Download Absolute value: Absolute value of a number is its distance from zero

MIAMI-DADE COLLEGE, HIALEAH CAMPUS
DEPARTME NT OF MA THEMA TICS, 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
Spring Term, 2009
Dr. M. Shakil & Prof. R. A. Sanchez
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.,
a a a . (ii) It‟s the distance between two numbers on the
number line, i.e., a b
3 5 , i.e., 2
5 3
b a (a and b are real numbers). Example:
2 , i.e., 2
2.
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., a bi
42
4 3i
3
2
16 9
25
a2
b 2 . Example:
5
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., a ( a)
Example: 5 ( 5) 0 , i.e., 5 and - 5 are additive inverses.
0.
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:
A
2
1 0
5 4
11
A
2 5
4
1 0 11
A ( A)
0 0 0
0 0 0
Algebraic expressions: Expressions that are made up of variables, numbers,
grouping symbols, operation signs, and exponents. Examples: 3x 2 y 7 z ;
2y2
7 z 34 p 1 ;
2 x 5 3x
4x 2 6
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 a 1 and the common
difference of successive members is d, then the nth term of the sequence is
given by: a n a1 n 1 d , and, in general, a n a m n m d .
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,
a1 2d .... a1 n 2 d
a1 n 1 d , can
the arithmetic series: a1 a1 d
n 1
be written using sigma notation as:
a id . Likewise, an arithmetic series
i 0
m
a1
a2
a3
.......
am
1
aj
a m , can be written as
j 1
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 3 2 (3 2) 5
6
Associative property of multiplication: It states that the way in which numbers
being multiplied are grouped does not change the product. Example:
5 3 2 (3 2) 5
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: 1.333333333 . 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: 53 5 5 5 . 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 m4 n2
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.
15
21
3 5
3 7
5
7
Example 2: When common factors exist on opposite sides of an equation, you
can also cancel as the following shows:
3x
3y
x
y
Example 3: We can also cancel using addition and subtraction.
x
5
x
y
5
y
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:
3x 6
3
x 6
1
3x
3x
3
6
3
6
3
x
2
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
Counting /
Natural
Definition
They are the positive numbers we use
to count objects
Whole Numbers
The whole numbers are the numbers
0, 1, 2, 3, 4, and so on (the natural
numbers and zero).
Integers
The integers are all the whole
numbers and their opposites (the
positive whole numbers, the negative
whole numbers, and zero).
Rational
Numbers
Irrationals
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.
An irrational number is a number with
a decimal that neither terminates nor
repeats. An irrational number cannot
a
be written as a fraction where a and
b
Examples
1,2,3,4,5,6,7,8,9,10
…
0,1,2,3,4,5,6,7,8,9,
10…
…-3,-2,-1, 0, 1,
2,3…
5 3 8
, , ,
1 4 9
1.33333,
0.25
3
2 1.41421...
= 3.14159...
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.
5
,1.33333,0.25
1
2 1.41421...
-1, 0, 1
Coefficient: A number multiplied by a variable in an algebraic expression.
Example: In 5x 3 6 x 2 10x , 5, 6 and 10 are coefficients.
Column Matrix: A matrix with only one column. Example:
1
5
7
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?
2 5
7
; and
3 4
6
First, find the multiples of the denominators, 3, 4 and 6.
Denominator
3
4
6
3
4
6
Multiples
6 9 12 15 18 21
8 12 16 20 24 28
12 18 24 30 36 42
24
32
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
Common
factors
Factors
36
1, 2, 3, 4, 6, 9, 12, 18, 36
24
1, 2, 3, 4, 6, 8, 12, 24
18
1, 2, 3, 6, 9,18
GCF
1, 2, 3, 6
6
Example of Relatively Prime Numbers:
Numbers
Factors
32
1, 2, 4, 8, 16, 32
65
1, 5, 13, 65
Common
factors
GCF
1
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
3
3,6,9,12,15,18,21,24,27,30,33,36,39...
Common
LCM
Multiples
15,30.....
5
15
5,10,15,20,25,30,35,40,45,50,55,60,65...
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 z a bi
where a and b are real numbers and i
1 is called the imaginary number.
The number a is called the real part of the complex number z a bi and b is
called the imaginary part of the number. Note that the imaginary is defined to be:
i
1 , and i 2
2
1
2
1 . .Now, you may think you can do this:
2
i2
1
1
1 1 . 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
Direct
Directly proportional to the
exponential function of x
Inverse
Inversely proportional to the
exponential function of x
Mathematical
Definition
y
Value of k
k
y
x
k
y
xn
k
y x
k
y xn
kx
n
y
kx
y
k
x
y
k
xn
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
y 2x 3 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.
a c
, b and c are the means and a and d are the extremes.
b d
So, the cross product is: ad bc . The cross product of the proportion shown is
5 15
5 36 12 15 .
12 36
In the proportion
14
Cube Root: One of three equal (identical) factors of a given number denoted
3
1
3
5 because.
or x , is a number a such that a 3 x . Example: 3 125
3
3
5
125 .
5
5
5
125 ; 27 3 because. 33 27 .
3 3 3 27
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.
x
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 10 2) + (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 10 0) + (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 f x a n x n a n 1 x n 1 ...... a1 x a 0 .
Thus, the degree of the polynomial is n, which is the largest exponent which
appears in the above polynomial. Examples: (i) 3x has degree 1. (ii) x 3 5 has
degree 3. (iii) x 2 2 x 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
7x
pieces to be considered (i.e., the part). For example, in the fraction
{7x is
16
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 betwee n 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
Direct
y
kx
Directly proportional to the
exponential function of x,
y
kxn
Value of k
k
y
x
k
y
xn
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 poi nts (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:
d
x2
x1
2
y2
y1
2
17
d
4
1
d
4 1
d
3
d
9 49
d
58
2
2
2
4
3
4 3
2
2
72
Distributive Property: The sum of two addends multiplied by a number is the
sum of the product of each addend and the number.
ab c d
a b c d
ab ac ad
ac ad
3x
bc bd
y 2z
3x 3 y 6 z
2x 1 x 2
2x 2
4x
x 2
Dividend: A number that is divided by another number. For example, given the
18
division
2 , 18 is the dividend, 9 is the divisor and 2 is the quotient.
9
Dividing Powers Property: To divide two powers of the same base, subtract the
exponent of the denominator from the exponent of the numerator.
am
an
a
m n
x3 y5
x2 y
x3 2 .y 5 1
xy 4
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:
18
9
2
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:
19
3 with remainder of 1. Here, we divide by 6, so 6 is called the divisor,
6
however, 19 is not divisible by 6 because the remainder is 1 (not zero.) The
18
number 3 Remainder 1 (or 3
1
) is called the quotient.
6
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:
3 x 18
3 x 18
3
3
x 6
Divisor: A number by which another number is to be divided. For example, given
19
the division
18
9
2 , 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
f x
1
x 1
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
,1
1,
.
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: 3 2 x 5 8 x 3 ; 3x 2 5x 4 12 .
Equation of a Straight line: Any equation that can be put in the form
A x B y C , where A , B, and C are real number and A and B are not both 0, is
called a linear equation in two variables x and y . The graph of any equation of
this form is called a straight line. The form A x B y C 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 ( x1 , y1 ) and ( x2 , y 2 ) is given by the following
formula:
20
To find the slope of a line we have to know:
1.
2.
Two points in the line, or
The equation of the line with y isolated. The coefficient of x is the slope.
(II) Slope Facts:
1.
Slope m
y2
x2
y1
.
x1
Horizontal lines have slope m = 0 .
Vertical lines have undefined slope.
Parallel lines have the same slope.
If m1 and m2 are slopes of the two perpendicular lines, then m1 x m2 = -1.
1
This also written as m1 =
.
m2
(III) Different Forms of the Equation of a Straight Line: The following table lists
different forms for the equation of a straight line.
2.
3.
4.
5.
Forms
Slopeintercept
Point-slope
Standard form
Equation
y = mx + b
y – y1 = m(x –
x1)
Ax + By = C
Two-intercept
Vertical
x=a
Horizontal
y=b
Application
Used when you have the slope and the yintercept.
(x1, y1) is a point on the line. Used to find
the equation.
If possible, A is nonnegative and A, B, and
C are relatively prime integers.
Used when you have both intercepts.
All points have x-coordinate a. Slope
undefined.
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:
(i ) x 6 5 and x 11 0;
(ii) 3 x 5 10 and 3x 5.
Equivalent Expressions: Expressions that simplify to an equal value when
numbers are substituted for the variables of the expression. Example:
3 2x 5
6 x 15
4 x 2 3 6 x 10 x 1
Equivalent Fractions: Fractions that have the same decimal form. Example:
24
6
and since both are equal to 2.0 in decimal form.
12
3
Equivalent Ratios: Ratios that represent the same fractional number, value, or
measure.
24
6
and are equivalent fractions since they have the same
12
3
24
6
decimal expansion. Thus,
and are equivalent ratios since they represent the
12
3
same amount.
Above we state that
Evaluate a Numerical Expression: To perform operations to obtain a single
number or value.
For example, "Evaluate 24 9 2 3 ” means to perform all the operations in the
given numerical expression. In other words, 24 9 2 3 24 9 6 24 3 8 .
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 x 3 2 xy y 2 for x 2 and y
1 means substituting
given values for x and y in the expression and calculate it. In other words,
x3
2 xy
y2
23
2 2
1
1
2
8 4 1 3.
Even Number: An even number is an integer of the form n 2k , 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:
358 300 50 8 3 102 5 101 8 100
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
x3
x
3
2
6
6
16
1
2
16
2
1
2
Expanded
x3
6
1
62
2
16
1
2
x x x
1
1
36
6 6
16
4
Exponential: A number written with an exponent. Example: 5 3 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 6 5 30 . 30 is a multiple of both
6 and 5 .
Numbers
Factors
36
1, 2, 3, 4, 6, 9, 12, 18, 36
24
1, 2, 3, 4, 6, 8, 12, 24
18
1, 2, 3, 6, 9,18
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 3a x c tells you to multiply 3a by x c .
When you do that, you get the sum 3a x c 3ax 3ac . When you do that in
reverse, by writing 3ax 3ac 3a x c , 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., a b c d ac ad bc bd
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)
(ii)
(iii)
The sum of three times a number and eight is equal to three:
3x 8 3 .
The product of a number and the same number less 3 is equal to
twice the number: x x 3 2 x .
A number divided by the same number less five is the same
x
number increased by 3:
x 3.
x 5
Fraction: A number that can be written as a quotient of two quantities. Example:
5 12 8
,
, .
3 7 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:
y f (x) , where x denotes an independent variable and y denotes a dependent
variable. Example: y
2 x 1 and y
x2
1 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:
51 3 17
130 2 5 13
550
2 5 2 11
5032 2 3 17 37
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
y
f x
5x 9
Figure 2
y
f x
x2
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 y 3x 2 .
25
Note that, in the line y
2.
3x 2 , the slope is equal to 3 and y-intercept is equal to
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
x 5 is read "x is greater than 5." This means that x can equal any real number
greater than 5.
(ii) x 5 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
36
1, 2, 3, 4, 6, 9, 12, 18, 36
24
1, 2, 3, 4, 6, 8, 12, 24
18
1, 2, 3, 6, 9,18
Common factors GCF
1, 2, 3, 6
6
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,
a 0 0 a a
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,
a 1 1 a a.
27
Improper Fraction: A fraction where the numerator is equal to or larger than the
denominator.
Fractions such as
15
7
and
are improper fractions
7
5
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
Inverse proportion ( Indirect)
Inversely proportional to the
exponential function of x,
y
k
x
y
k
xn
Value of k
k
y x
k
y xn
Inequality: Any mathematical sentence that contains the symbols >(greater
than), <(less than), <(less than or equal to), or >(greater than or equal to).
Examples:
3x 2
4x 6
5x
2 3x 1
5 x 14
Infinity: An expression increases to infinity if it continues to increase without
limit. Example:
Functions such as the following one y 5x 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.
a, b
x; a x b
a, b
x; a x b
[a, b) x; a x b
(a, b] {x; a x b}
(a, ) {x; x a}
[a, ) x; x a
( , b) {x; x b}
( , b] {x; x b}
( , )
 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 halfopen (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)
(ii)
The additive inverse states: a + (-a) = 0. An example is 5 + (-5)
= 0.
The multiplicati ve 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
Inverse proportion ( Indirect)
Inversely proportional to the
exponential function of x
Mathematical
Definition
y
k
x
y
k
xn
Value of k
k
y x
k
y xn
Irrational Number: A number whose decimal form is non-terminating and nonrepeating. 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 y k x z 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
y varies jointly with x and z .
z varies jointly with r and the square of
s.
V is directly proportional to T and is
inversely proportional to P .
F varies jointly with m1 and m 2 and
inversely with the square of r .
Algebraic
Equations
y k xz
z
V
F
k r s2
kT
P
k m1 m2
r2
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
1
given by m v 2 . (iii) The kinetic energy of a body rotating about an axis and
2
having angular velocity
and moment of inertia I about the axis is given by
31
1
I 2 . (iv) Note that the Potential Energy is defined as the energy a body
2
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
3
3,6,9,12,15,18,21,24,27,30,33,36,39...
5
5,10,15,20,25,30,35,40,45,50,55,60,65...
Common
Multiples
LCM
15, 30 ... 15
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:
3 5 and 7 5
(i)
they are like terms. They have the
same root.
(ii)
3 x 2 y and 8 x 2 y
same variables.
they are like terms. They have the
32
(iii)
5mn 2 and 3m 2 n
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
y
5x 9
A non-linear graph
y x2
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 (nonvertical) 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 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.
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.
12
18
12 6
18 6
2
3
34
Method 2:
Divide the numerator and denominator by any common factor. Keep dividing until
there are no more common factors.
12
18
12 2
18 2
6 3
9 3
2
3
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
4 4
4
2
6 8 14
3) 2,3,5,6,8,9,10,11 The median is
2
2
2) 1,2,3,4,4,5,6,7 The median is
7
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
x x2 y1 y 2
exactly between two other points. It is given by M x, y M 1
.
'
2
2
Example: Find the midpoint between the points (1, -7) and (-5, -3). Draw a sketch
in a rectangular Cartesian plane also
M x, y
M
x1
2
x2 y1
'
y2
2
1 ( 5) 7 ( 3)
M(
,
)
2
2
M(
4 10
,
)
2
2
M ( 2, 5)
Sketch:
N
Natural numbers (or counting numbers or positive integers ): It is defined as
the set of counting numbers 1, 2, 3, 4,  , denoted as N 1, 2, 3, 4,  .
Natural logarithm: The logarithm of a positive real number a with the base e is
called the natural logarithm, and is denoted as ln a . Note that the number e is
36
defined as e
lim 1
n
1
n
2.71828 , where n is a natural number.
Nature of the roots: A classification of the roots of a quadratic equation
ax 2 bx c 0 where a, b, and c are rational numbers and a 0 , which is
indicated by the discriminant, b2 4ac , 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 b 2 4ac 0, or 0, or 0 .
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
x a
, x 0.
xa
x is defined as the
Negative square root: Let x 0 be a real number. Then
5 is the negative square root of 5 .
negative square root of x . Example:
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 x 0 that can be written as a
fraction, which when expressed as a decimal is either a terminating or repeating
1
, and 0.333... 0. 3 are some negative
decimal. For example, 5, 0.5,
2
rational numbers.
Negative of a polynomial: If P x be a polynomial in x , then the negative of
P x , denoted by P x , is that polynomial which is obtained by changing the
sign of each term in P x . Example: Let P x x 2 x 1 . Then
P x
x 2 x 1.
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 F m a (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 P1 exerts a force on another particle
P2 , P2 simultaneously exerts a force on P1 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 m1 and m 2 with r as the distance between their centers, and
F the force of gravitational attraction between two bodies. Then, by Newton's
1
law of gravitation, we have F m1 m2 , and F
. Combining these, we obtain
r2
m1 m2
G m1 m2
F
, or F
, where G is a constant of proportionality called the
2
r
r2
gravitational constant. The value of G in si units, G 6.67 10 11 Newton m 2 kg 2 ,
and in cgs units, G 6.67 10 8 dyne cm 2 g 2 .
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: x 2 x 1 0 ; x 2 x y 1 0.
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 x n c when n is odd or the nonnegative solution of
x n c when n is even and nonnegative. For any real number c and any
1
positive integer n , the nth root of c is denoted by either
n
c or c n .
nth term: The final term of a finite sequence of elements a1 , a2 ,
arbitrary term of an infinite sequence.
, an , or an
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:
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
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 1, 2, 3, ... and those to the left of 0 with the
negative integers, 1, 2, 3, ... .
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
0,1, 2, 3, 4, 5, 6, 7, 8, 9 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.
a
Numerator: In a fraction , b 0 , the number a above the fraction bar is called
b
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 i nvolving numbers or a number system.
Numerical coefficient: The numerical factor in a term is defined as its numerical
1 2
x are
coefficient. Example: The numerical coefficients of the terms 15 x and
4
1
15 and
respectively.
4
Numerical equation: An equation in which the coefficients and constants are
numbers rather than letters. For example, the equation 15 x 1 29 is a
numerical equation, whereas a x b c is a literal equation.
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: 15 1 14 is a true
statement and 15 1 14 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., 5 x 14 y 29 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 90 0 .
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 90 0 and less than
1800 , i.e. 90 0
180 0 . Example:
41
Obtuse triangle: If one of the angles, say, , of a triangle is obtuse, i.e. if
90 0
180 0 , then it 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 y f (x) is called odd if and only if its graph is
symmetric with respect
to the origin. Thus a function is odd if a point x, y is on the graph of y f (x) ,
then
x, y is also on the graph of y f (x) , i.e. if f ( x)
f ( x) . Example:
3
y 2x .
Odd number: A whole number that is not divisible by 2 , such as 1, 3, 5, ... , is
called an odd number, i.e. any number that has a remainder when divided by 2 .
Notation: It is denoted by 2 n 1 , 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.
f x1
f x2 , x1 x2 X , where f x1 , f x2 Y . Example: y f x 2 x 1 .
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: y
f x
3x 2.
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 a, b , consists of all real numbers
x such that a x b . Note: A closed interval, denoted by a, b , consists of all
real numbers x such that a x b .
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 3 ( 3) 0 .
Opposite of a number A number that is the same distance from zero o n 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
B
A
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) 2 5.1 7 , and (ii) 12 5
4.
Ordered pair: An ordered pair, denoted as x, y , is a pair of numbers written
44
within parentheses in which the order of the numbers is important. An ordered
pair x, y 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 x, y
y, x as they represent
two different points. Example:
3, 4 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 x, y, z , is a set with three
numbers written within parentheses in which the order of the numbers is
important. An ordered triple x, y, z is used to locate a point in the threedimensional 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 x, y 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 x, y 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
C
C
B
B
A
A
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 90 0 .
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 meas ure.
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, y ax 2 bx c, where a , b , and c are real numbers and
1 2
x ,
a 0 . Example: The following figure represents the parabola, y
4a
i.e. x 2 4 a y , where a 0 , with Vertex 0, 0 , Focus 0, a , Directrix y
y -axis, i.e. x 0 as the Axis of Symmetry.
a , and
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 its 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 = a b c , where a , b , and c
are the lengths of its edges.
47
(c) Area of the entire surface of the rectangular parallelepiped =
2 ab bc ca .
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 y 2x 1 and 2 y 4 x 3 are parallel as they have the same
slope, i.e. m 2 .
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
C
D
B
A
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 ( x a) n is
given by
n
n n
n n
n n j j
n n
n n j j
( x a) n
x
x a 
x
a

a
x
a ,
0
1
j
n
j
j 0
n
def
n!
is
j
j! n j !
called the binomial coefficient. A triangular array of the binomial coefficients in
the above expansion of ( x a) n is called the Pascal triangle, named after the
French mathematician, Blaise Pascal (1623 – 1662), as provided in the following
figure.
where, for integers j and n with 0
j
n , the symbol
n
Cj
Pascal's triangle
1
1
1
1
1
1
1
2
3
4
5
1
3
10
1
4
6
10
1
5
1
etc
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
( x a) 4 are the numbers in the 4th row of the above Pascal‟s triangle, given by
(x
a) 4
1x4
4 x3 a
6 x2 a2
4 x a3
1a 4 .
49
Pattern: A design (geometric) or sequence (numeric or algebraic) that is
predictable because some aspect of it repeats. The following are some e xamples
of pattern.
(i)
Geometric pattern:
(ii)
Numeric pattern: 5, 10, 15, 20,  .
(iii)
Algebraic pattern: a 4 , a 8 , a 12 ,  .
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)
(ii)
(iii)
(iv)
Error: The difference between a number and the number it
approximates, i.e., the difference E A X , where A is an
approximation of X .
E
E
Relative error: It is defined as
, or sometimes
.
X
X
Percent error: It is defined as the relative error expressed in
E
percent, i.e., 100
.
X
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
S C
the cost of an article, then
100 is defined as the percent profit on cost (or
C
percent gain). Example: If an article costs $10 and sells for $12, then the percent
1
100 , or 20 %.
gain is
5
Percent profit on selling price: If S denotes the selling price and C the cost of
S C
100 is defined as the percent profit on selling price (or
an article, then
S
percent gain on selling price). Example: If an article costs $10 and sells for $12,
51
1
100 , or 16.67 %. Note: The
6
percent gain on the cost price is always greater than the percent gain on the
selling price.
then the percent gain on the selling price is
Perfect cube: It is defined as a number or polynomial that is an exact third power
(or cube) of some number or polynomial. Example: 8 2 3 , 27 33 , and
x3
3 x2 y
3x y2
y3
x
3
y .
Perfect number: An integer which is equal to the sum of all of its factors except
itself. Example: 28 is a perfect number, since 28 1 2 4 7 14 . 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 n 1 .
Perfect square: It is defined as a number or polynomial that is an exact second
power (or square) of some number or polynomial. Example: 4 2 2 , 16 4 2 , and
2
x2 2 x y y2
x y . Alternatively, a positive integer resulting from
multiplying an integer by itself, i.e. if a is a positive integer such that a n n ,
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
2
x y ;
factored) as the square of a binomial. Example: x 2 2 x y y 2
x2
4x
4
x
2
2 .
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
real constant p such that f ( ) f (
is called periodic if there exists some
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 k
sin , and cos
2 k
cos , where k is any int 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)
(ii)
(iii)
(iv)
(v)
All possible permutations of the letters a, b, and c are:
a, b, c, a b, a c, b a, b c, c a, c b, a b c, a c b, b a c, b c a, c a b, c b a .
A permutation of n different objects taken all at a time is n factorial, i.e., n ! , where n 0 is an integer. Note that
n! n (n 1) (n 2)  3. 2 .1 ; & 0 ! 1 .
A permutation of n different objects taken r at a time without
n!
n (n 1) (n 2)  (n r 1) ,
repetitions is given by n Pr
(n r ) !
where r n . Example: The number of permutations of the letters
a, b, and c taken 2 at a time without repetitions is 3 P2 6 . These
permutations are: a b, a c, b a, b c, c a, c b . are: a b, a c, b a, b c, c a, c b .
A permutation of n different objects taken r at a time with
repetitions is given by n r . Example: The number of permutations
of the letters a, b, and c taken 2 at a time with repetitions is 3 2 9 .
These permutations are: a a, a b, a c, b a, bb, b c, c a, cb, c c .
A circular permutation is an arrangement of objects (or things)
around a circle. Example: The total number of circular
n!
permutations of n different objects taken n at a time is
,
n
because each arrangement will be exactly like (n 1) 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 o f 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
1
y
x 1 and 2 y 4 x 3 are perpendicular to each other.
2
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)
(ii)
(iii)
Mathematical Phenomena: Problems related purely to
mathematics,
e.g. (negative number) (negative number) = positive
number.
Physical Phenomena: Problems in the physical world that
involve math,
e.g. acceleration due to gravity.
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
y
f ( x)
x
x
if x
0
x if x
0
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 ( x, y ) 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 ( x, y ) . 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: (2, 3)
and (3, 2) . In order to locate the point (2, 3) 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 (2, 3) 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 O (0, 0) 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., P (r , ) . 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 ( 4, 50 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 50 0 , 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 ( x, y ) , 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 ( x, y ) 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
( x, y ) given by x r cos , y r sin .
 Rectangular to polar: From the figure above, we obtain the formulas to
convert rectangular coordinates ( x, y ) of a point to its polar coordinates
(r , ) given by
r
x2
y2
and
tan
1
when the po int ( x, y ) is in Quadrant I
tan
1
tan
1
tan
where x
y
x
1
y
x
y
x
when the po int ( x, y ) is in Quadrant II
y
x
when the po int ( x, y ) is in Quadrant III
,
when the po int ( x, y ) is in Quadrant IV
0.
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 (2, 5) .
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 r1 :r2 ,
both internal and externally, are given by the following formulas. Let the
coordinates of the point P be x, y . Let the coordinates of the end-points A and
B of the line segment AB be x1 , y1 and x 2 , y 2 respectively.
 Internal division of a line segment: If the point P x, y lies between the
points A x1 , y1 and B x2 , y 2 on the line segment AB , it divides AB
____
____
internally in a given ratio, say, r1 :r2 , such that AP
the coordinates of the point P x, y are given by
r1 , and BP
x
r2 x1
r1
r1 x 2
,
r2
r2 y1
r1
y
r2 . Then,
r1 y 2
.
r2
 External division of a line segment: If the point P x, y lies outside of
the line segment AB , it is said to divide AB externally in a given ratio,
____
____
say, r1 :r2 , such that AP
r1 , and BP
r2 . Then, the coordinates of the
point P x, y are given by
x
r1 x 2
r1
r2 x1
,
r2
y
r1 y 2
r1
r2 y1
r2
.
 Mid-point formula: If the point P x, y is the middle point of the line
segment AB , with the points as A x1 , y1 and B x2 , y 2 , it bisects line
segment AB , i.e., r1 r2 . Then, the coordinates of the point P x, y are
given by
x
x1
x2
2
,
y
y1
y2
2
.
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
y y1 m x x1 , where m is the slope of the line and ( x1, y1 ) are the
59
coordinates of a point on the line. Example:
y 2
4 x 3 .
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
P
P
Figure 2
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, p1, p2 , p3 ,
, pn (the vertices),
n 3 is a positive integer, and of the line segments p1 p2 , p2 p3 , , pn 1 pn , pn p1 ,
(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
3
4
5
6
NAME
Triangle
Quadrilateral
Pentagon
Hexagon
60
7
8
9
10
12
15
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
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
1800 .
61
(xii)
A polygon is concave if at least one of its interior angles is greater
than 1800 .
(xiii) Summary of Angle Measures for Polygons (in degrees):
Sum of all
(any polygon)
Interior angle
(n - 2) 180
Exterior
angle
360
Measure of
one
(regular
polygon)
180 -
360
n
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 + S22 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
AB BC AC
(ii)
PQ QR PR
 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);
P = 2L + 2W
l
w
b) Square:
s
c) Parallelogram:
a h
A = s2 = (side squared);
P = 4s
A = b h = (base . height);
P = 2a + 2b
63
b
d) Triangle:
a
h
c
A = ½ b . h = (1/2 . base . height);
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 threedimensional 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
(4 faces)
(20 faces)
Cube
Octahedron
Icosahedron
(Hexahedron)
(6 faces)
(8 faces)
Dodecahedron
(12 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:
x 2 4 x 3 ; 5 x3 3 x 2 x 10 .
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 f x an xn an 1xn 1 ... a2 x2 a1x a0 , where n is a nonnegative
integer and an , an 1 , ..., a2 , a1 , a0 are real numbers. Note: The domain of a
polynomial is the set of all real numbers. Example: f ( x) x 2 4 x 3 ; and
g ( x) 5 x3 3 x 2 x 10 are polynomials (or polynomial functions) in x of
degrees 2 and 3 respectively.
A TABLE OF SOME WELL-KNOWN POLYNOMIAL FUNCTIONS (OR
POLYNOMIALS)
Degre
e
No
degree
0
Form
f ( x)
f ( x)
1
f ( x)
2
f ( x)
f ( x)
5
f ( x) a5 x5
f ( x)
a5
a4 x
4
a0 , a0
a1 x
a2 x 2
a3 x 3
3
4
3
a4 x 4
0
a0 , a1
a1 x
a2 x 2
a3 x
0
a1 x
a2 x
2
a3 x3
Graph
Zero
function
Constant
function
The x -axis
Linear
function
0
a0 , a2
Name
Quadratic
function
0
Cubic
a1 x a0 , a4 0 Quartic (or
biquadratic
)
2
Quintic
a2 x a1 x a0 ,
a0 , a3
Horizontal line with y intercept a 0
Nonvertical,
nonhorizontal straight
line with slope a1 and y intercept a 0
Parabola: Graph opens
up if a2 0 ; graph
opens down if a2 0
0
Example: f ( x)
x4
Example: f ( x)
x5
0
Polynomial equation in x : A polynomial in one variable x , set equal to zero,
i.e., f x an xn an 1xn 1 ... a2 x2 a1x a0 0 , where n is a nonnegative
integer and an , an 1 , ..., a2 , a1 , a0 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, or 5 .
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: x 2 4 x 3 0 , and 5 x3 3 x 2 x 10 0 are polynomial equations in
x of degrees 2 and 3 respectively.
65
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
f x a xn , where a is a real number, a 0 , and n 0 is an integer. Example:
f ( x) x 2 ; and g ( x) 5 x 3 are power functions in x of degrees 2 and 3
respectively.
Powers of imaginary unit i : The repetitive pattern when the imaginary unit,
i , i 4 1, i 5 i ,
1, i 3
i
1 , is raised to sequential powers. Example: i 2
i6
1, i 7
i , 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 Rl (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 e xamples of well-known prime
numbers.
A TABLE OF SOME WELL-KNOWN PRIME NUMBERS
Prime Numbers
Number Factor
s
2
1, 2
7
1, 7
11
1, 11
17
1, 17
Non-Prime Numbers
Number
Factors
6
8
15
25
1, 2, 3, 6
1, 2, 4, 8
1, 3, 5, 15
1, 5, 25
Prime polynomial: A polynomial which has no polynomial factors except itself
and constants. Example: The polynomials x 1 and x 2 x 1 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) 4 2 , 2 is the principal square root of 4 . (ii)
3
8
2,
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.
a , a 0 , is called negative square root of a . Example: The negative
Note: (i)
square root of 16 is 4, written as
a. a
a
2
a , and
a .
16
a
4 . (ii) For nonnegative a ,
a
2
a . (iii) For any real number a ,
67
a2
a . (iv) For any nonperfect square positive real number a ,
irrational number. (v) For any negative real number a ,
(vi) 0 0 .
a is an
a is not a real number.
Principal square root of a negative number: If N is a positive real number,
N , is defined as
then the principal square root of N , denoted by
N
N i , where i
1 is the imaginary unit and i 2
1 ..
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:
P(rolling a 3 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
a
of the distance from a given point with coordinate x1 to another point with
b
coordinate x 2 . Since the total distance from x1 to x 2 is
68
x2
x
x1 units, we start at x1 and move
x1
a
. x2
b
x1 .
a
. x2
b
x1 toward x 2 . Thus,
Example: The coordinate x of a point located
1
of the distance from a point with coordinate 4 to another
4
1
point with coordinate 16 is given by x 3
. 16 4 6 .
4
one - fourth , i.e.,
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 and 5 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.,
( x y ) ( x y ) ( x 2 y 2 ) . Example: ( x 5) ( x 5) ( x 2 25) .
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
Commutative
Property
Associative
Property
Identity
With respect to
addition (+)
a+b =b +a
With respect to
multiplication (.)
a.b = b.a
a + (b + c) = (a + b) + c
a.(b.c) = (a.b).c
a+0 =a
a
1=a
69
Property
Inverse
Property
Zero Property
a + (–a) = 0
1
1
a
a .0 0
a.
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).
a c
, the product of the
b d
means (b and c) equals the product of the extremes (a and d), or in other words:
b • c = a • d.
Product property of proportions: In a proportion
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.
Y
B
15
A
150
75
a
Z
X
C
15 a
75 150
a = 30
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
A right square pyramid
Figure 2
A hexagonal pyramid
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 f x ax 2 bx c , 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.
f x
2 2
x
5
3x 2 ; g x
2x 2
3
Quadratic formula: A quadratic formula is the solution of a quadratic
ax 2
equation f x
bx c , where a
0 given by x
b
b2
2a
4ac
Quadratic inequality: An inequality written in one of the forms
,
Quadratic term: A term
, or
,
is called a quadratic inequality.
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
called rational function.
, then
is
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 multiplyi ng 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:
5 3 8
, , ,1.33333, 0.25 . (See Classification of numbers also).
1 4 9
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
5
real numbers. Examples: , 1.33333, 0.25 , 2 1.41421..., -1, 0, 1.
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
written as
.
and 0.601260126012. . . . can be
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
- 1.
are 2 and
Solved Example on Roots of a Quadratic Equation:
The graph represents the equation
equation
. Find the roots of the quadratic
.
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)
(ii)
Two points in the line, or
The equation of the line with y isolated. The coefficient of x is the
slope. Example: In the equation y mx b , m slope of the line ,
and b y - intercept.
Slope Facts:
(i)
(ii)
(iii)
(iv)
(v)
y 2 y1
x 2 x1
Horizontal lines have slope m = 0
Vertical lines have undefined slope.
Parallel lines have the same slope.
If m1 and m2 are slopes of the two perpendicular lines m1 x m2 = -1,
1
i.e., m1 =m2
Slope 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 co nstant 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
-5
m
3
3
-5m
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 3Dimensional 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
Volume
Volume
Volume
Volume
Volume
of a cube = side3
of a rectangular prism = length x width x height
of a pyramid = 1/3 x area of the base x height
of a cylinder = pi x r2 x height
of a cone = 1/3 x pi x r2 x height
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 coordina te plane. The horizontal line is the xaxis 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 arithmetic, it is defined as the identity element of addition, i.e., the
number 0 for which x 0 x and 0 x x 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
a
 Division by zero:
is meaningless for all a .
0
0
 Division of zero:
0 for all k 0 , since 0 k 0 .
k
 Factorial zero: It is denoted by 0 ! , and is defined as 0 ! 1 .
 Multiplication by zero: 0 k k 0 0 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 f (x) is equal to 0 , i.e., f (x) 0 . Note the following properties of zero of
a function f (x) :
(i)
(ii)
Real Zero: A real zero of a function f (x) is a real number c for
which f (x) is equal to 0 , i.e., f (c) 0 .
Real Zero as an x -intercept of the curve y f (x) : If the
function f (x) has only real number values for real number
values of x , then the real zeros of f (x) are the values of x for
which the curve y f (x) meets the x -axis, which are also
called the x -intercepts of the curve y f (x) . Example: A
polynomial f (x) 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 f (x) with real numbers as coefficients is equal to 0 , i.e., f (c) 0 .
Example: x 1 is a zero of the polynomial f ( x) x 1 .
85
Appendix A
NOTES ON SET THEORY AND PROP ERTI ES OF REAL NUMBERS
By DR. M. Shakil
1.
DEFINITION OF A S ET:
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 if 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
infinit e.
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 not ation; for example: S = {x
N is the set of all natural numbers.
3.
a)
0 < x < 10 and x
N}, where
DIFFERENT TYPES OF SETS :
EQUAL (IDENTICAL) S ETS have ex actly the same elements.
b) The UNIVERS AL SET, U, is the set that contains all possible elements under consideration
in a problem.
d) SUBS ET: A is a subset of B if every element in set A is also in set B. The
notation is A B. NOTE: E very set is a subset of itself, i.e., A A, where A is any set.
e) PROP ER S UBS ET: 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 eleme nts. 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 S UBS ETS OF A SET: If a set contains “N” elements then the number of
N
N
subsets is equal to 2 and the number of proper subsets is equal to 2 - 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 CARDI NAL NUMBER OF A S ET: The number of elements in a given
finite set A is called its cardinality or cardinal number. The notation is n(A ) or #(A).
N
j) NOTE: If n(A ) = N, then n(P(A)) = 2 , 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 OP ERATIONS:
a) UNION OF S ETS: 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) INTERS ECTION 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 not ation is A B. Thus
A
B = {x x
A and x
B}.
c) DIFFERENCE OF TWO S ETS: 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 S ET: 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 US ED WITH S ETS:
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 subs et 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)
(ii)
(A
(A
/
/
B
/
B
/
C) = (A
C) = (A
B)
B)
B) = A
/
/
B) = A
(b) Di stributive Laws
(i) A
(ii) A
(B
(B
(A
(A
C)
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 APPLI CATION OF V ENN DI AGRAM:
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 the se 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
 Di stributive Property :
 a.(b + c) = a.b + a.c
 (a + b).c = a.c + b.c
 Trichotom y :
If a and b are real numbers then one of the following three must hold
a<b
a>b
a=b
 Transiti vity :
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 Symbol s :
The following are four inequality symbols used in solving mathematical problems.
SYMBOL
<
>
MEANI NG
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 distanc e of the number X from zero on
a number line. In ot her 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:
a
a
if a
0
a if a
0
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.
2.
|a| ≥ 0
|a| = 0 if and only if a = 0.
90
3.
4.
5.
6.
7.
8.
|ab| = |a||b|
|a / b| = |a| / |b| (if b ≠ 0)
|a + b| ≤ |a| + |b| (also known as the triangle inequality)
|a − b| ≥ ||a| − |b||
|a| ≤ b iff −b ≤ a ≤ b
|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,
5 , 15
(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 f (x) at x
c , i.e., lim f ( x) exists if and only if the left-hand
x
c
limit lim f ( x) and the right-hand limit lim f ( x) exist and are equal.
x
c
x
c
(B) CONTINUITY OF A FUNCTION AT A POINT
A function f (x) is continuous at x
c if the following conditions are satisfied:
(i) f (c) is defined;
(ii) lim f ( x) exists;
x
c
(iii) lim f ( x)
x
c
f (c) .
(C) THE DERIVATIVE OF A FUNCTION
(1) Definition: The derivative of the function y
function f / ( x) given by
f ( x h) f ( x )
f / ( x) lim
h 0
h
(1.1)
f (x) with respect to x is the
(read as “ f prime of x ”). The process of computing the derivative is called
differentiation, and f (x) is said to be differentiable at x c if f / ( x) exists; i.e., if
the above limit (1.1) that defines f / ( x) exists when x c .
Notation: The derivative of y
f (x) is denoted as: f / ( x) ,
df
dy
or
.
dx
dx
(2) Slope (m) of the tangent line to the graph of y f (x) at ( x0 , y 0 ) , where
y 0 f ( x0 ) , is given by the derivative of the function y f (x) at x 0 , i.e., by
m
f / ( x0 ) .
92
(3) Equation the tangent line to the graph of y
y y 0 m( x x 0 ) .
f (x) at ( x0 , y 0 ) is given by
(D) BASIC RULES OF DERIVATIVE
(i) The constant rule:
d
(c)
dx
0
(ii) The derivative of x is one:
(ii) The power rule:
d n
(x )
dx
d
(x) 1
dx
nx n
(iii) The constant multiple rule:
1
d
df
[cf ( x)] c
dx
dx
(iv) The sum and difference rules:
(v) The product rule:
(vi) The quotient rule:
d
[ f ( x) g ( x)]
dx
d f ( x)
dx g ( x)
(vii) The chain rule for y
g ( x)]
df
dx
f ( x)
g ( x)
g ( x)
f [u ( x)] :
(viii) The general power rule:
d
[ f ( x)
dx
dy du
.
du dx
n
n h( x )
(ix) Derivatives of exponential functions:
(a)
d x
e
dx
(b)
d u ( x)
e
dx
e x ; and
e u ( x)
du
dx
(x) Derivatives of logarithmic functions:
or
n 1
dg
dx
dg
dx
df
dg
f ( x)
dx
dx , g ( x)
2
[ g ( x)]
dy
dx
d
h( x )
dx
df
dx
dh
dx
0.
f (u ( x))
/
f / (u ) . u / ( x)
93
(a)
d
ln x
dx
1
, (x
x
(b)
d
ln u ( x)
dx
0) ; and
1
u ( x)
du
dx
(xi) Derivatives of b x and log b x for base b
Let b
e:
0 be a positive real number. Let b 1 . Then we have
(a)
d x
b
dx
(b)
d
log b x
dx
(ln b) . b x ; and
(xii) The second derivative of y
1 1
. , (x
ln b x
0)
f (x) denoted as: f // ( x)
f / ( x)
/
d2 f
dx 2
gives the rate of change of f / ( x) .
(xiii) Notation for the nth derivative of y
dny
f (x) :
dx n
f
(n)
( x)
dn f
dx n
(xiv) Demand Function, Supply Function and Market Equilibrium:
Demand Function: A demand function p D(x) 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 S (x) is defined as a function
which gives the corresponding price p S (x) at which producers are willing to
supply x units.
Law of Supply and Demand: In a competitive market environment,
if S ( x) D( x) , the market is said to be in “equilibrium.” Thus, if x e denote the
production level at which the market equilibrium occurs, and if p e denote the
corresponding unit price, then p e S ( xe ) D( xe ) . Here pe is called the
“equilibrium price” and xe , pe is called the “equilibrium point.”
Shortage and Surplus: When the market is not in equilibrium, it has a
(i)
a “shortage” when D( x)
S ( x) ; and
94
(ii)
a “surplus” when S ( x)
D( x) .
(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 p c denote a variable cost per unit. Then the total cost of
producing the x units denoted by C (x) is defined as a function given by
C ( x)
(var iable cost )
( fixed cost ) , where var iable cos t
x . pc .
Revenue Function: Let p D(x) 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 R(x) , is defined as a function given by
R( x )
(number of items sold ) . ( price per item) x . p
x . D( x) .
Profit Function: The profit from the production and sale of x units at the unit
price p , denoted by P(x) , is defined as a function given by
P( x )
R( x )
C ( x)
x . D( x)
C ( x) .
Break Even: When total revenue equals total cost, that is, R( x) C ( x) , that is,
P(x) 0 , 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: y R(x) and Cost Function: y C (x) intersect is called the “breakeven point.”
(xvi) Marginal Cost, Marginal Revenue and Marginal Profit: Let C (x) the
total cost of producing x units of a particular commodity, R(x) the revenue
generated when x units of a particular commodity are produced, and P(x) the
corresponding profit. Then, if x x0 denote the number of units being produced,
the marginal cost, marginal revenue and marginal profit of producing x 0 units are
defined as follows:
(a) Marginal Cost: It is given by the derivative C / ( x 0 ) of the cost
function C (x) at x x0 , which approximates the additional cost (that is,
extra cost) C ( x0 1) C ( x0 ) incurred when the level of production is
increased by one unit, from x 0 to x 0 1 , that is, of producing the ( x0 1) st
unit.
95
(b) Marginal Revenue: It is given by the derivative R / ( x 0 ) of the revenue
function R(x) at x x0 , which approximates R( x0 1) R( x0 ) , the
additional revenue from producing the ( x0 1) st unit.
(c) Marginal Profit: It is given by the derivative P / ( x 0 ) of the profit
function P(x) at x x0 , which approximates P( x0 1) P( x0 ) , the
additional profit from producing the ( x0 1) st unit.
(xvii) Approximation by increments: Let y f (x) be a function of x which is
differentiable at x x0 , and let x denote a small change in x , then we have
f ( x0
If
f
f ( x0
x)
x)
f / ( x0 ) . x .
f ( x0 )
f ( x0 ) in the above formula, then we have
f / ( x0 ) . x .
f
(xviii) Differentials: Sometimes the increment x is called the “differential” of x
and is denoted by dx . Thus, if the differential of x be dx
x , and if y f (x) be
/
a differentiable function of x , then dy f ( x) . dx is called the “differential” of y .
(ix) Approximation Formula for Percentage Change: Let y f (x) be a
differentiable function of x . Let x denote a small change in x . Then the
corresponding “percentage change” in the function f (x) is defined as
f / ( x) . x
f
Percentage change in the function f ( x) 100
% 100
%.
f ( x)
f ( x)
INTEGRAL CALCULUS
BASIC RULES OF INTEGRATION
(A) INDEFINITE INTEGRAL:
(i) Antiderivative; indefinite integral:
f ( x)dx
F ( x) C iff F / ( x)
where C is a constant of integration.
(ii) Power rule:
x n dx
(iii) Logarithmic rule:
xn 1
C for n
n 1
1
dx
x
(iv) Exponential rule: (a)
ln x
e x dx
1
C
ex
C ; (b) e k x dx
1 kx
e
k
C
f ( x) ,
96
(v) Constant multiple rule:
k f ( x) dx
(vi) Sum and difference rules:
f ( x)
k f ( x) dx
g ( x) dx
f ( x) dx
g ( x) dx
(vii) Integration by substitution:
g (u( x)) u / ( x) dx
(viii) Integration by parts:
u dv
g (u) du where u
uv
u( x) and du
u / ( x) dx
v du
(B) DEFINITE INTEGRAL:
(i) The Definite Integral: Let y f (x) be a function of x that is continuous on
the interval a x b . Let the interval a x b be subdivided into n equal parts,
b a
each of width x
. Let a number x k be chosen from the kth subinterval
n
for k 1, 2 ,  , n . Then the sum f ( x 1 ) f ( x 2 )  f ( x n) x is formed, which is
called a “Riemann sum.” The “definite Integral” of f (x) on the interval
b
a
x b , denoted by
f ( x) dx , is the limit of the Riemann sum as n
; that
a
is,
b
f ( x) dx
a
lim
n
f ( x1 )
f ( x2 )  f ( xn )
x .
The function f (x) 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 y f (x) is continuous and
f (x) 0 on the interval a x b , then the region R under the curve y f (x)
b
over the interval a
x b has area A given by the definite Integral A
f ( x) dx .
a
(iii) The Fundamental Theorem of Calculus: If the function y
f (x) is
b
continuous on the interval a
f ( x) dx
x b , then
F (b) F (a) , where F (x) is
a
any antiderivative (i.e., indefinite i ntegral) of f (x) over the interval a
x b.
97
(iv) The Area Between Two Curves: If f (x) and g (x) are continuous with
f ( x) g ( x) on the interval a x b , then the area A between the curves
y f (x) and y g (x) over the interval a x b is given by the definite Integral
b
f ( x) g ( x) dx .
A
a
(v) The Average Value of a Function: Let y
continuous on the interval a x b . Then the
" average valueV of f ( x) over a
f (x) be a function of x that is
1
b" is given by V
x
b
b aa
f ( x) dx .
(vi) RULES FOR DEFINITE INTEGRALS:
a
(a)
f ( x) dx
0
a
a
(b)
b
f ( x) dx
f ( x) dx
b
a
b
(c) Constant multiple rule:
b
k f ( x) dx for constant k
k f ( x) dx
a
b
(d) Sum rule:
a
b
b
f ( x) g ( x) dx
a
f ( x) dx
a
g ( x) dx
a
b
b
f ( x) g ( x) dx
(e) Difference rule:
a
b
f ( x) dx
a
b
c
f ( x) dx
(f) Subdivision rule:
a
b
f ( x) dx
a
g ( x) dx
a
f ( x) dx
c
98
A Calculus Formula Sheet
Derivatives Properties
Trigonometry Derivatives
Inverse Trig Derivatives
Hyperbolic Derivatives
99
Derivative Formula
Inverse Hyperbolic Derivatives Formula
Hyperbolic Formula Identities
Basic Trigonometric Properties
Linearization Formula
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 lite r
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.
72 c 1 pt 1 qt 1 gal 9
gal 4.5 gal
1
2 c 2 pt 4 qt 2
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 1 2 4 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.
Comple mentary angles: Two angles whose measures have a sum of 90 . If they
are adjacent they form a right angle.
Suppleme ntary 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 inte rior 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 supple mentary
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 60 0 ; 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
AB BC AC
(iv)
PQ QR PR
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
3
4
5
6
8
NAME
Triangle
Quadrilateral
Pentagon
Hexagon
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 Measure of one
(any
(regular
polygon)
polygon)
360
Inte rior
180 n
angle
(n - 2) 180
360
Exterior
n
angle
360
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
A = s2 (side squared)
2. Square
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
height)
b2
A = ½ (b1 + b2 ) h (1/2 sum of bases x
h
b1
A = r2 (pi x radius squared)
(Circumference: C = d or 2 r)
6. Circle
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 te rms 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
h
w
l
+ 2 wh)
2. Cube
Volume = base area height
=
e2 e
e
e
e
= e3
r
3. Cylinder
h
Volume = base area height
=
r2
h
= r2 h
(Surface Area = area of ends + area
of “label”)
=
4. Pyramid
h
1/3 lwh
h
r
2 r2 + 2 rh
Volume = 1/3 base area height
= 1/3
lw
h
=
108
5. Cone
Volume = 1/3 base area height
= 1/3
r2
h
=
1/3 r2 h
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 e mpty set: A set containing no elements. Symbols are
and { }.
Number of subsets: If a set contains “n” elements then the number of subsets is
n
equal to 2 NOTE: The empty set is considered a subset of every set and every
set is considered a subset of itself.
Set Operations:
Comple ment: The set of all members of U that are not in A is called the
complement of A. The notation is A .
The inte rsection 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 state ments:
Negation
P
~P
T
F
F
T
p
T
T
F
F
q
T
F
T
F
Conjunction
p q
T
F
F
F
Disjunction
p q
T
T
T
F
Conditional
p
q
T
F
T
T
Other Forms of the Conditional:
Original conditional
converse
inverse
contrapositive
p
q
~p
~q
equivalent
Not equivalent
Not equivalent
equivalent
q
p
~q
~p
Equivalences and negations of statements:
Statement
p
q
Equivalent
1) ~q
~p
(If not q, then not p)
(If p, then q)
2) ~p
Negation
1) (p
q)
(It is not true that if p, then q)
q
(Not p, or q)
2) p
q
(p AND not q)
1) ~p
p
q
(p or q)
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
1) (p
(q and 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
Some A’s are B’s
Some A’s are not B’s
No A’s are B’s
No A’s are B’s
All A’s are B’s
Some A’s are B’s
There are no A’s that are not
B’s.
At least one A is a B
Not all A’s are B’s.
All A’s are not B’s
Determining whether an argume nt 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
EXAMPLES
ENGLISH
No A’s are B’s
Snakes are not
No B’s are A’s
No math course
pretty.
is easy.
112
Some A’s are B’s
red.
(at least one A is a B)
Some roses are
Some B’s are A’s
polluted.
(at least one B is an A)
Some rivers are
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
p
p
q
p
q
q
q
p
p
q
q
r
p
r
r
or
p
p
q
p
q
r
r
Type 4
p
q
p
q
Type 5
p
q
q
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 pre mises 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 comme nts about pre mises:
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 nonnumerical) for drawing valid conclusions and making reasonable decisions. It has meaningful
applications in production engineering, in analysis of experiment al data, in economics, in law, in
medicines, in biology, etc. The import ance 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 STATISTI CS:
It consists of the collection, organization, summarization, and presentation of data. (It describes
the situation as it is).
(b) INFERENTI AL STATISTI CS:
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 TERMINOLOGI ES:
We shall now ex plain 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) VARI ABLES :
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 definit e 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 S ET:
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 rec eived
by a local office of the Internal Revenue Servic e 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 dat a 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 diffic ult and time-c onsuming to use all members, therefore
statisticians use subgroups called samples to get the necessary dat a 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 subs et 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 V ARI ABLES (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 plac ed 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) QUANTITATIV E VARI ABLES (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 discret e 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 mont h, etc.
(ii) CONTINUOUS VARIABLES (OR DATA):
Continuous variables c an assume all values between any t wo specific values. They are obtained
by measuring. For Example, “temperature” is a continuous variable, since the variable can
assume all values bet ween any two given temperatures. Other examples of continuous variables
are height, weight, length, time, etc.
(3) RECORDED V ALUES OF A CONTI NUOUS RANDOM VARIABLE AND ITS
BOUNDARI ES:
Since continuous data must be measured, rounding ans wers is necessary because of the limits of
the measuring device. Usually, ans wers 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 incl ude 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
Length
Temperat ure
Time
Weight
Recorded Value
15 cm
0
86 F
0.43 sec
1.6 gm
Boundaries (Cla ss)
14.5 – 15.5 cm
0
85.5 – 86.5 F
0.425 – 0.435 sec
1.55 – 1.65 gm
Note: The boundaries of a continuous variable in the above table are given in one additional
decimal plac e 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. Thes e are described
below:
TABLE II
MEASUREMENT S CALES (DEFINITIONS AND EX AMPLES)
Nominal-level Data
Definition: The nominallevel of measurement
classifies data into
mutually exclusive (nonoverlapping), exhaustive
categories in which no
ordering or ranking can be
imposed on the data.
Ordinal-Level Data
Definition: The ordinallevel of measurement
classifies data into
categories that can be
ordered or ranked.
However, precise
differenc es 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.
Interval-level Data
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 int erval scale. There is
a meaningful difference of
one point bet ween 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.
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)
Temperat ure
Ratio-level Data
Definition: The ratiolevel 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 ex ample,
if one person can lift
200 pounds and
another can lift 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)
Height
(ii)
Weight
(iii)
Time
(iv)
Salary
(v)
Age
(vi)
Number of
Phone
Calls
(5) BASI C 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
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. Not e
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.
Stratified
Sampling
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.
Systematic
Sampling
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.
Cluster
Sampling
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.
Convenience
Sampling
In convenience
sampling, we use
the results that
are readily
available.
*** *** ***
(6) STATISTICAL INFERENCE AND MEAS UREMENT 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 cont ained 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 DES CRIPTIV E AND INFERENTIAL STATISTICAL P ROBLEMS:
TABLE IV
FOUR ELEMENTS OF DESCRIPTIVE
STATISTICAL PROBLEMS
FIVE ELEMENTS OF INFERENTIAL
STATISTICAL PROBLEMS
1. The population of interest.
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.
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 2 N - 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)) = 2 N, 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
n(E ) the number of outcomes in E and n(F ) the number of outcomes in F . Let
P (E ) denote the probability of event E occurring, and P (F ) denote the probability of
event F occurring.
(i) PROBABILITY OF AN EVENT: P( E )
n( E )
(Classical Probability)
n( S )
(ii) PROBABILITY OF AN IMPOSSIBLE EVENT: If the event E cannot occur,
then P(E ) 0 .
(iii) PROBABILITY OF A CERTAIN EVENT: If the event E is certain to occur,
then P(E ) 1.
(iv) The probability of any event E is a number between 0 and 1 , inclusive, i.e.
0 P( E ) 1.
(v) ADDITION RULES:
(a) FOR MUTUALLY EXLUSIVE EVENTS: The events E and F are called
mutually exclusive (or disjoint) if E  F
, i.e. they cannot both occur together, i.e.
simultaneously. For mutually exclusive events, we have P( E  F ) 0 , and the addition
rule is defined as follows:
P( E or F )
P( E  F )
P( E ) P( F ) .
(b) FOR NON-MUTUALLY EXLUSIVE EVENTS: The events E and F are not
mutually exclusive if E  F
, i.e. they both occur simultaneously. In this case, the
addition rule is defined as follows:
P( E or F )
P( E  F )
P( E ) P ( F ) P( E  F ) .
(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:
P(event E occurs in a first trial and event F occurs in a sec ond trial )
P( E and F ) P( E  F ) P( E ).P( F ) .
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:
P( E and F )
P( E ).P( F | E ) ,
where F | E is read as “ F given E, ” and P( F | E ) 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
n( A) the number of outcomes in A and n(B) the number of outcomes in B . Let
P ( A) denote the probability of event A occurring, and P (B ) 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 P( B | A) .
(b) FORMULA: The conditional probability of event B occurring, given that event A
has already occurred, is denoted by P( B | A) , and is given by
P( B | A)
P( A  B)
P( A)
n( A  B )
, provided P(A)
n( A)
0 , or n( A)
0.
(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
P(E ) The probability that the event E will not occur
P(E ) 1 P ( E ) ; and P (E )
P( E )
1 P( E ) ;
P( S ) 1 .
(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
P( B | A)
Two events A and B are “dependent” if
P( B)
P( B | A)
OR
P( A  B)
P( B)
OR
P( A  B)
P( A).P( B)
P( A).P( B)
(v) BAYES’ FORMULA:
P( B | A)
P( B) . P( A | B)
P( B) . P( A | B) P( B) . P( A | B)
9. FORMULAS FOR ODDS:
Let E be any event from the sample space S of a probability experiment. Let
# F n( E ) the number (#) for (in favor of) the event E , and # A n( E ) the number
(#) against the event E . Let # T # F # A n(S ) = total number (#) of outcomes in S .
Then, we have the following definitions:
(a) Odds in favor (for) of the event E
#F
, or F : A , i.e.
#A
P( E )
Odds in favor (for) of the event E
;
P( E )
(b) Odds against the event E
Odds against the event E
#A
, or A : F , i.e.
#F
P( E )
;
P( E )
(c) P(E )
#F
, which is the probability in favor of the event E ;
#T
126
(d) P (E )
#A
, which represents the probability against the event E .
#T
(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 m . n 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 n
0 denotes an integer, then the factorial symbol n! is defined as follows:
0! 1 ;
1! 1 ;
n! n . (n 1) . (n 2) ... 3 . 2 . 1
(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 n P r , where r n , in which the order makes a difference.
(b) FORMULA: n P r
n!
, where r
(n r )!
n.
(c) SHORT-CUT FORMULA: n P r n . (n 1) . (n 2) ... (n r 1) , where r
(d) NOTE: n P n n!
n.
127
(v) COBINATION RULE:
(a) DEFINITION: A combination is a group (or set or collection or selection) of n
n
different (or distinct) items (or objects) taken r at a time, denoted by n C r or
, where
r
r
n , in which the order is not important.
(b) FORMULA: n C r
n!
, where r
r! . (n r )!
n.
(c) NOTE: n C n 1
(vi) PERMUTATION RULE (WHEN SOME ITEMS ARE IDENTICAL OR
SIMILAR OR ALIKE TO OTHERS):
If there are n items (or objects) with n1 alike, n 2 alike, … , n k alike, the number of
permutations of all n items (or objects) is given by
n!
, where n1
n1! n2 ! n3 !  nk !
n2
n3
 nk
n.
(vii) APPLICATIONS OF PERMUTATION AND COMBINATION RULES TO
PROBABILITY:
(a) The probability of a permutation (arrangement) is given by
the number of ways forming the permutatio n
the total number of arrangemen ts
(b) The probability of a selection (combination) is given by
the number of ways of making that selection
the total number of selections
(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
n N.
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:
N
Cn
N!
, where n
n! . ( N n)!
N.
129
Appendix G
PROBABILITY AND COUNTING TECHNIQUES
By
Dr. M. Shakil
FLOW-CHART FOR PROBABILITY RUL ES
Addition and Multiplication Rules


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
Yes
ARE TH E EVENTS
MUTUALLY EXCLUSIVE?
Use Addition Rules
NO
N
O
O
R
COMPOUND EVENTS:
Do the compound events contain ?
 “OR”
 “BO TH” (i.e., “AND”), or
 “AT LEAST O NE”
BOTH
(AND)
P (A or B) = P (A) + P (B)
“What’s the probability that
the last digit of a randomly
selecte d phone numbe r is 5 or
7?”
Solution: P (last digit 5 or 7)
= (1/10) + (1/10) = 1/5

ARE TH E EVENTS
INDEPENDENT OR
DEPENDENT?
INDEPE
NDENT
P (A or B) = P (A) + P (B)
- P (A and B)
Example: P [last digit of a
randomly selecte d phone
number is odd or (< 7)]
= (5/10) + (7/10) – (3/10) =
9/10
Use Multiplication Rules
AT LEAST
ONE
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 / 23 ) = 7 / 8 (Answer)
DEPENDEN
T
P (A & B) = P (A) . P (B)
Example: “From a deck of 52
playing cards, two cards are
randomly selecte d in succession.
One is replace d 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 selecte d.”
P (both cards are queen)
= (4/52) .(3/51) = 1/221
130
FLOW-CHART FOR COUNTING TECHNIQUES
Permutations and Combinations
PERMUTATIONS
nPr
FUNDAMENTAL
PRINCIPLE O F
COUNTING
YES
YES
m1. m2 … m n OR mn
m!
COUNTING RULES

NO
“IS THERE
REPETITIO N OF
THINGS?”
NO
ARE THINGS IN
ARRANGEMENTS?
ARE DIFFERENT
ARRANGEMENTS
TO BE COUNTED
SEPARATELY?
YES
NO
COMBINATION
S
nCr