their only common factor, 32 and 65 are relatively prime to each other.

Common Multiple: A number that is a multiple of two or more numbers.

Numbers Nonnegative multiples Common Multiples

LCM

3 ...39,36,33,30,27,24,21,18,15,12,9,6,3 .....30,15

15

5 ...65,60,55,50,45,40,35,30,25,20,15,10,5

LCM: Lowest Common Multiple.

Commutative Properties: Properties that denote an operation is independent of

the order of combination, as defined below: (i) The commutative property of addition states: a + b = b + a. An example is 5 +

2 = 2 + 5, because both sides of the equation equal 7.

12

(ii) The commutative property of multiplication states: ab = ba. An example is 5 x

2 = 2 x 5, because both sides of the equation equal 10.

Complex Numbers: A complex number is an expression of the form biaz

where a and b are real numbers and 1i is called the imaginary number.

The number a is called the real part of the complex number biaz and b is

called the imaginary part of the number. Note that the imaginary is defined to be:

1i , and 112

2i . .Now, you may think you can do this:

111122

2i . But this doesn't make any sense! You already have

two numbers that square to 1; namely –1 and +1. And i already squares to –1.

Composite Number: A composite number is a number that has factors in

addition to one and itself. Thus, all non-prime numbers are composite numbers. (i) Examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, … . (ii) Examples of

non-composite (prime) numbers: 2, 3, 5, 7, 11, 13, … .

Constant of proportionality : The constant value of the ratio of two proportional

quantities x and y . Example: The following table lists different types of

proportionality. Note that k denotes the constant of proportionality here.

Proportionality

Mathematical

Definition

Value of k

Direct

kxy

x

yk

Directly proportional to the exponential function of x

nkxy

nx

yk

Inverse

x

ky

xyk

Inversely proportional to the exponential function of x

nx

ky

nxyk

Constant Term: A quantity that does not change its value. Example: (i) The numbers 1, 2, 3, 4, etc. are constants. (ii) The number 3 in the algebraic equation

32xy is also a constant. This is because it does not change its value, even

when the values of the variables x and y changes.

13

(iii) The following table lists some common constants, their symbols, and

approximate values.

Constant Symbol Approx. value

Delian constant 1.25992...

Number e 2.71828...

Natural logarithm of 2 0.693147...

Number pi 3.14159...

Pythagoras's constant 1.41421...

Coordinate Plane: A two-dimensional region determined by a pair of axes and

that uses numerical values to represent the location of an object. Coordinates: An ordered pair, (x, y), that locates a point in the plane. Note that

each point graphed can be represented by a unique pair of numbers called an ordered pair. The intersection of the two axes on a coordinate plane is called

the origin and is represented by the ordered pair (0, 0). In the following figure,

the red dot is three lines to the right of the y-axis (positive direction) and is one line above the x-axis (positive direction). So the red dot is represented by the ordered pair (3, 1). The numbers in the ordered pair are called the coordinates.

The red point has an x-coordinate of 3 and a y-coordinate of 1.

Counting Numbers: The natural numbers, also called the counting numbers, are

the numbers 1, 2, 3, 4, and so on. They are the positive numbers we use to count objects. Zero is not considered a "natural number." (See Classification of numbers also).

Cross-product Property: Cross product property states that 'in a proportion,

product of the means is equal to the product of the extremes.

In the proportiond

c

b

a, b and c are the means and a and d are the extremes.

So, the cross product is: bcad . The cross product of the proportion shown is

36

15

12

5 1512365 .

14

Cube Root: One of three equal (identical) factors of a given number denoted 3 x

or 3

1

x , is a number a such that xa3 . Example: 51253 because.

12553

. 125555 ; 3273 because. 2733 .

27333

Cubic Unit: Standard measure of volume.

In the Metric System the units for volume are derived from units for length. For example, the milliliter is defined as a cubic centimeter. In other words, a mililiter is equivalent to the capacity of a cube with sides of length 1 centimeter. The

largest volume (capacity) this cube can hold is:

V = length x width x height

V = 1 cm x 1 cm x 1 cm V = 1 cubic centimeter. D

Data: The facts or numbers that describe something.

Qualitative Data:

Categorical, such as a person's gender, race, or religion

Qualitative Data:

Count, such as the number of televisions in a person's house, number of packs of cigarettes smoked per day, number of visits to the doctor per

year.

Measurement, such as a person's test score, height, weight.

15

Decimal: A symbol that uses a base-ten place-value system with multiples of

tenths to represent a number.

Decimal System: The base 10 number system that uses the digits 0, 1, 2, …, 9.

The decimal system is based around units of 10. Example: (i) The number 256 really means (2 x 100) + (5 x 10) + (6 x 1) = (2 x 102) + (5 x 101) + (6 x 100). (ii) The number 7.93 really means (7 x 1) + (9 x 0.1) + (3 x 0.01) = (7 x 100) + (9 x

1/101) + (3 x 1/102). Decompose: The process of factoring terms and numbers in an expression.

Decomposing an expression involves factoring out common factors until the expression is in simplest term

Degree: The exponent of a number or expression.

TERM COEFFICIENT VARIABLE DEGREE

3x 3 x 1

- 5m3 -5 m 3

Degree of a Polynomial: The largest exponent of x which appears in the

polynomial

The standard form of a polynomial is 01

1

1 ...... axaxaxaxf n

n

n

n .

Thus, the degree of the polynomial is n, which is the largest exponent which

appears in the above polynomial. Examples: (i) x3 has degree 1. (ii) 53x has

degree 3. (iii) xx 22 has degree 2.

Denominator: The number below the line in a fraction. The denominator

indicates what kind or size of parts the numerator counts. Numerator & Denominator: A "fraction" is used to indicate the number of parts

of a whole. The denominator of the fraction indicates the number of equivalent pieces the whole is divided into. The numerator indicates the number of these

pieces to be considered (i.e., the part). For example, in the fraction 16

7x {7x is

the numerator and 16 is the denominator. Descending Order: To sort in order from largest to smallest. If ordering a list in

descending order that is alphanumeric (A - Z), then the list begins with words that

start with the letter Z, then Y, then X, etc until the letter A. If ordering numeric values, then the order is from highest to lowest (345, 123, 56, 55, 3). If ordering dates, then the order is from latest to earliest.

16

Difference: The amount left after one number is subtracted from another

number. For example, the difference between 12 and 9 is 3, i.e. 12 - 9 = 3

Digit: The symbols used to write numerals. In the base ten system, the digits are

0,1,2,3,4,5,6,7,8, and 9. In numeration systems based on place value, the place the digit is written determines the actual value of the digit. For example, in the number 83, since the digit 8 is in the tens position, it represents 80. The 3 lies in

the ones position and represents 3. Direct Proportion (Direct Variation): The relation between two quantities

whose ratio remains constant.

Proportionality

Mathematical

Definition

Value of k

Direct

kxy

x

yk

Directly proportional to the exponential function of x,

nkxy

nx

yk

where k is a constant of variation.

Distance: The amount of space between two points or things. Distance is always

a non-negative number.

Distance Formula: Given the two points (x1, y1) and (x2, y2), the distance

between these points is given by the formula:

Example: Find the distance between the points (–1, –3) and (–4, 4).

Just plug them in to the Distance Formula:

2

12

2

12 yyxxd

17

58

499

73

3414

3414

22

22

22

d

d

d

d

d

Distributive Property: The sum of two addends multiplied by a number is the

sum of the product of each addend and the number.

zyxzyxadacabdcba 63323

242212 2 xxxxxbdbcadacdcba

Dividend: A number that is divided by another number. For example, given the

division 29

18, 18 is the dividend, 9 is the divisor and 2 is the quotient.

Dividing Powers Property: To divide two powers of the same base, subtract the

exponent of the denominator from the exponent of the numerator.

nm

n

m

aa

a 41523

2

53

. xyyxyx

yx

Divisible: One number is divisible by another number if the second number

divides "evenly" into the first. That is, when the first number is divided by the second number, there is a remainder of zero.

Examples of Divisibility:

29

18

We say 18 is divisible by 9, meaning the remainder is zero when 18 is divided by 9. The number 9 can be called a divisor of 18. The result of dividing 18 by 9 is 2 and is called the quotient.

Non-examples of Divisibility:

36

19 with remainder of 1. Here, we divide by 6, so 6 is called the divisor,

however, 19 is not divisible by 6 because the remainder is 1 (not zero.) The

18

number 3 Remainder 1 (or 6

13 ) is called the quotient.

Divisibility Rules:

A number is divisible by 2 if the number is even. In other words, if a

number ends in 0, 2, 4, 6, or 8, then it is divisible by 2.

A number is divisible by 3 if the sum of the digits is divisible by 3.

For example 141 is divisible by 3 because 1 + 4 + 1 = 6 and 6 is divisible by three.

364 is not divisible by 3 because 3 + 6 + 4 = 13 and 13 is not divisible by 3.

Note that this rule can be used multiple times. For example, to check if

135,593,334,384 is divisible by 3, we add 1 + 3 + 5 + 5 + 9 + 3 + 3 + 3 + 4 + 3 + 8 + 4 = 51. To check that 51 is divisible by 3, we add 5 + 1 = 6. 6 is divisible by 3,

so 51 is divisible by 3, so 135,593,334,384 is divisible by 3.

A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

For example, 2,356 is divisible by 4 since 4 divides 56 evenly. Alternatively, a

number is divisible by four if the quotient of the number and 2 is even. In the previous example 2,356 divide by 2 is equal to 1178 which is even.

A number is divisible by 5 if the number ends in a 0 or 5.

A number is divisible by 6 if the number is divisible by 2 and is

divisible by 3.

A number is divisible by 9 if the sum of the digits is divisible by 9.

This rule is similar to the divisibility rule for 3. A number is divisible by 10 if the number ends in a 0.

Division by Zero: Division by zero is not allowed, It is undefined.

Division Property of Equality: States that when both sides of an equation are

divided by the same number, the remaining expressions are still equal. Example:

6

3

18

3

3

183

x

x

x

Divisor: A number by which another number is to be divided. For example, given

19

the division 29

18, 18 is the dividend, 9 is the divisor and 2 is the quotient.

Domain: For a function f defined by an expression with variable x, the implied

domain of f is the

set of all real numbers variable x can take such that the expression defining the function is real. The domain can also be given explicitly. Example: Find the domain of function f defined by

1

1

xxf

Domain: x can take any real number except 1 since x = 1 would make the

denominator equal to zero and the division by zero is not allowed in

mathematics. Hence the domain in interval notation is given by ,11, .

E

Elimination Method: It is the process of eliminating one of the variables in a

system of equations using addition or subtraction in conjunction with

multiplication or division and solving the system of equations. Equation: It is a mathematical statement, in symbols, that two things are the

same (or equivalent). Equations are written with an equal sign and they are often used to state the equality of two expressions containing one or more variables.

Example: 38523 xx ; 12453 2 xx .

Equation of a Straight line: Any equation that can be put in the form

CyBxA , where CandBA ,, are real number and BandA are not both 0, is

called a linear equation in two variables yandx . The graph of any equation of

this form is called a straight line. The form CyBxA is called standard form.

The equation of a straight line can be easily determined if its slope is known,

which is defined as follows.

(I) Slope of a Straight Line: It defines the steepness of the line. Geometrically, the

slope of a straight line is defined as the ratio of the vertical change to the

horizontal change encountered when moving from one point to another point on the line, as shown in the figure given below. The vertical change is sometimes called the rise. The horizontal change is called the run. The slope of a straight

line passing through two points ),( 11 yx and ),( 22 yx is given by the following

formula:

20

To find the slope of a line we have to know:

1. Two points in the line, or 2. The equation of the line with y isolated. The coefficient of x is the slope.

(II) Slope Facts:

1. Slope 12

12

xx

yym .

2. Horizontal lines have slope m = 0. 3. Vertical lines have undefined slope. 4. Parallel lines have the same slope.

5. If m1 and m2 are slopes of the two perpendicular lines, then m1 x m2 = -1.

This also written as m1 = 2

1

m.

(III) Different Forms of the Equation of a Straight Line: The following table lists different forms for the equation of a straight line.

Forms Equation Application

Slope-

intercept

y = mx + b Used when you have the slope and the y-

intercept.

Point-slope y – y1 = m(x – x1)

(x1, y1) is a point on the line. Used to find the equation.

Standard form Ax + By = C If possible, A is nonnegative and A, B, and

C are relatively prime integers.

Two-intercept

Used when you have both intercepts.

Vertical x = a All points have x-coordinate a. Slope

undefined.

Horizontal y = b All points have y-coordinate b. Slope equal to zero.

Equations’ Properties: If an equation in algebra is known to be true, the

following operations may be used to produce another true equation:

1. Any quantity can be added to both sides. 2. Any quantity can be subtracted from both sides. 3. Any quantity can be multiplied to both sides.

21

4. Any nonzero quantity can divide both sides.

Equivalent Equations: Equations that have the same solutions. Examples:

.531053)(

;01156)(

xandxii

xandxi

Equivalent Expressions: Expressions that simplify to an equal value when

numbers are substituted for the variables of the expression. Example:

1106324

156523

xxx

xx

Equivalent Fractions: Fractions that have the same decimal form. Example:

3

6

12

24and since both are equal to 2.0 in decimal form.

Equivalent Ratios: Ratios that represent the same fractional number, value, or

measure.

Above we state that 3

6

12

24and are equivalent fractions since they have the same

decimal expansion. Thus, 3

6

12

24and are equivalent ratios since they represent the

same amount. Evaluate a Numerical Expression: To perform operations to obtain a single

number or value.

For example, "Evaluate 32924 ” means to perform all the operations in the

given numerical expression. In other words, 8324692432924 .

Evaluate an Algebraic Expression: To find the value of an expression by

replacing each variable in an expression with numbers. For example, evaluating

the algebraic expression 23 2 yxyx for 2x and 1y means substituting

given values for x and y in the expression and calculate it. In other words,

31481122222323 yxyx .

Even Number: An even number is an integer of the form kn 2 , where k is an

integer. The even numbers are therefore ..., -4, -2, 0, 2, 4, 6, 8, 10... All integers divisible by 2 are considered to be the even numbers

22

Expanded Form: A way of writing a numbers in which the numbers are written to

show the place value of each digit. Example: 012 108105103850300358

Exponent: The number of times a number or expression (called base) is used as

a factor of repeated multiplication. Also called the power.

Exponential Base Exponent Expanded

3x x 3 xxxx3

26 6 2 36

1

66

1

6

16

2

2

2

1

16 16 2

1

41616 2

1

Exponential: A number written with an exponent. Example: 35 is called an

exponential.

F

Factor: When two or more integers are multiplied, each integer is a factor of the

product. "To factor" means to write the number or term as a product of its factors .

Example: 6 and 5 are factors of 30 since 3056 . 30 is a multiple of both

6 and 5 .

Numbers Factors

36 36,18,12,9,6,4,3,2,1

24 24,12,8,6,4,3,2,1

18 18,9,6,3,2,1

Factoring: Using the factoring property lets us change an expression from a sum

to a product. Using the distributive property lets us change an expression from a product to a

sum. Example: An expression such as cxa3 tells you to multiply a3 by cx .

When you do that, you get the sum acaxcxa 333 . When you do that in

reverse, by writing cxaacax 333 , you are factoring, so that using the

factoring property lets us change an expression from a sum to a product.

FOIL Pattern : A method based on the Distributive Property that is often used to

multiply two binomials, i.e., bdbcadacdcba

23

Forming an Equation (Writing an Equation): Writing information presented in

words as a mathematical sentence with an equality sign. Example: Here are

some statements in English with its translation in algebra.

(i) The sum of three times a number and eight is equal to three:

383x .

(ii) The product of a number and the same number less 3 is equal to

twice the number: xxx 23 .

(iii) A number divided by the same number less five is the same

number increased by 3: 35

xx

x.

Fraction: A number that can be written as a quotient of two quantities. Example:

5

8,

7

12,

3

5.

Function: (i) A rule of matching elements of two sets of numbers in which an

input value (called the independent variable or argument) from the first set has only one output value (called the dependent variable) in the second set. (ii) A function is a pairing of two sets of numbers so that to each element in the first set, there corresponds exactly one number in the second set. (iii) Vertical Line

Test: If you can draw a vertical line on the same coordinate plane as your graph

of the equation (or relation) such that the vertical line intersects your graph in only one point, the equation (or relation) defines a function. (iv) Notation:

)(xfy , where x denotes an independent variable and y denotes a dependent

variable. Example: 12 xy and 12xy define functions.

Fundamental Theorem of Arithmetic: Every integer, N > 1, is either prime or

can be uniquely written as a product of primes. Note: This is also known as the

Unique Factorization Theorem. It essentially states that for all positive numbers larger than 1, its factorization is unique. Example:

371725032

1152550

1352130

17351

3

2

G

Graph of a Function: The set of all the points on a coordinate plane whose

coordinates makes the rule of function true. Example: The following figures are examples of graphs of a function.

24

Figure 1 Figure 2

95xxfy 2xxfy

Vertical Line Test: If you can draw a vertical line on the same coordinate plane as

your graph such that the vertical line intersects your graph in more than one point, the equation is not a function - it is called a relation. Example: The

following figure is not the graph of a function. It defines a relation.

Graph of a Linear Equation: The graph of all solutions of a linear equation,

resulting in a straight line. Example: The following figure is the graph of the linear equation 23xy .

25

Note that, in the line 23xy , the slope is equal to 3 and y-intercept is equal to

2.

Graph of a Linear Inequality: The solutions of a linear inequality; they form a

half-plane on one side of a line and may or may not also form the line itself. Example: (i) The following figure is the graph of the inequality y < x + 2. In other

words, any point in the shaded half-plane is a solution to the inequality. Notice the line y = x + 2 is NOT included in the graph (because there are no equal

marks in the inequality), meaning that points that fall on the line are NOT solutions to the inequality.

(ii) The following figure is the graph of the inequality y x + 2. In other words,

any point in the shaded half-plane is a solution to the inequality. Notice the line y = x + 2 IS included in the graph (because there are equal marks in the inequality), meaning that points that fall on the line are ALSO solutions to the

inequality.

Greater than and Greater than or equal to: Symbols that describe a

relationship in which the expression on the left are greater than the expression on the right. Example: (i) The expression

26

5x is read "x is greater than 5." This means that x can equal any real number

greater than 5. (ii) 5x is read "x is greater than or equal to 5." In this case, x is equal to any

real number greater than 5 as well as the number 5.

Greatest Common Factor (GCF): The largest factor that two or more numbers

have in common.

Numbers Factors Common factors GCF

36 36,18,12,9,6,4,3,2,1

6,3,2,1

6

24 24,12,8,6,4,3,2,1

18 18,9,6,3,2,1

H

Horizontal Line: A line that goes left and right.

Hypotenuse: The hypotenuse of a right triangle is the triangle's longest side; the

side opposite the right angle. Example: The hypotenuse of a right triangle is shown in the following right triangle.

I

Identity for Addition: A number that can be added to any second number

without changing the second number. Example: Identity for addition is 0 (zero) since adding zero to any number will give the number itself. In other words,

aaa 00 Identity for Multiplication: A number that can be multiplied by any second

number without changing the second number. Example: Identity for multiplication is "1," because multiplying any number by 1 will not change it. In other words,

aaa 11 .

27

Improper Fraction: A fraction where the numerator is equal to or larger than the

denominator.

Fractions such as 7

15 and

5

7 are improper fractions

Indirect proportion: A proportion in which two quantities are inversely related to

each other. Example: The following are some examples of indirect proportion.

Here k is a constant of proportionality.

Proportionality

Mathematical Definition

Value of k

Inverse proportion ( Indirect)

x

ky

xyk

Inversely proportional to the exponential function of x,

nx

ky

nxyk

Inequality: Any mathematical sentence that contains the symbols >(greater

than), <(less than), <(less than or equal to), or >(greater than or equal to). Examples:

14564

132523

xx

xxx

Infinity: An expression increases to infinity if it continues to increase without

limit. Example: Functions such as the following one xy 5 will continue to infinity since there is

no limit. The symbol for infinity is Integer: It is defined as the set of all the whole numbers and their opposites (i.e., the positive whole numbers, the negative whole numbers, and zero). Note: An

integer is also a rational as well as a real number. (See the Classification of numbers also, to distinguish an integer from other real numbers, which are not

28

integers).

Interval: A regular distance or space between values. The set of points between

two numbers.

Interval notations and their classifications:

Interval notations: These are given below.

};{),(

};{],(

;),[

;,

;,

axxa

bxaxba

bxaxba

bxaxba

bxaxba

),(

};{],(

};{),(

;),[

bxxb

bxxb

axxa

Classification of Intervals:

Open interval: The interval with endpoints not included is the open

interval. The open interval between two numbers a & b (a < b) is shown by (a, b).

Closed interval: The interval with endpoints included is the closed

interval. The closed interval between two numbers a & b (a < b) is

shown by [a, b].

Half-open interval: The interval with one endpoint included is half-

open (half-closed) interval. The half-open interval between two numbers a & b (a < b) is shown either by (a, b] or [a, b) depending

on which number will be included. Inverse Operation: Pairs of operations that undo each other. Example:

29

(i) Addition and subtraction are inverse operations. For example, 1 + 4 = 5 reversely 5 - 4 = 1.

(ii) Multiplication and division are inverse operations. For example, 2 x 3 = 6, reversely 6 ÷ 3 = 2. Inverse Properties: Properties that state a number combined with its inverse

equals the identity. Example:

(i) The additive inverse states: a + (-a) = 0. An example is 5 + (-5)

= 0. (ii) The multiplicative inverse states: a x 1/a = 1. An example is 7 x

1/7 = 1. Inverse Proportion (Inverse Variation): The relationship between two variables

in which the product is a constant. Example: The following are some examples of inverse proportion. Here k is a constant of proportionality.

Proportionality

Mathematical

Definition

Value of k

Inverse proportion ( Indirect)

x

ky

xyk

Inversely proportional to the

exponential function of x

nx

ky

nxyk

Irrational Number: A number whose decimal form is non-terminating and non-

repeating. Irrational numbers cannot be written in the form a/b, where a and b are integers (b cannot be zero). So all numbers, which are not rational numbers, are called irrational numbers. Note: An irrational number is also a real number. (See the Classification of numbers also, to distinguish an irrational number from

other real numbers, which are not irrational numbers).

J Join: Line segment joining two pints.

30

Join (or union or sum of sets): The union of two sets A and B is the set of all

elements which are either in A or in B or in both. It is denoted by A B. Thus A

B = {x x A or x B or x both A and B}. Example: If A = {a, b, c} and B =

{b, d}, then A B = { a, b, c, d}.

Joint Variation: If the variable y varies directly with two other variables, say, x

and z , then y is said to vary jointly with x and z , and is denoted by the

equation zxky and z , where k is called the constant of joint variation.

Example: The following table is a list of some joint variation statements and their equivalent algebraic equations.

Joint Variation Statement Algebraic

Equations

y varies jointly with x and z . zxky

z varies jointly with r and the square of s .

2srkz

V is directly proportional to T and is

inversely proportional to P . P

TkV

F varies jointly with 1m and 2m and

inversely with the square of r . 2

21

r

mmkF

K

Kilogram: A metric unit used to measure the weight of an object. A kilogram is

used to measure the weight of an object. The abbreviation for kilogram is kg. Common conversions: 1,000 grams = 1 kilogram. 1 kilogram = 2.20462262

pounds, approximately. Kilometer: A metric unit used for linear measure, e.g., length, width, depth,

distance around an object. The abbreviation for kilometer is km. Common conversions: 1,000 meters = 1 kilometer. 1 kilometer = 3280 feet = 1760 yards,

approximately. Kilowatt: A unit of measure of electrical power. Common conversion: 1,000

watts = 1 kilowatt. Kinetic Energy: (i) It is defined as the energy a body possesses by virtue of its

motion. (ii) The kinetic energy of a particle of mass m moving with velocity v is

given by 2

2

1vm . (iii) The kinetic energy of a body rotating about an axis and

having angular velocity and moment of inertia I about the axis is given by

31

2

2

1I . (iv) Note that the Potential Energy is defined as the energy a body

possesses by virtue of its position. It is defined as the negative of the work done

in displacing a particle from its standard position to any other position.

L Least Common Denominator (LCD): The least common multiple of the

denominators of two or more fractions. (See Least Common Multiple below)

Least Common Multiple (LCM): The smallest multiple (other than zero) that two

or more numbers have in common. Example:

Numbers Nonnegative multiples Common

Multiples LCM

3 ...39,36,33,30,27,24,21,18,15,12,9,6,3 ...30,15

15

5 ...65,60,55,50,45,40,35,30,25,20,15,10,5

Length: The distance between two ends of a line segment.

Less than and Less than or equal to : Symbols that describe a relationship in

which the expression on the left is less than the expression on the right. Example: (i) The expression x < 5 is read "x is less than 5." This means that x can equal any real number less than 5. (ii) X 5 is read “x is less than or equal to

5.” In this case, x is equal to any real number less than 5 as well as the number

5.

Like Terms: Like terms are terms that contain the same variables raised to the

same power. Only the numerical coefficients are different. In an expression, only like terms can be combined.

Example:

(i) 53 and 57 they are like terms. They have the

same root.

(ii) yx 23 and yx 28 they are like terms. They have the

same variables.

32

(iii) 25mn and nm23 they aren‟t like terms. The variables

have different exponents.

(iv) all constants that belong to the real number are like terms.

Line (or Linear) Graph: A graph that displays data by using points joined by line

segments, so that the graph looks like a line. Example: A line graph A non-linear graph

95xy 2xy

Linear Equation: An equation whose solution is a straight line. A linear equation

has no more than two variables and has a solution that is a single value or ordered pair.

For example:

4s + 6 = 18 is a linear equation with the solution s = 3.

x + 5 = -13 is a linear equation with the solution x = -18.

5x + 8y = 13 is a linear equation with a solution (1, 1)

Linear Function: An equation in which the graphs of the solutions form a (non-

vertical) line. Linear Inequality : An inequality in two variables for which the graphs of the

solutions form a half-plane on one side of a line and may or may not also form

the line itself. Example: (i) The following example is the graph of the inequali ty y < x + 2. In other words,

any point in the shaded half-plane is a solution to the inequality. Notice the line y = x + 2 is NOT included in the graph (because there are no equal marks in the inequality), meaning that points that fall on the line are NOT solutions to the

inequality.

33

(ii) The following example is the graph of the inequality y x + 2. In other words,

any point in the shaded half-plane is a solution to the inequality. Notice the line y

= x + 2 IS included in the graph (because there are equal marks in the inequality), meaning that points that fall on the line are ALSO solutions to the

inequality

Lowest Terms: A fraction is in lowest terms when the greatest common factor of

its numerator and denominator is 1. A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. There are two

methods of reducing a fraction to lowest terms.

Method 1:

Divide the numerator and denominator by their greatest common factor.

3

2

618

612

18

12

34

Method 2:

Divide the numerator and denominator by any common factor. Keep dividing until there are no more common factors.

3

2

39

36

218

212

18

12

M Matrix: A rectangular array (organization) of numbers in rows and columns.

Example:

Maximum: (i) The highest point on a graph. (ii) The largest number in a data set. Example: (i) The maximum point for the graph below is (0, 3).

(ii) In the data set {3, 4, 5, 6, 7, 8, 9, 10}, 10 is the maximum.

Mean (Average): It is the number that is found by dividing the sum of values in a

data set by the number of items in the data set. It is also called the average.

Example: The average height of students in Dr. Bestard's class is 142.5 cm. This average was found by adding the heights of all students in his class and then

dividing that sum by the total number of students. Median: The number in the middle of a set of data when the data are arranged in

order. When there are two middle numbers, the median is their mean. Example:

1) 1,2,3,4,5 The median is 3

35

2) 1,2,3,4,4,5,6,7 The median is 42

44

3) 2,3,5,6,8,9,10,11 The median is 72

14

2

86

Midpoint: The point on a line segment that divides it into two equal parts.

In the figure below, C is the midpoint of the segment.

Midpoint Formula: The Midpoint formula is used when you need the point that is

exactly between two other points. It is given by 2

'2

, 2121 yyxxMyxM .

Example: Find the midpoint between the points (1, -7) and (-5, -3). Draw a sketch

in a rectangular Cartesian plane also

)5,2()2

10,

2

4()

2

)3(7,

2

)5(1(

2'

2, 2121 MMM

yyxxMyxM

Sketch:

N

Natural numbers (or counting numbers or positive integers ): It is defined as

the set of counting numbers ,4,3,2,1 , denoted as ,4,3,2,1N .

Natural logarithm: The logarithm of a positive real number a with the base e is called the natural logarithm, and is denoted as aln . Note that the number e is

36

defined as 71828.21

1limn

en

, where n is a natural number.

Nature of the roots: A classification of the roots of a quadratic equation

2 0ax bx c where a, b, and c are rational numbers and 0a , which is

indicated by the discriminant, 2 4b ac , the quadratic equation whether the roots

are real (rational or irrational) and equal (repeated), or real (rational or irrational)

and unequal, or imaginary and unequal according as 2 4 0, 0, 0b ac or or .

Negation: The operation of putting not or it is not the case that, denoted by the symbol ~ or , in front of a proposition or statement, p , i.e. for any given

statement p , its negation is the statement , ~ p (not p ) whose truth value is the

opposite of the truth value of p . Note that the negation of a proposition p is

false if p is true and vice versa.

Negative exponent: An exponent that is a negative number. In general,

1, 0.a

ax x

x

Negative square root: Let 0x be a real number. Then x is defined as the

negative square root of x . Example: 5 is the negative square root of 5 .

Negative integer: Any integer that is less than zero. Negative number: Any real number that is less than zero, i.e. located to the left

of 0 on the number line is called a negative number. It is denoted as a , where a is a positive real number. Example: 5 is a negative number.

Negative rational number: A real number 0x that can be written as a

fraction, which when expressed as a decimal is either a terminating or repeating

decimal. For example, ,2

1,5.0,5 and 3.0...333.0 are some negative

rational numbers.

Negative of a polynomial: If xP be a polynomial in x , then the negative of

xP , denoted by xP , is that polynomial which is obtained by changing the

sign of each term in xP . Example: Let 12 xxxP . Then

12 xxxP .

Newton's laws of motion: Three laws of mechanics which provide relationships

between the forces acting on a body and the motion of the body, first formulated

37

by Sir Isaac Newton in 1687. These are stated as follows: Newton’s first law (law of inertia): An object remains in a state of rest or

constant velocity unless acted on by an external force.

Newton’s second law: The resultant force acting on an object is

proportional to the rate of change of linear momentum of the object, the

change of momentum being in the same direction as the force. Momentum is the product of mass and velocity. This law is often stated as amF (the

force on an object is equal to its mass multiplied by its acceleration). Newton’s third law: To every action there is an equal and opposite

reaction. Thus, whenever a particle 1P exerts a force on another particle

2P , 2P simultaneously exerts a force on

1P with the same magnitude in the

opposite direction. The strong form of the law further postulates that these

two forces act along the same line. Newton's law of universal gravitation: Every particle of matter in the universe

attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance

between them. This force of gravitational attraction between the two bodies acts along the line joining their centers. This force is hence mutual. Consider two

bodies of masses 1m and 2m with r as the distance between their centers, and

F the force of gravitational attraction between two bodies. Then, by Newton's

law of gravitation, we have 21 mmF , and 2

1

rF . Combining these, we obtain

2

21

r

mmF , or

2

21

r

mmGF , where G is a constant of proportionality called the

gravitational constant. The value of G in si units, 22111067.6 kgmNewtonG ,

and in cgs units, 2281067.6 gcmdyneG .

Nonadjacent side of an angle in a triangle: The side of the triangle that does

not make up either side of the angle, i.e. the side opposite the specified angle of

the triangle. n-gon: A polygon with n sides and n angles.

Nonagon: A polygon with nine sides and nine angles.

Non-collinear: Not lying on the same straight line.

Non- coplanar points: Four or more points that do not lie on the same plane.

38

Non-Euclidean geometry: A geometry that contains an axiom which is

equivalent to the negation of the Euclidean parallel postulate, i.e. any system of

geometry in which the parallel postulate of Euclid does not hold. There are two types of non-Euclidean geometry, namely:

Riemannian or elliptic geometry: A non-Euclidean geometry using as

its parallel postulate any statement equivalent to the following: If l is any

line and P is any point not on l , then there are no lines through P that

are parallel to l .

Hyperbolic geometry: A non-Euclidean geometry using as its parallel

postulate any statement equivalent to the following: If l is any line and P

is any point not on l , then there exists at least two lines through P that

are parallel to l .

Non-linear equation: An equation containing a variable with an exponent other

than one, or containing some terms having more than one variable. The graph of such an equation is not a straight line (e.g., circle, parabola, hyperbola, etc).

Example: 012 xx ; .012 yxx

Non-repeating decimal: A decimal that does not repeat; it either terminates or

continues in no discernible pattern. Non-standard measurement: The use of items as measurement tools that are

not uniform in size (e.g., using fingers to measure something; one person's fingers are not necessarily the same size as another person's fingers). Non-standard unit: Any tangible item that can be used to measure something

(e.g., paper clips, crayons).

Non-terminating decimal: A decimal that does not terminate; it either repeats or

continues in no discernible pattern. Normal: It is meant as perpendicular. If two straight lines or planes are

perpendicular to each other, they are called normal to each other.

nth root: The solution of nx c when n is odd or the nonnegative solution of nx c when n is even and nonnegative. For any real number c and any

positive integer n , the nth root of c is denoted by either n c or 1

nc .

nth term: The final term of a finite sequence of elements 1 2, , , ,na a a or an

arbitrary term of an infinite sequence. Null set (empty set): The set which has no elements (members). Notation:

(Phi) or .

39

Number line: A straight line on which each point corresponds to a real number.

Note that integers are points on the number line marked at unit distance apart. Example:

Number scale: The scale formed by making a point 0 on a straight line,

separating the line into equal parts, and labeling the separation points to the right of 0 with the positive integers ...,3,2,1 and those to the left of 0 with the

negative integers, ...,3,2,1 .

Number system: A system used to represent numbers. Example: base ten

number system. Number theory: The study of properties of integers, generalizations of integers,

and relations between them, especially Diophantine equations (kinds of

equations when the coefficients are integers) and prime numbers. Numerals: Symbols used to denote numbers. Example: The symbols

9,8,7,6,5,4,3,2,1,0 are known as Arabic numerals.

Numeration: It is defined as the process of writing or stating numbers in their

natural order, i.e. the act or process of counting or numbering, or a system of

counting or numbering.

Numerator: In a fraction 0, bb

a, the number a above the fraction bar is called

the numerator. It shows how many of equal parts are being considered in a whole, if it is divided into a number of equal parts. Numerical: Consisting of numbers, rather than letters; of the nature of numbers.

Numerically: Expressed in or involving numbers or a number system.

Numerical coefficient: The numerical factor in a term is defined as its numerical

coefficient. Example: The numerical coefficients of the terms x15 and 2

4

1x are

15 and 4

1 respectively.

Numerical equation: An equation in which the coefficients and constants are numbers rather than letters. For example, the equation 29115x is a

numerical equation, whereas cbxa is a literal equation.

0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

40

Numeric expression: Any combination of words, variables, constants, and/or

operators that result in a number; also known as an arithmetic expression. Numeric pattern: An arrangement of numbers that repeat or that follow a

specified rule.

Numerical sentence: A statement about numbers. Example: 14115 is a true

statement and 14115 is a false statement. Note that an open numerical

sentence is an expression which becomes a numerical sentence when values

are given to certain variables. E.g., 29145 yx is an open numerical sentence.

Numerical value: By numerical value of a number, it is meant as the absolute

value of the number. The absolute value of a positive number is the number itself. For example, the

absolute value of 15 is 15 . The absolute value of a negative number is the opposite of the number. For example, the absolute value of 15 is 15 .

O

Oblique: Neither perpendicular nor horizontal, i.e., slanting.

Oblique angle: Any angle which is not a multiple of 090 .

Oblique axes: In the plane, when the axes ( x -axis and y -axis) are not

perpendicular, they are called oblique axes. Note that when the axes ( x -axis and y -axis) are perpendicular, they are called rectangular axes. The coordinates are

then called oblique coordinates and rectangular coordinates, respectively. Oblique lines: Lines which are neither parallel nor perpendicular.

Oblique line and a plane: A line is oblique to a plane if it is neither parallel nor

perpendicular to the plane. Oblique triangle: Any triangle (plane or spherical) which does not contain a right

angle is called an oblique triangle.

Obtuse angle: An angle whose measure is greater than 090 and less than 0180 , i.e. 00 18090 . Example:

41

Obtuse triangle: If one of the angles, say, , of a triangle is obtuse, i.e. if 00 18090 , then i t is called an obtuse triangle.

Octagon: A polygon with eight sides and eight angles. Note: A regular octagon

is an octagon whose angles are all equal and whose sides are all equal. Examples of octagon:

Octahedron: A polyhedron having eight faces. Note: A regular octahedron is

an octahedron whose faces are all congruent regular equilateral triangles.

Example of octahedron:

Odd function: A function )(xfy is called odd if and only if its graph is

symmetric with respect

to the origin. Thus a function is odd if a point yx, is on the graph of )(xfy ,

then yx, is also on the graph of )(xfy , i.e. if )()( xfxf . Example: 32 xy .

Odd number: A whole number that is not divisible by 2 , such as ...,5,3,1 , is

called an odd number, i.e. any number that has a remainder when divided by 2 . Notation: It is denoted by 12n , where n represents a natural number.

One-digit number: A number consisting of just one digit; 0, 1, 2, 3, 4, 5, 6,

7, 8, or 9. One-step equation or inequality: An equation or inequality that can be solved

in one step.

One-to-one function (or one-to-one mapping): A mapping, :f X Y in which

different elements of set X corresponds to different elements of set Y , i.e.

1 2f x f x , 1 2x x X , where 1 2,f x f x Y . Example: 2 1y f x x .

42

Onto function (or onto mapping): A mapping, :f X Y in which each

element of set Y is the image of at least one element in set X , i.e. f is onto iff

the range of f is equal to Y .

One-to-one correspondence: A mapping, :f X Y which is both onto and

one-to-one is called a one-to-one correspondence. Example: 3 2y f x x .

Open curve: A curve in which the end points do not meet. Example: a parabola

or a hyperbola. Note: A curve, such as a circle or an ellipse that forms a complete loop is called a closed curve. It has no end points. A simple closed curve is a closed curve that does not cross itself.

Open figure: A figure that is not closed; i.e., it does not start and end at the

same point. Open sentence: A statement that contains at least one unknown. It becomes

true or false when a quantity is substituted for the unknown (e.g., 3 + n = 5 becomes true when n = 2).

Open interval: An open interval, denoted by ba, , consists of all real numbers

x such that bxa . Note: A closed interval, denoted by ba, , consists of all

real numbers x such that bxa .

Operation: In algebra (or arithmetic), by an operation we mean a process in

which a number, quantity, expression, etc is altered or manipulated according to formal rules, notable among them are operations of addition, subtraction,

multiplication, division, and square root, which are denoted by ,,,, and

respectively. The words to describe the results of these operations are called

sum, difference, product, quotient and square root respectively. Opposites (or additive inverses): Two real numbers whose sum is zero are

called opposites. The opposites are at the same distance from zero on the real number line. Example: 3 and 3 are opposites since 0)3(3 .

Opposite of a number A number that is the same distance from zero on the

number line as the given number, but on the opposite side of zero. Opposite rays: Two collinear rays that share the same endpoint and together

form a straight line. Example: If A is between B and C , then AB and AC are

opposite rays, as shown in the following figure:

43

A

B

C

Opposite side in a right triangle: The side across from an angle. In a right

triangle the hypotenuse is opposite the right angle and each leg is opposite one of the acute angles. Example: In the following figure ABC is a right triangle.

With respect to A , BC is the opposite side, and AC is the adjacent side. With

respect to side AC , B is the opposite angle.

A

C B Order of operations: In many algebraic (or arithmetical) expressions, more than

one operation(including absolute value and square root symbols), and often,

grouping symbols such as brackets , braces , parentheses and fraction

bars are involved. In such a situation, the following rules for the order of

operations are applied. Most calculators and computers use the order of

operations. If grouping symbols are present, apply any operation inside grouping symbols, like parentheses, absolute values, and square root symbols, or simplify

within them, the innermost first (and above and below fraction bars separately), in the following order:

(i) Step 1: Simplify terms with exponents.

(ii) Step 2: Multiply and divide in the order in which they occur, working

from left to right.

(iii) Step 3: Add and subtract in the order in which they occur, working

from left to right.

Note: If no grouping symbols are present, begin with Step 1.

Ordered numbers: The real numbers that are arranged in order from smallest to

largest or from largest to smallest are called ordered numbers. Examples: (i) 71.52 , and (ii) 4512 .

Ordered pair: An ordered pair, denoted as yx, , is a pair of numbers written

44

within parentheses in which the order of the numbers is important. An ordered

pair yx, is used to locate a point in the coordinate plane or the solution of an

equation in two variables, where the first term is called the x -coordinate, and the

second term is called the y -coordinate. Note that xyyx ,, as they represent

two different points. Example: 4,3 represents a point 3 units to the left from

the origin on the x -axis and 4 units up from the origin on the y -axis of a

rectangular coordinate system.

Ordered triple: An ordered triple, denoted as zyx ,, , is a set with three

numbers written within parentheses in which the order of the numbers is

important. An ordered triple zyx ,, is used to locate a point in the three-

dimensional coordinate space or the solution of an equation in three variables, where the first term is called the x -coordinate, the second term is called the y -

coordinate, and the third term is called the z -coordinate.

Ordinate: The coordinate y of a point, in the Cartesian coordinates yx, in the

plane, which is the distance from the x -axis to the point, measured along a line parallel to the y -axis, is called the ordinate. Note that the coordinate x of a

point, in the Cartesian coordinates yx, in the plane, which is the distance from

the y -axis to the point, measured along a line parallel to the x -axis, is called the

abscissa.

Ordinal numbers: Numbers used to specify position in a sequence (e.g., first,

second, third, fourth).

Orientation: The arrangement of the points, relative to one another, after a

transformation; the reference made to the direction traversed (clockwise or counterclockwise) when traveling around a geometric figure. Example: In the following figures, ACB has a clockwise orientation in Figure 1, and ABC has

a counterclockwise orientation in Figure 2. Figure 1 Figure 2

A

B

C

A

B

C

Origin: The point in a plane at which the x -axis and y -axis of a rectangular

coordinate system intersect is called the origin. Its coordinates is given by 0,0 .

Orthocenter of a triangle: It is defined as the point of intersection of the three

altitudes of the triangle.

45

Orthogonal vectors: Two vectors are said to be orthogonal if the angle between

them is 090 .

Ounce (oz): A customary unit used to measure mass; 1 ounce = 1

16 pound;

16 ounces = 1 pound.

Outcome: One of the possible results in a probability experiment (e.g., when

tossing a fair coin there are two possible outcomes, heads or tails).

Outer product: When using the FOIL method to multiply two binomials

A B C D , the outer product is A D .

Outlier: A data value that is far removed from the body of the data. Example:

Given the data set {2, 4, 5, 16, 22, 112}, 112 is the outlier. The value of the outlier will greatly effect on the value of the mean but not the median. P

PAIRS OF ANGLES: (i) Adjacent angles: angles that share a common vertex and a

common side, but have no other intersection. NOTE: In a polygon, adjacent angles share only a common side; they are also referred to as consecutive angles.

(ii) Complementary angles: Two angles whose measures have a

sum of 90 . If they are adjacent they form a right angle.

(iii) Supplementary angles: Two angles whose measures have a

sum of 180 . If they are adjacent they form a straight angle.

(iv) Vertical angles: Two non-adjacent angles formed by

intersecting lines. Vertical angles are equal in measure. Parabola: Any plane section of a circular conical surface by a plane parallel to

the slant height of

the cone determines a plane curve, which is called a parabola, as shown in the figure below:

Definition of a Parabola: It is a plane curve which is the locus (path) of all points

in the plane equidistant from a fixed point (called the focus of the parabola) and a fixed line (called the directrix of the parabola). A common form of a parabola in

46

the rectangular Cartesian coordinates with vertical line of symmetry is given by a

quadratic function, ,2 cbxaxy where a , b , and c are real numbers and

0a . Example: The following figure represents the parabola, 2

4

1x

ay ,

i.e. yax 42 , where 0a , with Vertex 0,0 , Focus a,0 , Directrix ay , and

y -axis, i.e. 0x as the Axis of Symmetry.

Parallelepiped: A polyhedron, all of whose faces are parallelograms, is called a

parallelepiped. It is also defined as a prism whose bases are parallelograms. The faces other than the bases are called the lateral faces; the sum of their areas, the lateral area; and their intersection, the lateral edges. A diagonal of a

parallelepiped is a line segment joining two vertices which are not in the same face. There are four of these, the principal diagonals, the other diagonals being the diagonals of the faces. An altitude of a parallelepiped is the perpendicular

distance from one face (the base) to the opposite face. Note: (i) A right parallelepiped is a parallelepiped whose bases are perpendicular i ts lateral faces. It is a special type of right prism. (ii) A rectangular parallelepiped is a

right parallelepiped whose bases are rectangles. (iii) An oblique parallelepiped

is a parallelepiped whose lateral edges are oblique to its bases.

Example: Figure of a Parallelepiped

(a) Volume of a parallelepiped = Length of one altitude X Area of one

base. (b) Volume of a rectangular parallelepiped = cba , where a , b , and c

are the lengths of its edges.

47

(c) Area of the entire surface of the rectangular parallelepiped =

accbba2 .

Parallel lines: Two or more straight lines in the same plane that never intersect

no matter how far they are extended are called parallel lines. They are equidistant from each other. Parallel line segments or rays are line segments or

rays that are subsets of parallel lines. In the rectangular Cartesian coordinates, two non-vertical straight lines are said to be parallel if and only if their slopes are equal and they have different y -intercepts. Symbol: „||‟. Example: The following

straight lines 12xy and 342 xy are parallel as they have the same

slope, i.e. 2m .

PARALLEL LINES CROSSED BY ONE TRANSVERSAL: When two parallel

lines are crossed by a transversal, two groups of four angles are formed as

shown in the following figure.

a) Pairs of corresponding angles are 1 and 5, 2 and 6, 3 and

7, 4 and 8.

b) Pairs of alternate interior angles are 4 and 6; 3 and 5.

c) NOTE: If corresponding angles or alternate interior angles are not

equal, then the lines are not parallel. d) If the lines are parallel, then

all acute angles are equal;

all obtuse angles are equal; any acute angle and any obtuse angle are supplementary.

Parallel planes: Two or more planes that do not intersect.

Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.

Example: Figure of a Parallelogram

48

A

B

C

D

||AB CD and ||AD BC

Parameter: A quantity or constant whose value varies in different situations of its

applications. Example: In the point-slope equation of a straight line, y m x b , m and b

represent parameters.

Pascal's triangle: Let x and a be real numbers. According to the Binomial

Theorem, for any positive integer n , the expansion of the expression nax )( is

given by n

j

jjnnjjnnnn axj

na

n

nax

j

nax

nx

nax

010)( ,

where, for integers j and n with nj0 , the symbol !!

!

jnj

nC

j

n def

jn is

called the binomial coefficient. A triangular array of the binomial coefficients in

the above expansion of nax )( is called the Pascal triangle, named after the

French mathematician, Blaise Pascal (1623 – 1662), as provided in the following

figure. Pascal's triangle

Note that in the above figure, each row begins and ends with 1. The other values

are obtained by adding the two numbers that are in the row above and on either side of that value. Example: The binomial coefficients in the expansion of

4)( ax are the numbers in the 4th row of the above Pascal‟s triangle, given by

4322344 14641)( aaxaxaxxax .

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10

10 5 1

etc

49

Pattern: A design (geometric) or sequence (numeric or algebraic) that is

predictable because some aspect of it repeats. The following are some examples of pattern.

(i) Geometric pattern:

(ii) Numeric pattern: ,20,15,10,5 .

(iii) Algebraic pattern: ,,, 1284 aaa .

Pedal triangle: The triangle formed within a given triangle by joining the feet of

the perpendiculars from any given point to the sides. Example: The triangle

formed within a given triangle by joining the feet of the three altitudes of the triangle.

Penny: A coin with a value of one cent or 1

100 of an U.S. dollar.

Pentagon: A polygon with five sides and five angles. Note: A regular pentagon

is a pentagon whose angles are all equal and whose sides are all equal. Examples of pentagon:

Percent (or Per Cent or Percentage or Hundredths): It means per hundred, or

parts per hundred, i.e., a number expressed as a fraction of one hundred (100). It is denoted by the symbol %. For example, 7 percent (or 7 %) is equal to 7/100, i.e., 7 parts out of 100 is 7 %. Note that:

a) Any fraction or decimal can be expressed as a percentage by

multiplying it by 100. For example, 0.72 X 100 = 72 % and (1/5) X 100 = 20 %.

b) In Finance, rate in hundredths, i.e. the rate per cent which gives a certain profit is called Rate Percent (or Yield).

Percentile: A score below which a certain percentage of the total number of

50

scores in a distribution falls. Example: If a test score of 76 is the 87th percentile of a distribution, then 87 % of the scores are less than 76 and 13 % of the scores

are greater than or equal to 76. Percent decrease or increase:

A. Percent decrease: The magnitude of decrease expressed as a percent of

the original quantity. That is, when the value of something changes from x to y, the percent decrease is [100 (x – y)] / x if y < x. E.g., if the price of oil

changes from $ 4.00 to $ 3.95 per gallon, the percent decrease is [100 (4.00 – 3.95)] / 4, i.e., 1.25 %.

B. Percent increase: The magnitude of increase as a percent of the original

quantity. That is, when the value of something changes from x to y, the

percent decrease is [100 (y – x)] / x if y > x. E.g., if the price of oil changes from $ 3.959 to $ 3.989 per gallon, the percent increase is [100 (3.989 – 3.959)] / 3.959, i.e., 0.76 %.

Percent error: It is defined as follows:

(i) Error: The difference between a number and the number it

approximates, i.e., the difference XAE , where A is an

approximation of X .

(ii) Relative error: It is defined as X

E, or sometimes

X

E.

(iii) Percent error: It is defined as the relative error expressed in

percent, i.e., X

E100 .

(iv) Example: If a weight of 10 lb. is measured as 10.2 lb., then the

error is 0.2, the relative error is 0.02, and the percent error is 2

%.

Percent profit on cost (or percent gain): If S denotes the selling price and C

the cost of an article, then 100C

CS is defined as the percent profit on cost (or

percent gain). Example: If an article costs $10 and sells for $12, then the percent

gain is 1005

1, or 20 %.

Percent profit on selling price: If S denotes the selling price and C the cost of

an article, then 100S

CS is defined as the percent profit on selling price (or

percent gain on selling price). Example: If an article costs $10 and sells for $12,

51

then the percent gain on the selling price is 1006

1, or 67.16 %. Note: The

percent gain on the cost price is always greater than the percent gain on the

selling price.

Perfect cube: It is defined as a number or polynomial that is an exact third power

(or cube) of some number or polynomial. Example: 328 , 3327 , and 33223 33 yxyyxyxx .

Perfect number: An integer which is equal to the sum of all of its factors except itself. Example: 28 is a perfect number, since 14742128 . Note: (i) If

the sum of the factors of a number (except itself) is less than the number, then the number is said to be defective (or deficient). (ii) If the sum of the factors of a

number (except itself) is greater than the number, then the number is said to be abundant.

Perfect power: It is defined as a number or polynomial that is an exact nth power of some number or polynomial for some positive integer 1n .

Perfect square: It is defined as a number or polynomial that is an exact second

power (or square) of some number or polynomial. Example: 224 , 2416 , and 222 2 yxyyxx . Alternatively, a positive integer resulting from

multiplying an integer by itself, i.e. if a is a positive integer such that nna ,

where n is an integer, then a is called a perfect square. Example: 16 = 4 4,

and 121 = (–11) (–11).

Perfect trinomial square: It is defined as a trinomial which can be expressed (or

factored) as the square of a binomial. Example: 222 2 yxyyxx ;

22 244 xxx .

Perimeter: The distance around a closed figure (i.e., the length of a closed

curve). Example: The perimeter of a circle, the perimeter of an ellipse, or the sum of the lengths of the sides of a polygon.

Perimeter of a polygon: The sum of the lengths of all the sides of the polygon.

Periodic function: A function y f is called periodic if there exists some

real constant p such that ( ) ( )f f p for every number in the domain of

y f . In other words, an oscillating function that repeats its values at regular

intervals is called a periodic function.

52

Period of a function: It is defined as the horizontal distance after which the

graph of a function y f starts repeating itself. Thus, in the above definition

of a periodic function, the smallest value of the number p such that

( ) ( )f f p for every number in the domain of y f , is called the

(fundamental) period of the function y f . Example: In Trigonometry, the

sine and cosine functions are periodic having period 2 , i.e.

sin 2 sin , cos 2 cos , int .k and k where k is any eger

Periodicity (or periodicity of a function or a curve): The property of having

periods or being periodic. Permutation: An ordered arrangement or sequence of all or part of a set of

objects (or things). The following are some examples of permutation:

(i) All possible permutations of the letters candba ,, are:

abcbacacbcabbcacbabcaccbabcabacba ,,,,,,,,,,,,,, .

(ii) A permutation of n different objects taken all at a time is n -factorial, i.e., !n , where 0n is an integer. Note that

1.2.3)2()1(! nnnn ; & 1!0 .

(iii) A permutation of n different objects taken r at a time without

repetitions is given by )1()2()1(!)(

!rnnnn

rn

nPrn ,

where nr . Example: The number of permutations of the letters

candba ,, taken 2 at a time without repetitions is 623 P . These

permutations are: bcaccbabcaba ,,,,, . are: bcaccbabcaba ,,,,, .

(iv) A permutation of n different objects taken r at a time with

repetitions is given by rn . Example: The number of permutations

of the letters candba ,, taken 2 at a time with repetitions is 932 .

These permutations are: cccbaccbbbabcabaaa ,,,,,,,, .

(v) A circular permutation is an arrangement of objects (or things)

around a circle. Example: The total number of circular

permutations of n different objects taken n at a time is n

n!,

because each arrangement will be exactly like )1(n others except

for a shift of the places around the circle.

Perpendicular: At right angles. Symbol: „ ‟. Thus, two lines, segments, or rays

that intersect to form right angles are said to be perpendicular, as shown in the following figure:

53

Perpendicular bisector: A line, segment, or ray that is perpendicular to a line

segment at its midpoint, i .e., bisects it. Also, for a line segment in space, the perpendicular bisector is the plane perpendicular to the line segment at its midpoint. Note: In either case, the perpendicular bisector is the set of all points

equidistant from the end points of the line segment. Perpendicular lines: When two straight lines intersect to form right angles, they

are said to be perpendicular to each other. In the rectangular Cartesian

coordinates, two straight lines are said to be perpendicular if and only if the product of their slopes is equal to 1 . Example: The following straight lines

12

1xy and 342 xy are perpendicular to each other.

Perpendicular planes: Two planes are said to be perpendicular to each other if

a line in one plane, that is perpendicular to their line of intersection, is perpendicular to the other plane. Phenomena: Something that is observable. The following are some examples

of phenomena:

(i) Mathematical Phenomena: Problems related purely to

mathematics, e.g. (negative number) (negative number) = positive number.

(ii) Physical Phenomena: Problems in the physical world that

involve math,

e.g. acceleration due to gravity. (iii) Social phenomena: Problems in the real world relating

mathematics and social concepts or events, e.g. sharing 6

cookies among 3 friends.

Physical model: A representation of something using objects.

Pi ( π ): The symbol π (a Greek letter) is used to denote the ratio of the

circumference of a circle to its diameter. Its approximate value is 3.141592654

54

upto nine decimal places. Note: In 1770, Lambert proved that π is an irrational

number. Lindemann proved in 1882 that π is a transcendental number.

Piecewise-defined function: A function which is defined by more than one

equation is called a piecewise-defined function. Example: The absolute value function given by

0

0)(

xifx

xifxxxfy

is an example of a piecewise-defined function.

Pint (pt): A customary unit used to measure capacity; 2 cups = 1 pint; 2 pints = 1

quart. Place value (or local value): The value given to a digit by virtue of the place it

occupies in the number relative to the units place, i.e., the value of a digit in a number based on its position. Example: (i) In the number 28, the 2 is in the tens

place and the 8 is in the ones place. (ii) In 534.6, 4 denotes merely 4 units, 3 denotes 30 units, 5 denotes 500 units, and 6 denotes 6/10 of a unit. We also say that 4 is in unit‟s place, 3 is in ten‟s place, 5 is in hundred‟s place, and 6 is in

tenth‟s place, etc.

Plane: (i) A set of points forming a flat surface that extends without end in all directions. (ii) An undefined term in geometry usually visualized as a flat surface with no thickness that extends indefinitely in two dimensions. (iii) A flat surface

such that a straight line joining any two of its points lies entirely in the surface. Plane curve (figure, surface, etc): A curve (figure, surface, etc) lying entirely in

a plane. Plane figure: A figure that lies on a flat surface; it has length, width, perimeter,

and area. Example: triangle, circle, etc. Plane section: The intersection of a plane and a surface or a solid.

Platonic solids: The five regular polyhedra: tetrahedron, cube, octahedron,

dodecahedron and icosahedron, as shown in the figures below.

55

Tetrahedron Cube Octahedron Dodecahedron

Icosahedron Plot: (i) Plot a point: To locate a point geometrically, either in the plane or in

space, when its coordinates are given in some coordinate system. For example, to plot a point represented by an ordered pair such as ),( yx is to locate it on a

rectangular coordinate plane by standard techniques. (ii) Point-by-point plotting (or graphing) of a curve: Finding an ordered set of points which lie on

a curve and drawing through these points a curve which is assumed to resemble

the required curve.

Plotting ordered pairs on a graph: An ordered pair is simply two numbers x and y in a certain order, and is denoted as ),( yx . A very important application of

ordered pairs is to locate points on a grid or map. When an ordered pair is used

to locate a point on a grid, the two numbers are called the coordinates of the point. On a graph grid, the point (0, 0) is called the origin. The first coordinate of

a plotted point is called the x coordinate. The x coordinate is the horizontal

distance from the origin to the plotted point. The second coordinate of a plotted point is called the y coordinate. The y coordinate is the vertical distance from

the origin to the plotted point. When locating points, positive x values are to the

right of the origin, while negative x values are to the left of the origin. Also, positive y values are above the origin, while negative y values are below the

origin. Example: The numbers 2 and 3 form two different ordered pairs: )3,2(

and )2,3( . In order to locate the point )3,2( on a graph grid, we start at the

origin, move 2 units horizontally to the right and then 3 units vertically up, as shown in the figure below. The point )3,2( has been marked with a red dot. The

x and y coordinates of this point are 2 and 3 , respectively.

56

Plotting polar coordinates on a graph: Polar coordinates locate a point, say,

P on a plane with a distance r measured from the origin )0,0(O to the point P

and an angle measured from the positive part of the x -axis as the starting

point for measuring angles to the point P . The polar coordinates of the point P is denoted as an ordered pair, i.e., ),(rP . The x -axis is called the polar axis

(initial line). The line OP is called the terminal line. The origin O is called the

pole. When the angle is measured anticlockwise from the positive x -axis, it

is considered as positive, and when is measured clockwise from the positive x -axis, it is considered as negative. Example: Consider a point in polar

coordinates )50,4( 0 . In order to locate the point, we measure the distance to a

point 4 units from the origin along the positive x -axis , and then rotate the point

anticlockwise about the origin through angle 050 , as shown in the figure given

below.

Polar- rectangular coordinates conversion: Rectangular coordinates and

polar coordinates are two different ways of using two numbers to locate a point on a plane. Rectangular coordinates are expressed in the form ),( yx , where x

and y are the horizontal and vertical distances from the origin. Polar coordinates

are expressed in the form ),(r , where r is the distance from the origin to the

point, and is the angle measured from the positive x -axis to the point. To

convert between polar and rectangular coordinates, we make a right triangle to the point ),( yx as shown in the following figure.

57

Polar to rectangular: From the figure above, we obtain the formulas to

convert polar coordinates ),(r of a point to its rectangular coordinates

),( yx given by cosrx , sinry .

Rectangular to polar: From the figure above, we obtain the formulas to

convert rectangular coordinates ),( yx of a point to its polar coordinates

),(r given by

22 yxr

and

IVQuadrantinisyxpothewhenx

y

IIIQuadrantinisyxpothewhenx

y

IIQuadrantinisyxpothewhenx

y

IQuadrantinisyxpothewhenx

y

),(inttan

),(inttan

),(inttan

),(inttan

1

1

1

1

,

where 0x .

Plus: A term that refers to addition or the symbol for addition.

Plus sign: The symbol (+) used to indicate addition.

Point: (i) An undefined term in geometry usually visualized as a dot representing

a non-dimensional location in space. According to Euclid, it is that which has position but no nonzero dimensions. (ii) An element of geometry defined by its coordinates, such as the point )5,2( .

58

Point of concurrency: A point that is the intersection of three or more lines.

Point of division of a line segment (Section-formula): It is defined as the point

which divides

the line segment joining two given points in a given ratio. In the Cartesian (rectangular) plane, the

coordinates of a point P that divides a line segment AB in a given ratio 21 :rr ,

both internal and externally, are given by the following formulas. Let the

coordinates of the point P be yx, . Let the coordinates of the end-points A and

B of the line segment AB be 11 , yx and

22 , yx respectively.

Internal division of a line segment: If the point yxP , lies between the

points 11 , yxA and

22 , yxB on the line segment AB , it divides AB

internally in a given ratio, say, 21 :rr , such that

2

____

1

____

, rBPandrAP . Then,

the coordinates of the point yxP , are given by

21

2112

21

2112 ,rr

yryry

rr

xrxrx .

External division of a line segment: If the point yxP , lies outside of

the line segment AB , it is said to divide AB externally in a given ratio,

say, 21 :rr , such that 2

____

1

____

, rBPandrAP . Then, the coordinates of the

point yxP , are given by

21

1221

21

1221 ,rr

yryry

rr

xrxrx .

Mid-point formula: If the point yxP , is the middle point of the line

segment AB , with the points as 11 , yxA and 22 , yxB , it bisects line

segment AB , i.e., 21 rr . Then, the coordinates of the point yxP , are

given by

2

,2

2121 yyy

xxx .

Point of tangency: A point where a straight line is tangent to a curve.

Point-slope form: It is defined by the equation of a straight line by using its

slope and the coordinates of a point on the line, and expressed as

1 1y y m x x , where m is the slope of the line and 1, 1( )x y are the

59

coordinates of a point on the line. Example: 2 4 3y x .

Point symmetry: (i) A graph is said to be symmetric with respect to the x -axis if, for every point ( , )x y on the graph, the point ( , )x y is also on the graph. (ii) A

graph is said to be symmetric with respect to the y -axis if, for every point ( , )x y

on the graph, the point ( , )x y is also on the graph. (iii) A graph is said to be

symmetric with respect to the origin if, for every point ( , )x y on the graph, the

point ( , )x y is also on the graph.

Point symmetry of a geometric figure: A geometric figure has point symmetry

if every point on the figure is the image of itself under a rotation of 180° about

some fixed point. For examples, as shown in Figure 1 and Figure 2 below: Figure 1 Figure 2

P

P'

P

P'

P

P'

P

P'

A regular hexagon has point A pentagon does not have

symmetry about its center. point symmetry

Poll: The results of a question or questions answered by a group of people.

Polygon: (i) A plane figure consisting of n points, 1, 2 3, , , np p p p (the vertices),

3n is a positive integer, and of the line segments 1 2 2 3 1 1, , , ,n n np p p p p p p p ,

(the sides). (ii) In elementary geometry, it is defined as a closed plane figure

formed by three or more line segments (the sides) that have no common point

except their end points. Examples: Some well-known polygons are given in the following table and figures. Note that a triangle is a 3-sided polygon, a quadrilateral is a 4-sided polygon, etc. A TABLE OF SOME POLYGONS

NUMBER OF SIDES

NAME

3 Triangle

4 Quadri lateral 5 Pentagon

6 Hexagon

60

7 Heptagon 8 Octagon

9 Nonagon 10 Decagon

12 Dodecagon 15 Penta-decagon

FIGURES OF SOME COMMON POLYGONS

Properties of Polygons:

(i) AN INTERIOR ANGLE OF A POLYGON is formed by adjacent

sides of the polygon and lying within the polygon. (ii) AN EXTERIOR ANGLE OF A POLYGON is formed by one side

and the extension of the other side at that vertex. (iii) The exterior angle and the interior angle at any vertex of any

polygon are supplementary.

(iv) The exterior angle of a triangle is equal to the sum of the two remote interior angles.

(v) A polygon is equiangular if its interior angles are congruent. (vi) A polygon is equilateral if its sides are congruent. (vii) Note: A triangle is equiangular if and only if it is equilateral, but this

is not true for polygons of more than three sides. (viii) A polygon is regular if its sides are congruent and its interior

angles are congruent. (ix) Polygons having their corresponding angles equal and their

corresponding sides proportional are called similar polygons.

(x) A line segment joining any two nonadjacent vertices of a polygon is called a diagonal of the polygon.

(xi) A polygon is convex if each interior angle is less than or equal to 0180 .

61

(xii) A polygon is concave if at least one of its interior angles is greater

than 0180 .

(xiii) Summary of Angle Measures for Polygons (in degrees):

Sum of all (any polygon)

Measure of one

(regular

polygon)

Interior angle (n - 2) 180

180 - 360

n

Exterior

angle

360

360

n

(xiv) QUADRILATERALS:

(xv) TRIANGLES:

The sum of the interior angles of a triangle is 180 .

Since all triangles have at least 2 acute angles, a triangle may be classified by the nature of its third angle, that is:

an acute triangle if the third angle is also acute.

an obtuse triangle if the third angle is obtuse.

a right triangle if the third angle is a right angle.

Triangles may be classified by sides: Scalene -- no sides are equal (and no angles are

equal).

62

Isosceles -- two sides are equal (and the angles

opposite the equal sides are equal).

Equilateral -- all three sides are equal (each angle

measures 600; thus, the triangle is also equiangular).

The shortest side of a triangle is opposite the smallest angle; the longest side is opposite the longest side.

(xvi) PYTHAGOREAN THEOREM (applies to right triangles): The

square of the hypotenuse equals the sum of the squares of the other two sides.

H2 = S12 + S2

2 or a2 + b2 = c2 (where c is the hypotenuse).

(xvii) SIMILAR TRIANGLES:

When two triangles are similar, the measures of corresponding angles are equal, and the lengths of corresponding sides are in

proportion. That is, if ABC PQR, then:

(i) A = P, B = Q and C = R, and

(ii) PR

AC

QR

BC

PQ

AB

Two triangles will be similar if: (i) two angles of one triangle are equal to two angles of

the other triangle, or (ii) three pairs of corresponding sides are proportional,

or (iii) two pairs of sides are proportional and the included

angles are equal.

If the triangles are similar and corresponding sides are equal, the triangles are congruent.

(xviii) AREA AND PERIMETER FORMULAS OF SOME WELL-KNOWN

POLYGONS:

(Note: Area measures the amount of surface enclosed in a region

and is measured in square units. E.g., sq. in. = in2 ; sq. cm. = cm2 ;

9 ft2 = 1 yd2.)

a) Rectangle: A = l w = (length . width);

l P = 2L + 2W

w

b) Square: A = s2 = (side squared);

s P = 4s

c) Parallelogram: A = b h = (base . height); a h P = 2a + 2b

63

b

d) Triangle: A = ½ b . h = (1/2 . base . height);

a h c P = a + b + c

b

e) Trapezoid:

A = 1/2 . h . (a + b) = (1/2 . height . sum of bases);

P = a + b + c + d Polyhedron: (i) A solid figure bounded by plane polygons. (ii) A three-

dimensional figure that is bounded by four or more polygonal faces. Note: The bounding polygons are called the faces; the intersections of the faces are called

the edges; and the points where three or more edges intersect are called the vertices. Some examples of polyhedron are shown in the figures below.

Tetrahedron Cube Octahedron Dodecahedron

Icosahedron (Hexahedron) (4 faces) (6 faces) (8 faces) (12 faces)

(20 faces)

Polynomial: (i) A monomial or sum of monomials. (ii) A polynomial is a term or a

finite sum of terms in which all coefficients are real, all variables have whole number exponents, and no variables appear in denominators. Example:

2 4 3x x ; 3 25 3 10x x x .

64

Polynomial in x (or Polynomial function or polynomial expression in x ): A

polynomial in one variable x (or a polynomial expression or a polynomial

function in one variable x ) of degree n is a rational integral algebraic expression

of the form 1 2

1 2 1 0...n n

n nf x a x a x a x a x a , where n is a nonnegative

integer and 1 2 1 0, , ..., , ,n na a a a a are real numbers. Note: The domain of a

polynomial is the set of all real numbers. Example: 2( ) 4 3f x x x ; and 3 2( ) 5 3 10g x x x x are polynomials (or polynomial functions) in x of

degrees 2 and 3 respectively.

A TABLE OF SOME WELL-KNOWN POLYNOMIAL FUNCTIONS (OR

POLYNOMIALS)

Degree

Form Name Graph

No

degree

( ) 0f x Zero

function

The x -axis

0 0 0( ) , 0f x a a Constant

function

Horizontal line with y -

intercept 0a

1 1 0 1( ) , 0f x a x a a Linear

function Nonvertical,

nonhorizontal straight

line with slope 1a and y -

intercept 0a

2 2

2 1 0 2( ) , 0f x a x a x a a Quadratic function

Parabola: Graph opens

up if 2 0a ; graph

opens down if 2 0a

3 3 2

3 2 1 0 3( ) , 0f x a x a x a x a a Cubic

4 4 3 2

4 3 2 1 0 4( ) , 0f x a x a x a x a x a a

Quartic (or

biquadratic)

Example: 4( )f x x

5 5 4 3 2

5 4 3 2 1 0

5

( ) ,

0

f x a x a x a x a x a x a

a

Quintic Example: 5( )f x x

Polynomial equation in x : A polynomial in one variable x , set equal to zero,

i.e., 1 2

1 2 1 0... 0n n

n nf x a x a x a x a x a , where n is a nonnegative

integer and 1 2 1 0, , ..., , ,n na a a a a are real numbers, is called a polynomial equation

in x of degree n . A polynomial equation in x is linear, quadratic, cubic, quartic (or biquadratic), or quintic, according as its degree is 1, 2, 3, 4, 5or .

Note: A root (or solution) of a polynomial equation in one variable x is a value

of x which reduces the polynomial equation to a true equality.

Example: 2 4 3 0x x , and 3 25 3 10 0x x x are polynomial equations in

x of degrees 2 and 3 respectively.

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Positive angle: An angle formed by counter clockwise rotation about its

endpoint. Post meridiem (p.m.): After noon, used after times of day between 12 noon and

12 midnight. Example: 2:30 p.m. Positive number: Any number greater than zero or to the right of zero on the

number line.

Position vector: A coordinate vector whose initial point is the origin. Any vector

can be expressed as an equivalent position vector by translating the vector so

that it originates at the origin.

Postulate: (i) A statement assumed to be true without proof. (ii) An accepted

statement of fact that can be used to prove theorems. Example: Two points define exactly one straight line.

Pound (lb): A customary unit used to measure mass. Example: 1 pound = 16

ounces. Power: An exponent. Example: (i) In the expression 38, 8 is the power and 3 is

the base. (ii) By power of ten, it means power with a base of ten.

Power function: A power function of degree n is a function of the form nf x a x , where a is a real number, 0a , and 0n is an integer. Example:

2( )f x x ; and 3( ) 5g x x are power functions in x of degrees 2 and 3

respectively.

Powers of imaginary unit i : The repetitive pattern when the imaginary unit,

1i , is raised to sequential powers. Example: ,12i ,3 ii ,14i ,5 ii

,16i ,7 ii etc.

Precise: Exact in measuring; accurate.

Precision: A property of measurement that is related to the unit of measure

used; the smaller the unit of measure used, the more precise the measurement is. Example: 19 mm is more precise than 2 cm. Predict: To be able to determine the next step or value (to make an educated

guess), based on evidence or a pattern. Prediction: An educated guess about an outcome.

66

Pre-image: In transformational geometry, the figure before a transformation is

applied. Example: If /

lR F F( ) = denotes a transformation of a figure F in to

another figure /F , then F is the pre-image of /F . Premise: A proposition upon which an argument is based or from which a

conclusion is drawn. Preserved: In transformational geometry, a property that is kept or maintained.

Example: In a translation, the shape and size (property of congruence) is preserved. Prime factorization: A method of expressing a composite number as a product

of its prime factors. Example: 12 = 2 2 3 = 22 3.

Prime number (or prime): A number greater than 1 that has exactly 1 and itself

as factors. The following table gives some examples of well-known prime

numbers.

A TABLE OF SOME WELL-KNOWN PRIME NUMBERS

Prime Numbers Non-Prime Numbers

Number Factor

s

Number Factors

2 1, 2 6 1, 2, 3, 6

7 1, 7 8 1, 2, 4, 8

11 1, 11 15 1, 3, 5, 15

17 1, 17 25 1, 5, 25

Prime polynomial: A polynomial which has no polynomial factors except itself

and constants. Example: The polynomials 1x and 2 1x x are prime.

Principal root (Principal nth root): The principal nth root (or principal root) of a

number a is denoted by n a , and is defined as the positive real root in the case

of roots of positive a , and the negative real root in the case of odd roots of

negative a . Example: (i) 24 , 2 is the principal square root of 4 . (ii)

283 , 2 is the principal cube root of 8 .

Principal square root: The positive square root of a positive number a is called

its principal square root, and is denoted by a . Example: The principal square

root of 4 is 2.

Note: (i) 0, aa , is called negative square root of a . Example: The negative

square root of 16 is 4, written as 416 . (ii) For nonnegative a ,

aaaa2

. , and aaaa2

. . (iii) For any real number a ,

67

aa2 . (iv) For any nonperfect square positive real number a , a is an

irrational number. (v) For any negative real number a , a is not a real number.

(vi) 00 .

Principal square root of a negative number: If N is a positive real number,

then the principal square root of N , denoted by N , is defined as

iNN , where 1i is the imaginary unit and 12i ..

Prism: A three-dimensional figure (solid) that has two congruent and parallel

faces that are polygons; these are the bases; the remaining faces are parallelograms. In the following figures, some well-known prisms are shown.

FIGURES OF SOME WELL-KNOWN PRISMS

Probability of an event:

(i) The chance of an event occurring. (ii) The ratio of the number of favorable outcomes to the total number of possible outcomes. Note: The probability of an event must be greater than or equal to 0 and less than or

equal to 1. Example:

( 3 )P rolling a when a die is rolled

the number of 3's on the faces

the total number of faces

1

6

Problem solving strategies: Various methods used to solve word problems.

E.g., strategies may include, but are not limited to: acting it out, drawing a picture or graph, using logical reasoning, looking for a pattern, using a process of

elimination, creating an organized chart or list, solving a simpler but related problem, using trial and error (guess and check), working backwards, writing an

equation. Process of finding the coordinate of a point located somewhere between

two given points: Suppose we want to find the coordinate x of a point located

b

a of the distance from a given point with coordinate 1x to another point with

coordinate 2x . Since the total distance from 1x to 2x is

68

12 xx units, we start at 1x and move

12. xxb

a toward

2x . Thus,

121 . xxb

axx . Example: The coordinate x of a point located

one - fourth, i.e., 4

1 of the distance from a point with coordinate 4 to another

point with coordinate 16 is given by 6416.4

13x .

Product: The number that is obtained when two or more factors are multiplied. Example: In (4)(5) 20 , 20 is called the product, and 4 5and are called the

factors. Product of the Sum and Difference: The product of the sum and difference of

two terms is the difference of the squares of the terms, i.e., 2 2( ) ( ) ( )x y x y x y . Example: 2( 5) ( 5) ( 25)x x x .

Profit: The amount of money left after expenses have been subtracted from

income.

Proof : (i) A logical argument that establishes the truth of a statement. (ii) A valid

argument, expressed in written form, justified by axioms, definitions, and

theorems. Proof by contradiction: A method of proof which demonstrates the truth of an

implication by proving that the negation of the conclusion of that implication leads to a contradiction is called a proof by contradiction, or an indirect proof.

Proper fraction: A fraction whose numerator is less than its denominator.

Properties: Characteristics of a shape or object (e.g., size, shape, number of

faces, or ability to be stacked or rolled). Properties of real numbers: Rules that apply to the operations with real

numbers. The following table gives some properties of real numbers.

A TABLE OF SOME WELL-KNOWN PROPERTIES OF REAL NUMBERS

Property With respect to

addition (+)

With respect to

multiplication (.)

Commutative Property

a + b = b + a a.b = b.a

Associative Property

a + (b + c) = (a + b) + c a.(b.c) = (a.b).c

Identity a + 0 = a a 1 = a 11a

a

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Property Inverse

Property

a + (–a) = 0 11a.

a

Zero Property 0 0a.

Distributive Property: a.(b + c) = a.b + a.c

Proportion: An equation which states that two ratios are equivalent. Example:

5 1

10 2, or

5:10 = 1:2).

Product property of proportions: In a proportion a c

b d, the product of the

means (b and c) equals the product of the extremes (a and d), or in other words: b • c = a • d.

Proportional reasoning: Using the concept of proportions when analyzing and

solving a mathematical situation. Example: If triangle ABC is similar to triangle XYZ and AB = 15 when side XY = 75, find BC when YZ = 150.

Proportionality: The quality, character, or fact of being proportional.

Protractor: An instrument used to find the degree measure of an angle. Pyramid: A polyhedron having a polygonal base and triangles as lateral faces.

For example, a pyramid whose base is a pentagon is called a pentagonal pyramid. Example: In the following figures, some well-known pyramids are

shown. Figure 1 Figure 2

A right square pyramid A hexagonal pyramid

X

Y

Z

75 150

A C

B

15 a

15

75 150

a

a = 30

70

Pythagorean formula (or identities or theorem): It states that that in any right

triangle the sum of the squares of the two legs is equal to the square of the

hypotenuse, i.e., if a and b are the lengths of the legs and c is the length of the hypotenuse, then a2 + b2 = c2.

Q

Quadrants: The x- and y- axes divide the coordinate plane into four regions.

These regions are called the quadrants.

The figure below shows the four quadrants of the coordinate plane.

Quadratic Function: Quadratic function is a function that can be described by

an equation of the form cbxaxxf 2 , where a 0

In a quadratic function, the greatest power of the variable is 2. The graph of a quadratic function is a parabola.

The following are few examples of quadratic functions.

235

2 2 xxxf ; 32 2xxg

Quadratic formula: A quadratic formula is the solution of a quadratic

equation cbxaxxf 2 , where a 0 given bya

acbbx

2

42

.

Quadratic inequality: An inequality written in one of the forms ,

, , or is called a quadratic inequality.

Quadratic term: A term is the quadratic term in the equation

Quadrilateral: A quadrilateral is a four-sided polygon.

More about Quadrilaterals:

71

Parallelogram: A quadrilateral with two pairs of opposite sides parallel. Rhombus: A parallelogram with four congruent sides.

Rectangle: A parallelogram with four right angles. Square: A parallelogram with four congruent sides and four right angles.

Trapezoid: A quadrilateral with only one pair of parallel sides.

R

Radical Expression: A radical expression is an expression containing a square

root.

are examples of radical expression.

More about Radical Expression:

Radical: The symbol that is used to denote square root or nth roots. Radicand: Radicand is a number or expression inside the radical symbol.

For example, 5 is the radicand in .

Radical equation: An equation containing radical expressions with

variables in the radicands. Radical inequality: An inequality containing a radical expression with the

variable in the radicand.

Radius: It is the distance from the center of a circle or a sphere to any point on

the circle or a sphere. In other words, radius is a line segment joining the center

of a circle with any point on the circle. Example: is a radius of the circle shown

in the following figure.

Range of a Function: The set of all output values or the y-values of a function or

a relation is called the range of the function or the relation.

72

Examples of Range of a Function:

The table shows some of the input and the corresponding output values of the function y = x + 1.

The range of the function is the set of all real numbers {. . . 1, 2, 3, 4, 5, 6, 7, . . . .}.

Rate: Rate is a ratio that compares two quantities of different units.

Examples of Rate:

20 oz of juice for $4, miles per hour, cost per pound etc. are examples of rate.

More about Rate:

Unit rate: Unit rate is a rate in which the second term is 1.

For example, Jake types 10 words in 5 seconds.

Jake‟s unit rate is the number of words he can type in a second. His unit rate is 2 words per second.

Ratio: A ratio is a comparison of two numbers by division.

Examples of Ratio:

4 : 7, 1 : 6, 10 : 3 etc. are examples of ratio.

Any ratio a : b can also be written as „a to b‟ or .

Rational Equation: A rational equation is an equation containing rational

expressions.

More about Rational Equation:

Rational Expression: A rational expression is an expression of the

form where P and Q are nonzero polynomials.

73

Rational Function: A function written, as a quotient of polynomials is a

rational function.

That is, if p(x) and q(x) are polynomial functions and , then is

called rational function.

Rationalizing the Denominator: It is the process by which a fraction containing

radicals in the denominator is rewritten to have only rational numbers in the

denominator.

Examples of Rationalizing the Denominator

In the example above, the radical in the denominator is converted into a

rational number by multiplying and dividing the fraction by using the fact that

. This process is called „rationalizing the denominator.‟

Here in the denominator is converted into a rational number by

multiplying and dividing the fraction by the conjugate using the fact that

.

Rational Numbers: Fractions, repeating and terminating decimals. All natural

numbers, whole numbers, and integers are rationales, but not a ll rational numbers are natural numbers, whole numbers, or integers. Examples:

25.0,33333.1,9

8,

4

3,

1

5. (See Classification of numbers also).

Real Numbers: A number which can be plotted on a straight line is called a real

number. All the rational numbers and all the irrational numbers together form the

real numbers. Examples: 25.0,33333.1,1

5, 2 1.41421..., -1, 0, 1.

(See Classification of numbers also).

Reciprocal: If the product of two numbers is 1, then the two numbers are said to

be reciprocals of each other. In other words, a reciprocal is the multiplicative

inverse of a number. The reciprocal of „a‟ is .

74

Examples of Reciprocal:

Consider .

9 is the reciprocal of

is the reciprocal of 9.

The following are few examples of reciprocals.

Repeating Decimal: Repeating decimals, also called recurring decimals are

decimals in which a digit or a sequence of digits keeps repeating.

Examples of Repeating Decimal:

The following are examples of repeating decimals.

0.23232323 ..., 0.601260126012 ..., 0.11111111 ..., 2.22222222 ... . The symbol for a repeating decimal is a bar over the digit or the sequence of

digits that repeat.

So, 2.22222222. . . can be written as and 0.601260126012. . . . can be

written as .

Right Angle: Right angle is an angle that has a measure of 90°.

Examples of Right Angle:

The figure below shows a right angle.

Roots of a Quadratic Equation: The solutions of a quadratic equation are

called the roots of a quadratic equation.

Example on Roots of a Quadratic Equation:

75

The solutions or the roots of the quadratic equation are 2 and

- 1.

Solved Example on Roots of a Quadratic Equation:

The graph represents the equation . Find the roots of the quadratic

equation .

S

Scalar: A quantity that has size but no direction.

Scalene triangle: A triangle with three unequal sides.

Secant: A line that intersects a circle or a curve in two places.

Secant: The reciprocal of the cosine.

Second: The unit of measure of an angle that is 1/60 of a minute.

Sector: A region bounded by two radii of a circle and the arc whose endpoints lie

on those radii.

Segment: The union of a point, A, and a point, B, and all the points between

them.

Series: The sum of a sequence.

Set: A well defined group of objects.

Similar: Two polygons are similar if their corresponding sides are proportional.

Simultaneous equations: A group of equations that are all true at the same

time.

Sine: In a right triangle, the length of a side opposite an angle divided by the

length of the hypotenuse of the triangle.

76

Slope: The slope of a line is the change in the vertical coordinates/the change in

the horizontal coordinates of any two points on the line .

Slope of a Line

To find the slope of a line we have to know:

(i) Two points in the line, or

(ii) The equation of the line with y isolated. The coefficient of x is the slope. Example: In the equation bmxy , linetheofslopem ,

yband - intercept.

Slope Facts:

(i) Slope 12

12

xx

yym

(ii) Horizontal lines have slope m = 0 (iii) Vertical lines have undefined slope.

(iv) Parallel lines have the same slope. (v) If m1 and m2 are slopes of the two perpendicular lines m1 x m2 = -1,

i.e., m1 =-2

1

m

Solid: A three dimensional object that completely encloses a volume of space.

Sphere: The set of all points in space that are a fixed distance from a given

point.

Square: A quadrilateral with four equal sides and four 90 degree angles.

Square root: Square root of a number, x, is the number that, when multiplied by

itself gives the number, x.

Subset: A set, B, is a subset of another set, A, if every element in B is also an

element of A.

Sum: The result of adding.

77

Supplementary: Two angles are supplementary if their sum is 180 degrees.

Symmetric: Two points are symmetric with respect to a third point if the

segments joining them to the third point are equal. Two points are symmetric

with respect to a line if the line is the perpendicular bisector of the segment joining the points.

T

Tangent: A line that intersects a circle in one point.

Tangent: In a triangle, (the side opposite an angle)

(the side adjacent the same angle).

Term: A part of a sum in an algebraic expression.

Terminating decimal: A fraction whose decimal representation contains a finite

number of digits.

Tetrahedron: A polyhedron with four faces.

Trajectory: The path that a body makes as it moves through space.

Transitive property: The property that states that if a = b, and b = c, then a = c.

Translation: A shift of the axes of the Cartesian Coordinate System.

Transversal: A line that intersects two other lines.

Trapezoid: A quadrilateral that has exactly two sides parallel.

Triangle: A three sided polygon.

Trinomial: A polynomial with exactly three terms.

U

Uniform Rate: Movement at a constant speed. For example, when your parents

set the cruise control on their car at 55 miles per hour, the computer in the car makes sure that the car actually travels at a uniform rate of 55 miles per hour.

Otherwise, if your parents were using the gas pedal to try to maintain a uniform rate of 55 miles per hour, it probably wouldn't happen. The car's speed would probably fluctuate between about 52 and 58 miles per hour, depending on the

amount of pressure that your mom or dad puts on the gas pedal. Therefore, the car would not really be traveling at a constant speed.

78

Union of Sets: The set of all elements that belong to at least one of the given

two or more sets. In other words, the union is formed by putting all the elements

in the given sets together. For example, if H = {0, 3, 6, 9} and J = {0, 6, 12} then the union of H and J is {0, 3, 6, 9, 12} and is written H J. Notice in the example

above that 0 and 6 are in both H and J, but they are only listed once in the union set. Unit: A fixed amount that is used as a standard of measurement. For example:

A mile is a unit of distance. An hour is a unit of time.

A liter is a unit of volume

Unit Fraction: A fraction whose numerator is 1 and whose denominator is a

positive integer. For example, 1/2, 1/100, and 1/x are all unit fractions

Unit Rate: A comparison of two measurements in which the second term has a

value of 1. Unit rates are used to compare the costs of items in a grocery store.

For example if eight ounces of orange juice cost $2.25 and twelve ounces of juice costs $3.25, you would calculate the unit rate to determine which container

was the cheapest. Unknown: A variable whose values are solutions of the equation that they are

written in. For example: (i) In the equation 3x + 4 = 13, "x" is the unknown. It has the value of x = 3 found

by solving the equation for x.

(ii) In the system of equations

2x - y = 9 3x + y = 1 the unknowns are "x" and "y" which have the values x = 2 and y = -5 found by

solving the system for x and y. V Variable: A letter or symbol used to represent a number.

TERM COEFFICIENT VARIABLE DEGREE

3x 3 x 1

-5m3 -5 m 3

Venn Diagram: A picture that illustrates the relationship between two or more

sets. Venn diagrams use circles to enclose members of sets. If sets contain the same element(s) the circles overlap or intersect. If sets do not contain the same

elements, there is no intersection or overlap.

79

Vertex of A 3-Dimensional Object: The point(s) where the edges of a 3-

Dimensional object intersect.

The above figure shows three sides of a cube. The faces of the cube are the

colored squares. The edges are labeled with lowercase letters and are the line segments where the faces intersect. The vertices are labeled with capital letters

and are point(s) where the edges intersect. Vertex of a Polygon: The common endpoint of two line segments that serve as

two sides of a polygon.

Vertex of an Angle: The common endpoint of the two rays that serve as the

sides of an angle.

Vertical Line: A line that goes up and down. Example: In the figure below, Line l

is a vertical line.

80

Vertical Line Test: A test used to determine if a relation is a function. A relation

is a function if there are no vertical lines that intersect the graph at more than one point. Example: The following graph is not a function, because vertical line

intersect the graph more than once.

Volume: The amount of space occupied by an object.

Some common volume formulas:

81

Volume of a cube = side3 Volume of a rectangular prism = length x width x height

Volume of a pyramid = 1/3 x area of the base x height Volume of a cylinder = pi x r2 x height

Volume of a cone = 1/3 x pi x r2 x height Volume of a sphere = 4/3 x pi x r3

W

Weight: The vertical force exerted by a mass as a result of gravity. The weight of

an object is also referred to as the mass of an object. Weight can be measured

using both customary units (e.g., ounces, pounds, tons) and metric units (e.g., grams, kilograms). Whole Numbers: The set of numbers 0, 1, 2, 3, 4, … . Note that this set differs

from the natural numbers by including 0. A whole number is also a real number.

The following table distinguishes a whole number from other real numbers (which are not whole numbers). (See Classification of numbers also).

Whole-Part: One meaning of a ratio or fraction that represents a relationship

between a whole and part of the whole. For example, there are 150 students in the eighth grade at Smith Middle School. 40 of them are in the band. Therefore,

the ratio of all eighth graders at Smith Middle School to those eighth graders in the school band could be represented by the whole-part ratio: 150/40.

X

X-axis: The horizontal number line on the Cartesian coordinate plane. The

following figure is an example of a Cartesian coordinate plane. The horizontal line is the x-axis and the vertical line is the y-axis

X-coordinate: The first number of an ordered pair; the

position of a point relative to the vertical axis. The point on the plane is represented by the ordered pair (3, 1). It has an x-coordinate of 3, indicating that it is 3 units to the right of the vertical axis.

82

X-intercept: The value on the x-axis where a graph crosses the x-axis.

Y

Y-axis: The vertical number line on the Cartesian coordinate plane. The following figure is an example of a Cartesian coordinate plane. The horizontal line is the x-axis and the vertical line is the y-axis.

83

Y-coordinate: The second number in an ordered pair; the position of a point

relative to the horizontal axis. The point on the plane is represented by the ordered pair (3, 1). It has a y-coordinate of 1, indicating that it is 1 unit above the

horizontal axis.

Y-intercept: The value on the y-axis where a graph crosses the y-axis.

Z

Zero: In ari thmetic, it is defined as the identity element of addition, i.e., the

number 0 for which xx 0 and xx0 for all numbers x . Note the following

84

properties of the number zero 0 :

Zero (i.e., 0 ) is also called the cardinal number of the empty set .

Division by zero: 0

a is meaningless for all a .

Division of zero: 00

k for all 0k , since 00 k .

Factorial zero: It is denoted by !0 , and is defined as 1!0 .

Multiplication by zero: 000 kk for all k .

Zero-factor (or Zero product) property: If ab = 0, then a = 0, or b = 0, or both a

= 0 and b = 0.

For example:

If x(y - 2) = 0, the zero product property states that either x = 0 or y - 2 = 0, or both equal zero. Thus, x = 0 and y = 2 would be solutions to this equation.

If (x + 4)(x - 3) = 0, the zero product property states that either x + 4 = 0 or x - 3 =

0, or both equal zero. Thus, x = -4 and x = 3 would be solutions to this equation.

Zero of a function: It is defined as a value of the argument x for which a function )(xf is equal to 0 , i.e., 0)(xf . Note the following properties of zero of

a function )(xf :

(i) Real Zero: A real zero of a function )(xf is a real number c for

which )(xf is equal to 0 , i.e., 0)(cf .

(ii) Real Zero as an x -intercept of the curve )(xfy : If the

function )(xf has only real number values for real number

values of x , then the real zeros of )(xf are the values of x for

which the curve )(xfy meets the x -axis, which are also

called the x -intercepts of the curve )(xfy . Example: A

polynomial )(xf with real numbers as coefficients is an

example of such a function.

Zero of a polynomial: It is defined as a value c of the argument x for which a polynomial )(xf with real numbers as coefficients is equal to 0 , i.e., 0)(cf .

Example: 1x is a zero of the polynomial 1)( xxf .

85

Appendix A

NOTES ON SET THEORY AND PROPERTIES OF REAL NUMBERS

By DR. M. Shakil

1. DEFINITION OF A SET:

A set is a well-defined collection of distinct objects, called the elements or members of the set. A set is well-defined if it is possible to determine whether or not a given element belongs to it. NOTE: Sets can be finite or infinite. A set is called finite i f it consists of “n” different elements

where “n” is some natural number (i.e., positive integer). A set which is not finite is said to be infinite.

2. DESCRIPTION OF A SET: A set may be described by: a) listing its elements in a “Roster;” for example: A = {a, e, i, o, u};

X = {0, 8, 2, 3, 7, 10}; etc. NOTE: When the list is extensive and the pattern is obvious, an ellipsis may be used; for

example: B = {a, b, c, …, z}. NOTE: The above form is also called the Tabular Form of the Set. The elements of a set are

separated by commas and are enclosed in brackets { }. b) giving a description of the elements that belong in the set; for example:

A = {all vowels in the English alphabet} and B = { all letters in the English alphabet}.

c) using set-builder notation; for example: S = {x 0 < x < 10 and x N}, where

N is the set of all natural numbers.

3. DIFFERENT TYPES OF SETS:

a) EQUAL (IDENTICAL) SETS have exactly the same elements.

b) The UNIVERSAL SET, U, is the set that contains all possible elements under consideration in a problem.

d) SUBSET: A is a subset of B if every element in set A is also in set B. The

notation is A B. NOTE: Every set is a subset of itself, i.e., A A, where A is any set.

e) PROPER SUBSET: A is a proper subset of B (notation: A B) if every element in set A is also in set B, and at least one element in set B is not in set A. In other words,

A B if and only if A B and A B.

f) NULL OR EMPTY SET: The null or empty set is the set with no elements. The notation is

or { }. NOTE: The empty set is a subset of every set, i.e., A, where A is any set.

g) NUMBER OF SUBSETS OF A SET: If a set contains “N” elements then the number of subsets is equal to 2

N and the number of proper subsets is equal to 2

N - 1.

h) POWER SET: The set of all the distinct subsets of a set A is called the power set of A. The notation is P(A).

86

i) CARDINALITY OR CARDINAL NUMBER OF A SET: The number of elements in a given finite set A is called its cardinality or cardinal number. The notation is n(A) or #(A).

j) NOTE: If n(A) = N, then n(P(A)) = 2N, where A is any finite set.

k) EQUIVALENT SETS have the same cardinality, i. e., same number of distinct elements.

Thus, if A and B are any two finite sets with the same number of distinct elements, then n(A) = n(B).

4. SET OPERATIONS: a) UNION OF SETS: The union of sets A and B is the set of all elements which

are either in A or in B. The notation is A B. Thus

A B = {x x A or x B or x both A and B}.

b) INTERSECTION OF SETS: The intersection of sets A and B is the set of all

those elements, which are common to both A and B. The notation is A B. Thus

A B = {x x A and x B}.

c) DIFFERENCE OF TWO SETS: Let A and B be two sets. Then the di fference of sets A and B, denoted by A – B, is the set of all those elements of A which do not belong to B. Thus

A – B = {x x A and x B}.

Similarly, the difference of sets B and A, denoted by B – A, is the set of all those elements of B which are not in A, i.e.

B – A = {x x B and x A}. d) COMPLEMENT OF A SET: Let A be a set and U the universal set. Then the

complement of set A with respect to the universal set U is the difference of sets U _

and A, i.e., U – A, and is denoted by A or A. Thus

A = {x x U and x A}.

e) DISJOINT SETS: Any two sets A and B are said to be disjoint if and only if

A B = .

5. NOTATIONS USED WITH SETS:

i) “is an element of”

ii) “is not an element of”

iii) A B “A is a subset of B”

iv) A B “A is a proper subset of B”

v) A B “A is not a subset of B” vi) U “Universal set”

vii) “Empty set”

viii) A “Complement of set A”

ix) A B “A union B”

x) A B “A intersection B”

xi) or means “such that”

87

6. VENN DIAGRAMS:

(i) Geometric pictures are extremely helpful (but not necessary) for mathematical thinking and discussions. In particular, we can visualize sets and operations on sets and guess the truth of a number of propositions on sets with the help of geometric diagrams, known as Venn diagrams.

These diagrams are named in honor of the English mathematician, John Venn (1831 – 1923), who invented these and used them to illustrate ideas in his text on symbolic logic, published in 1881.

(ii) A convenient way to use Venn Diagrams is to represent the universal set U by rectangular area in a plane and the elements which make up U by the points of this area. Sets can be

visualized as parts of the rectangular area. In particular, simple plane areas bounded by circles or their parts drawn within the rectangular area can give a simple and instructive picture of sets and operations on sets. We can think of each set as consisting of all points within the corresponding

circle. (iii) A subset A of set B is represented by drawing the circle inside the circle representing the set

B. A single circle is used to represent equal sets. Two circles, drawn in such a way that they have no common area, represent two disjoint sets. Two sets having common elements are represented by closed figures (generally circles) having some area common to both.

7. Some Algebra of Sets:

(a) De Morgan’s Laws

(i) (A B)/ = A

/ B

/

(ii) (A B)/ = A

/ B

/

(b) Distributive Laws

(i) A (B C) = (A B) (A C)

(ii) A (B C) = (A B) (A C)

8. SOME EXAMPLES ON SET OPERATIONS:

Let U = {2, 3, 4, 5, 7, 9} denote the universal set in a problem under consideration, and let X = {2, 3, 4, 5}; Y = {3, 5, 7, 9}; and Z = {2, 4, 5, 7, 9} be any three sets of the said problem. List the members of each of the following sets, using set braces. Draw Venn Diagrams also.

(i) X/ Z; (ii) X (Y Z)

9. AN EXAMPLE ON APPLICATION OF VENN DIAGRAM:

A survey of 53 business executives shows that: 10 play golf and tennis; 9 play all three; 25 play

bridge; 30 play golf; 14 play bridge and tennis; 22 play tennis; 17 play golf and bridge. (A) Use the following Venn Diagram to represent these data appropriately.

88

(B) Answer the following questions: (a) How many executives played none of the three games?

(b) How many played Tennis only? (c) How many played Golf and Bridge, but not Tennis?

The Number Line and Real Numbers:

To draw a number line we draw a line with several dashes in it and ordered numbers below the line, both positive and negative. The number corresponding to the point on the number line is called the coordinate of the number line.

Any number which can be plotted on a number line by a point is called a real number.

Properties of Real Numbers with respect to (w. r. t.) Addition (+) and Multiplication (.) :

Commutative Property w. r. t. :

Addition: a + b = b + a Multiplication: a.b = b.a

Associative Property w. r. t. :

Addition: (a + b) + c = a + (b + c) Multiplication: (a.b).c = a.(b.c)

Identity Property w. r. t. :

Addition: there exists a 0 such that a + 0 = 0 + a = a; (0 is called Additive Identity) Multiplication: there exists a 1 such that a.1 = 1.a = a; (1 is called Multiplicative Identity)

Inverse Property w. r. t. :

Addition: for any a, there exists an additive inverse - a of a such that a + (-a) = 0 Multiplication: for any a not 0, there exists a multiplicative inverse 1/a of a such that

a.(1/a) = 1

Distributive Property : a.(b + c) = a.b + a.c (a + b).c = a.c + b.c

Trichotomy : If a and b are real numbers then one of the following three must hold

a < b

a > b a = b

Transitivity : If a, b, and c are real numbers and a < b and b < c then a < c

Some Examples on Properties of Real Numbers : (i) (4 + 5) + 7 = 4 + (5 + 7) (Associative Property of Addition)

(ii) (a - b).(a + b) = (a + b).(a - b) (Commutative Property of Multiplication)

89

1 (iii) (2 - u) . = 1 (Multiplicative inverse) (2 – u)

Inequality Symbols :

The following are four inequality symbols used in solving mathematical problems.

SYMBOL MEANING

< Less Than

> Greater Than

Less Than or Equal to

Greater Than or Equal to

Interval Notations :

When we graph an inequality on a number line we use "[" or "]" to include the point and "(" or ")" to not include the point.

An Example :

[4, 10) means all the numbers (denoted as points on a number line) between 4 and 10 including 4 but not including 10.

Absolute Values :

The absolute value of a number X can be regarded as the distance of the number X from zero on a number line. In other words, by the absolute value of a number X, we mean the number X without a negative sign. For example, 6 is the absolute value of both 6 and −6.

Definition :

For any real number a, the absolute value (also called modulus) of a, denoted by |a|, is is defined as follows:

0

0

aifa

aifaa

Note that |a| is never negative, as absolute values are always either positive or zero. That is, |a| < 0 has no solution for a.

Properties of Absolute Value of a Number :

The absolute value has the following properties:

1. |a| ≥ 0

2. |a| = 0 if and only if a = 0.

90

3. |ab| = |a||b| 4. |a / b| = |a| / |b| (if b ≠ 0)

5. |a + b| ≤ |a| + |b| (also known as the triangle inequality) 6. |a − b| ≥ ||a| − |b|| 7. |a| ≤ b iff −b ≤ a ≤ b 8. |a| ≥ b iff a ≤ −b or b ≤ a

We can use properties (7) and (8) to solve inequalities.

Example: Solve the following inequality.

|x − 5| ≤ 10

or, − 10 ≤ x − 5 ≤ 10

or, − 5 ≤ x ≤ 15

or, 15,5 (Interval Notation)

91

Appendix B

SOME USEFUL FORMULAS FOR BUSINESS CALCULUS AND CALCULUS I, II, II

By DR. M. Shakil

DIFFERENTIAL CALCULUS

(A) EXISTENCE OF LIMIT OF A FUNCTION

The limit of a function )(xf at cx , i.e., )(lim xfcx

exists if and only if the left-hand

limit )(lim xfcx

and the right-hand limit )(lim xfcx

exist and are equal.

(B) CONTINUITY OF A FUNCTION AT A POINT

A function )(xf is continuous at cx if the following conditions are satisfied:

(i) )(cf is defined;

(ii) )(lim xfcx

exists;

(iii) )()(lim cfxfcx

.

(C) THE DERIVATIVE OF A FUNCTION

(1) Definition: The derivative of the function )(xfy with respect to x is the

function )(/ xf given by

h

xfhxfxf

h

)()(lim)(

0

/

(1.1) (read as “ f prime of x ”). The process of computing the derivative is called

differentiation, and )(xf is said to be differentiable at cx if )(/ xf exists; i.e., if

the above limit (1.1) that defines )(/ xf exists when cx .

Notation: The derivative of )(xfy is denoted as: .,)(/

dx

dyor

dx

dfxf

(2) Slope )(m of the tangent line to the graph of )(xfy at ),( 00 yx , where

)( 00 xfy , is given by the derivative of the function )(xfy at 0x , i.e., by

)( 0

/ xfm .

92

(3) Equation the tangent line to the graph of )(xfy at ),( 00 yx is given by

)( 00 xxmyy .

(D) BASIC RULES OF DERIVATIVE

(i) The constant rule: 0)(cdx

d

(ii) The derivative of x is one: 1)(xdx

d

(ii) The power rule: 1)( nn nxxdx

d

(iii) The constant multiple rule: dx

dfcxcf

dx

d)]([

(iv) The sum and difference rules: dx

dg

dx

dfxgxf

dx

d)]()([

(v) The product rule: dx

dgxf

dx

dfxgxgxf

dx

d)()()]()([

(vi) The quotient rule: .0)(,)]([

)()(

)(

)(2

xgxg

dx

dgxf

dx

dfxg

xg

xf

dx

d

(vii) The chain rule for )]([ xufy : )(.)())((. ///xuufxufor

dx

du

du

dy

dx

dy

(viii) The general power rule: dx

dhxhnxh

dx

d nn 1)()(

(ix) Derivatives of exponential functions:

(a) xx eedx

d; and

(b) dx

duee

dx

d xuxu )()(

(x) Derivatives of logarithmic functions:

93

(a) )0(,1

ln xx

xdx

d; and

(b) dx

du

xuxu

dx

d

)(

1)(ln

(xi) Derivatives of xb and xblog for base eb :

Let 0b be a positive real number. Let 1b . Then we have

(a) xx bbbdx

d.)(ln ; and

(b) xb

xdx

db

1.

ln

1log )0(, x

(xii) The second derivative of )(xfy denoted as: 2

2//// )()(

dx

fdxfxf

gives the rate of change of .)(/ xf

(xiii) Notation for the nth derivative of )(xfy : n

nn

n

n

dx

fdxf

dx

yd)()(

(xiv) Demand Function, Supply Function and Market Equilibrium:

Demand Function: A demand function )(xDp is a function that relates the

unit price p for a particular commodity to the number of units x demanded by

consumers at that price or sold in the marketplace.

Supply Function: A supply function denoted by )(xS is defined as a function

which gives the corresponding price )(xSp at which producers are willing to

supply x units.

Law of Supply and Demand: In a competitive market environment,

if )()( xDxS , the market is said to be in “equilibrium.” Thus, if ex denote the

production level at which the market equilibrium occurs, and if ep denote the

corresponding unit price, then )()( eee xDxSp . Here ep is called the

“equilibrium price” and ee px , is called the “equilibrium point.”

Shortage and Surplus: When the market is not in equilibrium, it has a

(i) a “shortage” when )()( xSxD ; and

94

(ii) a “surplus” when )()( xDxS .

(xv) Cost Function, Revenue Function, Profit Function and Break Even:

Cost Function: Let x represent the number of units produced and sold in the

marketplace. Let cp denote a variable cost per unit. Then the total cost of

producing the x units denoted by )(xC is defined as a function given by

,)cos()cos(var)( tfixedtiablexC where cpxtiable .cosvar .

Revenue Function: Let )(xDp denote a demand function that relates the unit

price p for a particular commodity to the number of units x demanded by

consumers at that price or sold in the marketplace. Then the total revenue from the sale of x units, denoted by )(xR , is defined as a function given by

)(..)(.)()( xDxpxitemperpricesolditemsofnumberxR .

Profit Function: The profit from the production and sale of x units at the unit price p , denoted by )(xP , is defined as a function given by

)()(.)()()( xCxDxxCxRxP .

Break Even: When total revenue equals total cost, that is, )()( xCxR , that is,

0)(xP , we say that the manufacturer “breaks even,” experiencing neither a

profit nor a loss. The point at which the graphs of the two functions, Revenue Function: )(xRy and Cost Function: )(xCy intersect is called the “break-

even point.”

(xvi) Marginal Cost, Marginal Revenue and Marginal Profit: Let )(xC the

total cost of producing x units of a particular commodity, )(xR the revenue

generated when x units of a particular commodity are produced, and )(xP the

corresponding profit. Then, if 0xx denote the number of units being produced,

the marginal cost, marginal revenue and marginal profi t of producing 0x units are

defined as follows:

(a) Marginal Cost: It is given by the derivative )( 0

/ xC of the cost

function )(xC at 0xx , which approximates the additional cost (that is,

extra cost) )()1( 00 xCxC incurred when the level of production is

increased by one unit, from 100 xtox , that is, of producing the stx )1( 0

unit.

95

(b) Marginal Revenue: It is given by the derivative )( 0

/ xR of the revenue

function )(xR at 0xx , which approximates )()1( 00 xRxR , the

additional revenue from producing the stx )1( 0 unit.

(c) Marginal Profit: It is given by the derivative )( 0

/ xP of the profit

function )(xP at 0xx , which approximates )()1( 00 xPxP , the

additional profit from producing the stx )1( 0 unit.

(xvii) Approximation by increments: Let )(xfy be a function of x which is

differentiable at 0xx , and let x denote a small change in x , then we have

xxfxfxxf .)()()( 0

/

00 .

If )()( 00 xfxxff in the above formula, then we have xxff .)( 0

/ .

(xviii) Differentials: Sometimes the increment x is called the “differential” of x and is denoted by dx . Thus, if the differential of x be xdx , and if )(xfy be

a differentiable function of x , then dxxfdy .)(/ is called the “differential” of y .

(ix) Approximation Formula for Percentage Change: Let )(xfy be a

differentiable function of x . Let x denote a small change in x . Then the corresponding “percentage change” in the function )(xf is defined as

%)(

.)(100%

)(100)(

/

xf

xxf

xf

fxffunctiontheinchangePercentage .

INTEGRAL CALCULUS

BASIC RULES OF INTEGRATION

(A) INDEFINITE INTEGRAL:

(i) Antiderivative; indefinite integral: )()()()( / xfxFiffCxFdxxf ,

where C is a constant of integration.

(ii) Power rule: 11

1

nforCn

xdxx

nn

(iii) Logarithmic rule: Cxdxx

ln1

(iv) Exponential rule: (a) Cedxe xx ; (b) Cek

dxe xkxk 1

96

(v) Constant multiple rule: dxxfkdxxfk )()(

(vi) Sum and difference rules: dxxgdxxfdxxgxf )()()()(

(vii) Integration by substitution:

dxxuduandxuuwhereduugdxxuxug )()()()())(( //

(viii) Integration by parts: duvvudvu

(B) DEFINITE INTEGRAL:

(i) The Definite Integral: Let )(xfy be a function of x that is continuous on

the interval bxa . Let the interval bxa be subdivided into n equal parts,

each of width n

abx . Let a number kx be chosen from the kth subinterval

for nk ,,2,1 . Then the sum xxfxfxf n)21 ()()( is formed, which is

called a “Riemann sum.” The “definite Integral” of )(xf on the interval

bxa , denoted by

b

a

dxxf )( , is the limit of the Riemann sum as n ; that

is,

xxfxfxfdxxf nn

b

a

)()()(lim)( 21 .

The function )(xf is called the “integrand,” and the numbers a and b are

called the “lower and upper limits of integration,” respectively. The process of finding a definite integral is called “definite integration.”

(ii) Area as a Definite Integral: If the function )(xfy is continuous and

0)(xf on the interval bxa , then the region R under the curve )(xfy

over the interval bxa has area A given by the definite Integral

b

a

dxxfA )( .

(iii) The Fundamental Theorem of Calculus: If the function )(xfy is

continuous on the interval bxa , then )()()( aFbFdxxf

b

a

, where )(xF is

any antiderivative (i.e., indefinite i ntegral) of )(xf over the interval bxa .

97

(iv) The Area Between Two Curves: If )(xf and )(xg are continuous with

)()( xgxf on the interval bxa , then the area A between the curves

)(xfy and )(xgy over the interval bxa is given by the definite Integral b

a

dxxgxfA )()( .

(v) The Average Value of a Function: Let )(xfy be a function of x that is

continuous on the interval bxa . Then the

")(" bxaoverxfofVvalueaverage is given by b

a

dxxfab

V )(1

.

(vi) RULES FOR DEFINITE INTEGRALS:

(a) 0)(

a

a

dxxf

(b)

b

a

a

b

dxxfdxxf )()(

(c) Constant multiple rule: dxxfkdxxfk

b

a

b

a

)()( for constant k

(d) Sum rule: dxxgdxxfdxxgxf

b

a

b

a

b

a

)()()()(

(e) Difference rule: dxxgdxxfdxxgxf

b

a

b

a

b

a

)()()()(

(f) Subdivision rule: dxxfdxxfdxxf

b

c

c

a

b

a

)()()(

98

A Calculus Formula Sheet Derivatives Properties

Trigonometry Derivatives

Inverse Trig Derivatives

Hyperbolic Derivatives

99

Derivative Formula Linearization Formula

Inverse Hyperbolic Derivatives Formula

Hyperbolic Formula Identities

Basic Trigonometric Properties

100

TABLE OF INTEGRALS FOR QUICK REFERENCES

(From World Wide Web Site At The Address:

http://www.jessschwartz.com/~swalker/calculus/General_Integration_rules.htm)

101

102

Appendix C

SUMMARY OF GEOMETRY (Courtesy: Dept. of Math., MDC-North Campus)

MEASUREMENT SYSTEMS:

METRIC SYSTEM BASIC units: linear: m is a meter; weight or mass: g is a gram capacity: l is

a liter

PREFIXES: m- is a milli- (key number is 1000); c- is a centi- (key number is 100);

k- is a kilo- (key number is 1000); CONVERSION:

Using concept that if you are converting units to smaller units, you will need

more of the smaller units for the same measurement and if you are converting to larger units you will need fewer of the larger units:

a) If converting from x large units to smaller units: multiply x by the key number of the prefix.

b) If converting from x small units to larger units: divide x by the key number of the prefix

Alternate method: Since you are multiplying or dividing by a power of 10, the

conversion can be accomplished by moving the decimal point to the right (if multiplying) or to the left (if dividing) the appropriate number of places.

To help arrive at the correct directions and appropriate number of places, you can use the memory aid of: K H D M D C M (King Henry died miserably doing college math). Place the

number under the letter representing its unit and note the direction and places to get to the letter of the unit you want to convert to.

For example: Convert 5400 cm to kilometers: K H D M D C M

.054

5400. move decimal 5 places to the left.

ENGLISH SYSTEM

You must use “conversion fractions” to get from one measurement to another.

Ex: Convert 72 cups to gallons.

galgalqt

gal

pt

qt

c

ptc5.4

2

9

4

1

2

1

2

1

1

72

When rounding feet and inches to nearest foot, you must have 6 or more inches to round up.

When rounding pounds and ounces to the nearest pound, you must have 8

ounces or more to round up.

When rounding a ruler measurement to the nearest fraction of an inch, the

final answer may be in any larger unit. For example, given something that measures between 1½ in. and 1¾ in. and asked to find the measure to the

103

nearest quarter inch, the answer could be 1½ inches, since 1½ inches is the same as 4

21 inches.

ANGLES are measured in degrees using a curved ruler called a protractor:

acute angle < 90

right angle = 90

obtuse angle > 90

straight angle = 180

PAIRS OF ANGLES: Adjacent angles: angles that share a common vertex and a common side, but have

no other intersection. NOTE: in a polygon, adjacent angles share only a common side; they are also

referred to as consecutive angles.

Complementary angles: Two angles whose measures have a sum of 90 . If they

are adjacent they form a right angle.

Supplementary angles: Two angles whose measures have a sum of 180 . If they are adjacent they form a straight angle.

Vertical angles: Two non-adjacent angles formed by intersecting lines. Vertical

angles are equal in measure.

PARALLEL LINES CROSSED BY ONE TRANSVERSAL

Two groups of four angles are formed:

Pairs of corresponding angles are 1 and 5, 2 and 6, 3 and

7, 4 and 8.

Pairs of alternate interior angles are 4 and 6; 3 and 5

NOTE: If corresponding angles or alternate interior angles are NOT equal,

then the lines are NOT parallel

If the lines are parallel, then

all acute angles are equal

all obtuse angles are equal

any acute angle and any obtuse angle are supplementary

104

TRIANGLES

The sum of the interior angles of a triangle is 180

Since all triangles have at least 2 acute angles, a triangle may be classified by the nature of its third angle, that is:

an acute triangle if the third angle is also acute.

an obtuse triangle if the third angle is obtuse.

a right triangle if the third angle is a right angle.

Triangles may be classified by sides:

Scalene -- no sides are equal (and no angles are equal).

Isosceles -- two sides are equal (and the angles opposite the equal sides are equal).

Equilateral -- all three sides are equal (each angle measures 600; thus, the triangle is also equiangular).

The shortest side of a triangle is opposite the smallest angle; the longest side is

opposite the longest side.

PYTHAGOREAN THEOREM (applies to right triangles):

The square of the hypotenuse equals the sum of the squares of the other two sides.

H2 = S12 + S2

2 or a2 + b2 = c2 (where c is the hypotenuse) SIMILAR TRIANGLES:

When two triangles are similar, the measures of corresponding angles are equal, and

the lengths of corresponding sides are in proportion. That is, if ABC PQR, then:

(iii) A = P, B = Q and C = R, and

(iv) PR

AC

QR

BC

PQ

AB

Two triangles will be similar if:

two angles of one triangle are equal to two angles of the other triangle, or

three pairs of corresponding sides are proportional, or

two pairs of sides are proportional and the included angles are equal.

If the triangles are similar and corresponding sides are equal, the triangles are congruent.

POLYGONS:

NUMBER OF SIDES

NAME

3 Triangle

4 Quadrilateral 5 Pentagon

6 Hexagon 8 Octagon

105

AN EXTERIOR ANGLE OF A POLYGON is formed by one side and the extension of the other side at that vertex.

The exterior angle and the interior angle at any vertex of any polygon are supplementary.

The exterior angle of a triangle is equal to the sum of the two remote interior angles.

Summary of Angle Measures for Polygons (in degrees):

Sum of all

(any

polygon)

Measure of one

(regular

polygon)

Interior

angle

(n - 2) 180

180 - 360

n

Exterior

angle

360

360

n

QUADRILATERALS

DISTANCE AROUND AN OBJECT:

PERIMETER is the linear measure of the distance around a figure. To find the perimeter of a polygon, add the lengths of all of its sides.

CIRCUMFERENCE is the linear measure of the distance around a circle. Circumference is Pi times the diameter or Pi times twice the radius.

C = d or C = 2 r

106

MEASUREMENT OF GEOMETRIC FIGURES:

1) Linear: used when measuring length, width, depth, distance around an object (in., ft., yd., m., cm, km, etc.)

2) Square: used when measuring area (in2., ft2., yd2., m2., cm2, km2, etc.) 3) Cubic: used when measuring volume (in3., ft3., yd3., m3., cm3, km3, etc.)

Area Formulas

1. Rectangle A = l w (length x width)

l

w

2. Square A = s2 (side squared)

s

3. Parallelogram A = b h (base x height)

h

b

4. Triangle A = ½ b h (1/2 base x height)

h

b

5. Trapezoid b2 A = ½ (b1 + b2) h (1/2 sum of bases x

height)

h

b1

6. Circle A = r2 (pi x radius squared)

(Circumference: C = d or 2 r)

d

r

107

Notes: 1. Area measures the amount of surface enclosed in a region and is measured in

square units: sq. in. = in2 or sq. cm = cm2 NOTE: 9 sq. ft = 1 sq. yd.

2. The area of any figure made up of a combination of the above figures can be

found by adding up the individual parts.

3. The area of a shaded region can be found by first finding the figure’s total

area and then subtracting out the unshaded area.

4. Leave area and circumference in terms of unless otherwise indicated.

Volume Formulas

In general, the volume of a solid figure (in cubic units) will be:

a) the area of the base height , if the figure goes “straight up” from the

base.

b) 1/3 the area of the base height, if the figure rises to a point (e.g. cone or

pyramid).

1. Rectangular Solid (box) Volume = base area height

= lw h = lwh

(Surface Area = area of 6 faces = 2 lw + 2 lh

+ 2 wh)

2. Cube Volume = base area height

= e2 e = e3

3. Cylinder Volume = base area height

= r2 h = r2h

(Surface Area = area of ends + area of “label”)

= 2 r2 + 2 rh

4. Pyramid h Volume = 1/3 base area height

= 1/3 lw h =

1/3 lwh

l

h

w

e

e e

r

h

r

h

108

5. Cone Volume = 1/3 base area height

= 1/3 r2 h =

1/3 r2h

6. Sphere Volume = 4/3 r3

109

Appendix D

SUMMARY: “SETS and LOGIC”

(Courtesy: Dept. of Math., MDC-North Campus)

SETS

Notation:

A set may be defined by:

1. listing its elements in a “Roster”

for example: A={a,e,i,o,u}. NOTE: when the list is extensive and the pattern is obvious, an ellipsis may be used; for example: B={a,b,c,. . . ,z}

2. giving a description of the elements that belong in the set

for example: A={all vowels in English alphabet} and B={all letters in the English alphabet}

3. using set builder notation for example: A={x| x > 0}

Special sets:

Universal set: The set of all elements under discussion. The notation for the

universal set is set U.

Subset: Set A is a subset of Set B if Set A contains no element that is not also an

element of Set B. The notation is A B.

Null or empty set: A set containing no elements. Symbols are and { }. Number of subsets: If a set contains “n” elements then the number of subsets is

equal to 2n NOTE: The empty set is considered a subset of every set and every

set is considered a subset of itself.

Set Operations:

Complement: The set of all members of U that are not in A is called the

complement of A. The notation is A . The intersection of sets A and B is the set of elements that are members of both

A and B. The notation is A B. The union of sets A and B is the set of elements that are in at least one of the sets

A or B. That is, all elements that are in A or B or both. The notation is A B.

DeMorgan’s Laws:

(A B) = A B

(A B) = A B

LOGIC

Basic Definitions: Negation “Not p”: The notation for the negation of p is ~p A statement and its negation always have opposite truth values.

Conjunction “p and q” The notation is p q: The conjunction of two statements is true if both statements are true..

110

Disjunction “p or q” The notation is p q: The disjunction of two statements

is true if at least one of the statements is true [that is: one or the other or both are true]

Conditional “If p, then q.” The notation is p q: A conditional statement is

false only when the “If-clause” is true and the “Then-clause” is false

Truth tables for compound statements:

Negation

P ~P

T F

F T

Conjunction Disjunction Conditional

p q p q p q p q

T T T T T

T F F T F

F T F T T

F F F F T

Other Forms of the Conditional:

Original conditional p q equivalent

converse q p Not equivalent

inverse ~p ~q Not equivalent

contrapositive ~q ~p equivalent

Equivalences and negations of statements:

Statement Equivalent Negation

p q (If p, then q)

1) ~q ~p

(If not q, then not p)

2) ~p q (Not p, or q)

1) (p q) (It is not true that if p, then q)

2) p q (p AND not q)

p q (p or q)

1) ~p q (If not p, then q)

2) ~q p (If not q, then p)

3) q p

1) (p q) (It is not true that p or q)

2) p q (Not p AND not q)

111

(q or p)

p q (p and q)

1) q p (q and p)

1) (p q) (It is not true that p and q)

2) p q (not p OR not q)

Equivalences and Negations for statements involving Quantifiers

STATEMENT NEGATION EQUIVALENT

All A’s are B’s Some A’s are not B’s There are no A’s that are not

B’s.

Some A’s are B’s No A’s are B’s At least one A is a B

Some A’s are not B’s All A’s are B’s Not all A’s are B’s.

No A’s are B’s Some A’s are B’s All A’s are not B’s

Determining whether an argument is invalid or valid:

I. If premises contain “all,” “some,” or “no,” use Euler circles:

YOU TRY TO MAKE A DRAWING THAT SHOWS THE PREMISES BUT NOT

THE CONCLUSION. IF YOU CAN DO THAT, THEN THE ARGUMENT IS INVALID.

DIAGRAM ENGLISH

EXAMPLES

No A’s are B’s Snakes are not

pretty.

No B’s are A’s No math course

is easy.

112

Some A’s are B’s Some roses are red.

(at least one A is a B)

Some B’s are A’s Some rivers are polluted.

(at least one B is an A)

All A’s are B’s All sharks scare

me.

Apples are good

for you.

NOTE: This is not commutative since only some B’s are A’s

x B Marta is a

scientist.

Bob swims.

II. If at least one of the premises is an “if. . . then” statement or an “or”

statement, use symbolic logic:

In general, an argument is valid if P1 P2 C is a tautology. In most cases arguments will be

recognizable as fitting one of the valid or invalid forms.

Summary of Valid Forms of Arguments

Type 1 Type 2 Type 3 Type 3a Type 4 Type 5

p q

p q p q p q q r p q p q

p q q r p p q

q p p r r q p

or

r p

113

Common errors made in drawing conclusions or determining the validity of an

argument are: 1. Arguing the converse (assuming in a conditional that if q happens, p must happen).

2. Arguing the inverse (assuming in a conditional that if p didn’t happen, q can’t happen).

3. Using the exclusive “or” (assuming in a disjunction that if p happens, q can’t happen).

Use of premises that are conjunctions, and conclusions of valid arguments:

If a premise is a conjunction, then both its parts must be true and each can be used separately as a

premise.

If an argument has been shown to be VALID then its conclusion must be true, and it

can be used as a premise for another argument.

Miscellaneous comments about premises:

1. A premise such as “mathematicians enjoy puzzles” can be read as either:

a) “all mathematicians enjoy puzzles” (Euler circles) or, b) “if you are a mathematician, then you enjoy puzzles” (symbolic logic)

In other words, “all A are B” is equivalent to “if A then B”.

2. A premise such as “Bob is a mathematician” can be read as either:

a) “Bob is an element of the set of mathematicians” (Euler circles) or, b) The event of having a mathematician has occurred. (symbolic logic)

114

Appendix E

NOTES ON STATISTICS: AN INTRODUCTION

By

Dr. M. Shakil

(1) STATISTICS:

Statistics deals with the methods of classification and analysis of data (numerical and non-numerical) for drawing valid conclusions and making reasonable decisions. It has meaningful applications in production engineering, in analysis of experimental data, in economics, in law, in

medicines, in biology, etc. The importance of statistical methods whether it be in engineering, in social sciences, in biological sciences, in medical sciences, in health sciences, or, in physical sciences, is on the increase. As such we shall now study this interesting and important field and

its applications. Depending on how data are used, the two major areas of statistics are descriptive statistics and inferential statistics.

(a) DESCRIPTIVE STATISTICS: It consists of the collection, organization, summarization, and presentation of data. (It describes

the situation as it is). (b) INFERENTIAL STATISTICS:

It consists of making inferences from samples to populations, hypothesis testing, determining relationships among variables, and making predictions. (Inferential statistics is based on

probability theory. It goes beyond what is known). (NOTE: By probability, we mean the chance of an event occurring. For example, people who play

cards, dice, bingo, and lotteries are using the concepts of probability theory. It is also used in the insurance industry and other areas such as genetics, etc.).

*** *** ***

(2) SOME USEFUL TERMINOLOGIES:

We shall now explain certain terminologies which will be useful in the study of various statistical techniques and their applications.

(A) In order to gain information about seemingly haphazard events, statisticians study random variables. These are defined as follows:

(i) VARIABLES:

A variable is a characteristic or an attribute that can assume different values. Height, weight, temperature, number of phone calls received, etc. are examples of variables.

(ii) RANDOM VARIABLES:

Variables whose values are determined by chance are called random variables. (B) COLLECTION OF DATA:

115

The collection of data constitutes the starting point of any statistical investigation. It should be

conducted systematically with a definite aim in view and with as much accuracy as is desired in the final results, for detailed analysis would not compensate for the bias and inaccuracies in the original data. The definition of data is given below.

(i) DATA:

The measurements or observations (values) for a variable are called data. (ii) DATA SET:

A collection of data values forms a data set.

(iii) DATA VALUE OR DATUM: Each value in the data set is called a data value or a datum.

Example: Suppose a researcher selects a specific day and records the number of calls received by a local office of the Internal Revenue Service each hour as follows: {8, 10, 12, 12, 15, 11, 13,

6}, where 8 is the number of calls received during the first hour, 10 the number of calls received during the second hour, and so on. The collection of these numbers is an example of a data set, and each number in the data set is a data value.

(C) Data may be collected for each and every unit of the whole lot (called population), for it would ensure greater accuracy. But, however, since in most cases the populations under study are

usually very large, and it would be difficult and time-consuming to use all members, therefore statisticians use subgroups called samples to get the necessary data for their studies. The conclusions drawn on the basis of this sample are taken to hold for the population. The definitions

of a population and a sample are given below. (i) POPULATION:

A population is the totality of all subjects possessing certain common characteristics that are being studied.

(ii) SAMPLE:

A sample is defined as a subgroup or subset of the population.

(iii) RANDOM SAMPLE: A sample obtained without bias or showing preferences in selecting items of the population is

called a random sample. (D) CLASSIFICATION OF VARIABLES (AND DATA):

(a) Random Variables (or Data) can be classified as qualitative or quantitative as follows:

(i) QUALITATIVE VARIABLES (OR DATA): Qualitative variables are variables that can be placed into distinct categories, according to some

characteristic or attribute. For example, if subjects are classified according to gender (male or female), then the variable “gender” is qualitative. Other examples of qualitative variables are religious preferences, geographic locations, grades of a student, etc.

116

(ii) QUANTITATIVE VARIABLES (OR DATA):

Quantitative variables are numerical in nature and can be ordered or ranked. For example, the variable “age” is numerical, and people can be ranked in order according to the value of their ages. Other examples of quantitative variables are heights, weights, body temperatures, etc.

(b) Quantitative random variables (or data) can be further classified either as discrete or continuous, depending on the values it can assume. These are defined as follows:

(i) DISCRETE VARIABLES (OR DATA): Discrete variables assume values that can be counted (such as, 0, 1, 2, 3, etc.). They are

obtained by counting. Examples of discrete variables are the number of children in a family, the number of students in a class-room, the number of calls received by a switchboard operator each day for one month, etc.

(ii) CONTINUOUS VARIABLES (OR DATA):

Continuous variables can assume all values between any two specific values. They are obtained by measuring. For Example, “temperature” is a continuous variable, since the variable can assume all values between any two given temperatures. Other examples of continuous variables

are height, weight, length, time, etc. (3) RECORDED VALUES OF A CONTINUOUS RANDOM VARIABLE AND ITS

BOUNDARIES: Since continuous data must be measured, rounding answers is necessary because of the limits of

the measuring device. Usually, answers are rounded to the nearest given unit. For example, heights must be rounded to the nearest inch, weights to the nearest ounce, etc. Hence, a recorded height of 73 inches would mean any measure of 72.5 inches up to but not including 73.5

inches. Thus, the boundary of this measure is given as 72.5 – 73.5 inches. (We have taken 72.5 as one of the boundaries since it could be rounded to 73. But, we can not include 73.5 because it would be 74 when rounded). Sometimes 72.5 – 73.5 is called a class which will contain the

recorded height of 73 inches. The concept of the boundaries of a continuous variable is illustrated in the following Table I:

TABLE I

Variable Recorded Value Boundaries (Class)

Length 15 cm 14.5 – 15.5 cm

Temperature 860 F 85.5 – 86.5

0 F

Time 0.43 sec 0.425 – 0.435 sec

Weight 1.6 gm 1.55 – 1.65 gm

Note: The boundaries of a continuous variable in the above table are given in one additional

decimal place and always end with the digit 5. The concept of the class (or boundaries) of a continuous variable will be discussed again in Chapter2.

117

(4) MEASUREMENT SCALES OF A DATA: Data can also be measured by various scales. The four basic levels of measurements are nominal, ordinal, interval, and ratio. These are described

below:

TABLE II

MEASUREMENT SCALES (DEFINITIONS AND EXAMPLES)

Nominal-level Data Ordinal-Level Data Interval-level Data Ratio-level Data

Definition: The nominal-

level of measurement classifies data into mutually exclusive (non-

overlapping), exhaustive categories in which no ordering or ranking can be

imposed on the data.

Definition: The ordinal-

level of measurement classifies data into categories that can be

ordered or ranked. However, precise differences between the

ranks do not exist. (For example, when people are classified according

to their build (small, medium, or large), or when students are

classified according to their grades (A, B, C, or D), a large variation

exists among the individuals in each class.

Definition: The interval-level

of measurement ranks data, and precise differences between units of measure do

exist. However, there is no meaningful zero (i.e., starting point). For example, many

standardized psychological tests yield values measured on an interval scale. There is

a meaningful difference of one point between an IQ of 109 and an IQ of 110. There

is no true zero (i.e., no starting point) because IQ tests do not measure people

who have no intelligence.

Definition: The ratio-

level of measurement possesses all the characteristics of

interval measurement (i.e., data can be ranked, and there exists

a true zero or starting point). In addition, true ratios exist between

different units of measure. For example, if one person can li ft

200 pounds and another can li ft 100 pounds, then the ratio

between them is 2 to 1. In other words, the first person can lift twice as

much as the second person.

Examples: (i) Zip Code

(ii) Gender (Male, Female)

(iii) Eye Color

(Blue, Brown, Green, Hazel)

(iv) Political

Affiliation (v) Religious

Affiliation

(vi) Major Field of Study (Math., Comp. Sc.)

(vii) Nationality (viii) Marital Status

Examples: (i) Grade (A,

B, C, D, F) (ii) Judging (1st

place, 2nd

place, etc.) (iii) Rating

Scale

(Poor, Good, Excellent)

(iv) Ranking of Tennis Players

Examples: (i) SAT Score

(ii) IQ (iii) Temperature

Examples: (i) Height

(ii) Weight (iii) Time (iv) Salary

(v) Age (vi) Number of

Phone

Calls

(5) BASIC METHODS OF SAMPLING:

When the population is large and diverse, a sampling method must be designed so that the sample is representative, unbiased and random, i.e. every subject (or element) in the population

has an equal chance of being selected for the sample. The following sampling methods are commonly used for obtaining a random sample.

118

TABLE III

Random

Sampling

Stratified

Sampling

Systematic

Sampling

Cluster

Sampling

Convenience

Sampling

This method requires that each member of

the population be identified and assigned a

number. Then a set of numbers drawn randomly

from this list forms the required random

sample. Note that each member of the

population has an equal chance of being

selected. For a large population, computers are

used to generate random numbers which contain

series of numbers arranged in

random order.

This method requires that the population be

classified into a number of smaller

homogeneous strata or subgroups. A

sample is drawn randomly from each stratum.

For example, a population could be stratified by

age, sex, marital status, education,

religion, occupation, ethnic

background or virtually any characteristic.

This method requires that every kth

member (or item) of the population be selected to

form the required random sample. For example, we

might select every 10th house on a city block

for the random sample.

The population area is first divided into a

number of sections (or subpopulations)

called clusters. A few of those clusters are

randomly selected, and sampling is

carried out only in those clusters. For example, a

community can be divided into city blocks as its

clusters. Several blocks are then randomly

selected. After this, residents on the selected

blocks are randomly chosen,

providing a sampling of the entire

community.

In convenience sampling, we use the results that

are readily available.

*** *** ***

(6) STATISTICAL INFERENCE AND MEASUREMENT OF RELIABILITY: Definition 1: A statistical inference is an estimate or prediction or some other generalization

about a population based on information contained in a random sample of the population. That is, the information contained in the random sample is used to learn about the population.

Definition 2: A measure of reliability is a statement (usually quantified) about the degree of uncertainty associated with a statistical inference.

119

(7) ELEMENTS OF DESCRIPTIVE AND INFERENTIAL STATISTICAL PROBLEMS:

TABLE IV

FOUR ELEMENTS OF DESCRIPTIVE STATISTICAL PROBLEMS

FIVE ELEMENTS OF INFERENTIAL STATISTICAL PROBLEMS

1. The population or sample of interest. 2. One or more variables (characteristics of the

population or sample units) that are to be investigated.

3. Tables, graphs, numerical summary tools. 4. Identification of patterns in the data.

1. The population of interest.

2. One or more variables that are to be investigated.

3. The sample of population units.

4. The statistical inference about the population based on information contained in the random sample of the population.

5. A measure of reliability for the statistical inference.

120

Appendix F

PROBABILITY AND COUNTING TECHNIQUES

By

Dr. M. Shakil

“KEY CONCEPTS AND FORMULAS”

(A) SETS

DEFINITION OF A SET: A set is a well-defined collection of distinct objects, called the elements or

members of the set. One way of describing a set is by listing its elements in a “Roster.” For example: A = {a, e, i, o, u}; X = {0, 8, 2, 3, 7, 10}; etc.

NOTE: When the list is extensive and the pattern is obvious, an ellipsis may be used.

For example: B = {a, b, c, …, z}.

NOTE: The above form is also called the Tabular Form of the Set. The elements of a set are separated by commas and are enclosed in brackets { }.

SUBSET: A is a subset of B if every element in set A is also in set B. The notation is A

B. NOTE:

Every set is a subset of itself, i.e., A A, where A is any set.

NULL OR EMPTY SET: The null or empty set is the set with no elements. The

notation is or { }.

NOTE: The empty set is a subset of every set, i.e., A, where A is any set.

UNIVERSAL SET: The UNIVERSAL SET, U, is the set that contains all possible

elements under consideration in a problem.

NUMBER OF SUBSETS OF A SET: If a set contains “N” elements then the number of subsets is equal to 2N and the number of proper subsets is equal to 2N - 1.

POWER SET: The set of all the distinct subsets of a set A is called the power set of A. The notation is P(A). Thus, if n(A) = N, then n(P(A)) = 2N, where A is any finite set.

UNION OF SETS: The union of sets A and B is the set of all elements which are either

in A or in B. The notation is A B. Thus A B = {x x A or x B or x both A and

B}.

121

INTERSECTION OF SETS: The intersection of sets A and B is the set of all those

elements, which are common to both A and B. The notation is A B. Thus A B = {x

x A and x B}.

COMPLEMENT OF A SET: Let A be a set and U the universal set. Then the complement of set A with respect to the universal set U is the difference of sets U and A,

i.e., U – A, and is denoted by _

A or A. Thus A = {x x U and x A}.

DISJOINT SETS: Any two sets A and B are said to be disjoint if and only if A B =

.

VENN DIAGRAMS: We can visualize sets and operations on sets and guess the truth of

a number of propositions on sets with the help of geometric diagrams, known as Venn diagrams. A convenient way to use Venn diagrams is to represent the universal set U by rectangular area in a plane and the elements which make up U by the points of this area.

Simple plane areas bounded by circles or their parts drawn within the rectangular area can give a simple and instructive picture of sets and operations on sets. We can think of

each set as consisting of all points within the corresponding circle.

(B) PROBABILITY

1. PROBABILITY EXPERIMENT: A probability experiment refers to any act or

process or procedure that can be performed that yields a collection of outcomes (or results). The outcome of the probability experiment is not known in advance of the act. For example:

(i) The act of tossing a fair coin.

(ii) The act of rolling a fair die.

(iii) The act of drawing a single card from an ordinary deck of 52 playing cards.

2. SAMPLE SPACE: A sample space is defined as the set of all possible outcomes (or results) that can occur in a probability experiment such that exactly one outcome occurs at a time. The letter S will be used to denote a sample space. In a Venn diagram, a

rectangle will be used to represent a sample space and the outcomes which make up S by the points drawn within this rectangle.

For example:

(i) If a fair coin is tossed, there are two possible, equally likely outcomes in the sample space that could occur in this experiment (Heads, H, or Tails, T). Thus S = {H, T}.

(ii) If a fair die is rolled, then S = {1, 2, 3, 4, 5, 6}.

122

3. EVENT: An event is any collection of results or outcomes of a probability experiment. Thus, an event may be defined as any subset of the sample space. For example:

(i) If a fair coin is tossed, each of the two individual outcomes (Heads, H, or Tails, T)

may be referred to as events since the sets {H} and {T} are subsets of S = {H, T}. (ii) If a fair die is rolled, then each of the individual outcomes in this probability

experiment is an event because {1}, {2}, {3}, {4}, {5}, and {6} are subsets of S = {1, 2, 3, 4, 5, 6}.

Similarly, {1, 3, 5} and {2, 4, 6} represent the events of rolling an odd number and an even number on the die respectively. Note that no two outcomes can occur at the same

time.

The event of rolling a 7 on a die is or { } because is a subset of S, i.e. S.

Notations: The capital letters such as A, B, C, D, E, etc. will be used to denote an event. In a Venn diagram, a circle drawn within the rectangle will be used to represent an event and the outcomes which make up the event by the points of this circle.

4. SIMPLE EVENT: A simple event is an outcome or an event that cannot be further

broken down. For example: (i) If a fair coin is tossed, each of the two individual outcomes (Heads, H, or Tails, T)

may be referred to as simple events.

(ii) If a fair die is rolled, then each of the individual outcomes {1}, {2}, {3}, {4}, {5}, and {6} is a simple event.

5. COMPOUND EVENT: A compound event is any event combining two or more simple events. For example:

(i) If a fair die is rolled, then {1, 3, 5} and {2, 4, 6} may be referred to as compound events.

6. COMPLEMENT OF AN EVENT: Let E be any event and S the sample space of a

probability experiment. Then the complement _

of the event E with respect to the sample space S, denoted by E or E, consists of those

outcomes which are not in E, i.e.

E = {x x S and x E}. For example:

(i) If a fair die is rolled, then {1, 3, 5} and {2, 4, 6} may be referred to as complementary events of each other.

7. FORMULAS:

123

Let E and F be any two events from the sample space S of a probability experiment. Let

)(En the number of outcomes in E and )(Fn the number of outcomes in F . Let

)(EP denote the probability of event E occurring, and )(FP denote the probability of

event F occurring.

(i) PROBABILITY OF AN EVENT: )(

)()(

Sn

EnEP (Classical Probability)

(ii) PROBABILITY OF AN IMPOSSIBLE EVENT: If the event E cannot occur, then 0)(EP .

(iii) PROBABILITY OF A CERTAIN EVENT: If the event E is certain to occur,

then 1)(EP .

(iv) The probability of any event E is a number between 0 and 1, inclusive, i.e.

.1)(0 EP

(v) ADDITION RULES:

(a) FOR MUTUALLY EXLUSIVE EVENTS: The events E and F are called mutually exclusive (or disjoint) if FE , i.e. they cannot both occur together, i.e.

simultaneously. For mutually exclusive events, we have 0)( FEP , and the addition

rule is defined as follows:

)()()()( FPEPFEPForEP .

(b) FOR NON-MUTUALLY EXLUSIVE EVENTS: The events E and F are not

mutually exclusive if FE , i.e. they both occur simultaneously. In this case, the

addition rule is defined as follows:

)()()()()( FEPFPEPFEPForEP .

(vi) MULTIPLICATION RULES:

(a) FOR INDEPENDENT EVENTS: The events E and F are called independent if the occurrence of one event, say, E does not affect the probability of the occurrence of

the other event F . The multiplication rule is defined as follows:

)sec( trialondainoccursFeventandtrialfirstainoccursEeventP

)().()()( FPEPFEPFandEP .

124

(b) FOR DEPENDENT EVENTS: The events E and F are called dependent if the occurrence of one event, say, E affects the probability of the occurrence of the other

event F . In this case, the multiplication rule is defined as follows:

)|().()( EFPEPFandEP ,

where EF | is read as “ ,EgivenF ” and )|( EFP represents the probability of event F

occurring after it is assumed that event E has already occurred.

8. FORMULAS (CONTINUED):

Let A and B be any two events from the sample space S of a probability experiment. Let

)(An the number of outcomes in A and )(Bn the number of outcomes in B . Let

)(AP denote the probability of event A occurring, and )(BP denote the probability of

event B occurring.

(i) CONDITIONAL PROBABILITY RULE:

(a) DEFINITION: A conditional probability of an event B represents the probability of event B occurring after it is assumed that some other event A has already occurred. It is denoted by )|( ABP .

(b) FORMULA: The conditional probability of event B occurring, given that event A

has already occurred, is denoted by )|( ABP , and is given by

)(

)(

)(

)()|(

An

BAn

AP

BAPABP

, provided 0)(AP , or 0)(An .

(ii) PROBABILITY OF THE COMPLEMENT OF AN EVENT: Let E denote the

complement of the event E with respect to the sample space S . Then

)(EP The probability that the event E will not occur )(1 EP ;

)(EP )(1 EP ; and )(EP 1)()( SPEP .

(iii PROBABILITY OF “AT LEAST ONE”:

(a) “At least one” is equivalent to “one or more.”

(b) “None” and “at least one” are complement of each other.

(c) P(at least one) = 1 – P(none) and P(none) = 1 – P(at least one)

125

(iv) TESTING FOR INDEPENDENCE:

Two events A and B are “independent”

if

Two events A and B are “dependent” if

)()|( BPABP

OR

)().()( BPAPBAP

)()|( BPABP

OR

)().()( BPAPBAP

(v) BAYES’ FORMULA:

)|(.)()|(.)(

)|(.)()|(

BAPBPBAPBP

BAPBPABP

9. FORMULAS FOR ODDS:

Let E be any event from the sample space S of a probability experiment. Let

)(# EnF the number (#) for (in favor of) the event E , and )(# EnA the number

(#) against the event E . Let )(### SnAFT = total number (#) of outcomes in S .

Then, we have the following definitions:

(a) Odds in favor (for) of the event E A

F

#

#, or AF : , i.e.

Odds in favor (for) of the event E

)(

)(

EP

EP;

(b) Odds against the event E F

A

#

#, or FA : , i.e.

Odds against the event E

)(

)(

EP

EP;

(c) )(EP T

F

#

#, which is the probability in favor of the event E ;

126

(d) )(EP T

A

#

#, which represents the probability against the event E .

(C) COUNTING TECHNIQUES

(i) FUNDAMENTAL PRINCIPLE OF COUNTING:

For a sequence of two events (or choices or tasks), say, A and B , in which the first event

A can occur m ways and the second event B can occur n ways, the events A and B together can occur a total of nm . ways. Here, none of the two events (or choices or

tasks) depends on another.

NOTE: The fundamental counting principle easily extends to situations involving more than two events.

(ii) FACTORIAL NOTATION:

If 0n denotes an integer, then the factorial symbol !n is defined as follows:

1!0 ;

1!1 ;

1.2.3...)2(.)1(.! nnnn

(iii) FACTORIAL RULE:

A collection of n different items (or objects) can be arranged in order !n different ways. (iv) PERMUTATION RULE (WHEN ITEMS ARE ALL DIFFERENT):

(a) DEFINITION: A permutation is a sequential arrangement of n different (or

distinct) items (or objects) taken r at a time, without replacement or repetition, denoted

by rn P , where nr , in which the order makes a difference.

(b) FORMULA: )!(

!

rn

nP rn , where nr .

(c) SHORT-CUT FORMULA: )1(...)2(.)1(. rnnnnP rn , where nr .

(d) NOTE: !nP nn

127

(v) COBINATION RULE:

(a) DEFINITION: A combination is a group (or set or collection or selection) of n

different (or distinct) items (or objects) taken r at a time, denoted by rn C or n

r

, where

nr , in which the order is not important.

(b) FORMULA: )!(.!

!

rnr

nC rn , where nr .

(c) NOTE: 1nn C

(vi) PERMUTATION RULE (WHEN SOME ITEMS ARE IDENTICAL OR

SIMILAR OR ALIKE TO OTHERS):

If there are n items (or objects) with 1n alike,

2n alike, … , kn alike, the number of

permutations of all n items (or objects) is given by

!!!!

!

321 knnnn

n

, where nnnnn k321 .

(vii) APPLICATIONS OF PERMUTATION AND COMBINATION RULES TO

PROBABILITY:

(a) The probability of a permutation (arrangement) is given by

tsarrangemen ofnumber totalthe

npermutatiotheformingwaysofnumber the

(b) The probability of a selection (combination) is given by

selections ofnumber totalthe

selection that makingof waysofnumber the

(viii) AN USEFUL FORMULA ON SIMPLE RANDOM SAMPLING:

(a) DEFINITION OF A SIMPLE RANDOM SAMPLE:

A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring.

The sample is then called a simple random sample.

NOTE: In the above definition, the sample is always a subset of the population with

Nn .

128

(b) FORMULA: The number of different simple random samples of size n from a

population of size N is defined as a combination of N objects by selecting n at a time without replacement, and is given by the following formula:

)!(.!

!

nNn

NC nN , where Nn .

129

FLOW-CHART FOR PROBABILITY RULES Addition and Multiplication Rules

N

O

SIMPLE EVENT: Sample Problems:

“What’s the probability that an answer is correct on

a multiple choice test question, with 5 possible choices, when the question is answered randomly?”

“What’s the probability that a couple, who wants to have a total of three children, will have two children

as girls? Assume that the probability of having a girl is the same as the probability of having a boy.”

Yes

P (E) = n (E) / n (S) Sample Answers:

P (correct answer) = 1/5 P (2girls out of 3 children) = 3/8

COMPOUND EVENTS:

Do the compound events contain ? “OR” “BO TH” (i.e ., “AND”), or

“AT LEAST O NE”

ARE TH E EVENTS MUTUALLY EXCLUSIVE?

Use Addition Rules

ARE TH E EVENTS

INDEPENDENT OR DEPENDENT?

Use Multiplication Rules

P (AT LEAST O NE) = 1 – P (NONE) “What’s the probability that the

couple will have at least one boy out

of three children? Solution: P (AT LEAST O NE BOY) = 1 – P (NO BOYS) = 1 – P (ALL GIRLS) = 1 – (1 / 2

3) = 7 / 8 (Answer)

OR

BOTH

(AND)

AT LEAST

ONE

P (A or B) = P (A) + P (B)

“What’s the probability that the last digit of a randomly selected phone number is 5 or 7?”

Solution: P (last digit 5 or 7) = (1/10) + (1/10) = 1/5

P (A & B) = P (A) . P (B)

Example: “From a deck of 52 playing cards, two cards are randomly selected in succession. One is replaced before another is

selected.” P (both cards are queen) = (4/52) .(4/52) = 1/169

P (A & B) = P (A) . P (B A) Example: “From a deck of 52 playing cards, two

cards are randomly selected in succession. One is not replaced before another is selected.” P (both cards are queen) = (4/52) .(3/51) = 1/221

Yes

NO

P (A or B) = P (A) + P (B)

- P (A and B) Example: P [last digit of a randomly selected phone number is odd or (< 7)]

= (5/10) + (7/10) – (3/10) = 9/10

DEPE- NDEN

T

INDEPE-

NDENT

Appendix G

PROBABILITY AND COUNTING TECHNIQUES

By

Dr. M. Shakil

130

FUNDAMENTAL

PRINCIPLE O F COUNTING

m1. m2 … mn OR mn

ARE THINGS IN ARRANGEMENTS?

YES

NO

NO

m!

ARE DIFFERENT ARRANGEMENTS

TO BE COUNTED SEPARATELY?

YES

PERMUTATIONS

nPr

YES

NO

FLOW-CHART FOR COUNTING TECHNIQUES Permutations and Combinations

COMBINATIONS

nCr

COUNTING RULES “IS THERE

REPETITIO N OF

THINGS?”