CONCEPT GUIDE FOR COLLEGE MATHEMATICS I

Introduction:Introduction:

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Index Concepts Introduction Concept Explanations, Examples (with visuals and application scenarios as per

necessity) Summary

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Chapter P

Prerequisites: Fundamental Concepts of Algebra

P.1:Real Numbers and Algebraic Expressions.

ObjectivesAt the end of this session, you will be able to:

Recognize subsets of real numbers.Use inequality symbols.Evaluate absolute value.Use absolute values to express distance.Evaluate algebraic expressions.Identify properties of real numbers.Simplify algebraic expressions.

Index1. Sets2. Real Number System3. Real Number Line4. Ordering the Real Numbers5. Absolute Value6. Distance Between Points on a Number Line7. Properties of Real Numbers8. Algebraic Expressions9. Properties of Negatives10. Summary

In every day life we usually speak of ‘a pack of cards’, ‘a bunch of keys’, ‘a collection of articles’ etc., to denote groups of different objects. In mathematics, we use the word ‘set’ to denote such collections.

Therefore, we can say that a set is a collection or an aggregate of well-defined objects.

By ‘well defined’ we mean that it must be possible to determine whether or not a given object belongs to the specified collection.

A set is a collection of objects whose contents can be clearly determined.

Therefore, we can state the following definition:

Set:

A well defined collection of objects is known as a set.

The objects which belong to the set, are called elements or members of the set. Example:

The collection of vowels in English alphabets is a set containing five elements, namely a, e, i, o, u.

The collection of the first four prime numbers is a set containing the elements 2, 3, 5, 7.

The collection of all honest persons or good players does not form sets because the terms ‘ honest’ or ‘good’ are vague and are not well defined.

1. Sets

1. Sets (Cont…) Set Notation: We use capital letters A, B, C, D,…..to describe a set.The elements of a set are

denoted by small letters a, b, c, x, y etc. A set can be expressed in the following two ways:

1. Tabulation or Roster Method:

In this method, the elements of the set are listed together and are placed within braces { }.

This form of representation uses commas to separate the elements of the sets.

Example: If A is a set of natural numbers less than 5, then A = {1, 2, 3, 4}

If B is a set of vowels in English alphabet, then B = {a, e, i, o, u}

Set of letters of the word MATHEMATICS is written as {M, A, T, H, E, I, C, S}.

Set of whole numbers is written as {0, 1, 2, 3, 4,…}.

NOTE: The three dots after number 4 indicate that there is no final element and the listing continues till infinity.

2. Set builder Form :

Sometimes listing of all the elements is not possible. Can we write all the elements of the set of rational numbers? Certainly not! So, we write a set by using one or if necessary more variables x, y etc. representing a typical element of the set followed by a rule or property satisfied by all the elements of the set.

Example: Set of natural numbers less than 5 can be written as {x | x is a natural number < 5}

Set of prime numbers can be written as {x | x is prime}

1. Sets (Cont…)Subsets: If A and B are two sets such that every element in set A also belongs to set B, we say that A is a subset of B and write this as A B.

If there exists even a single element in set A, which is not in set B, then A is not a subset of B and we write this as A B.

Example: Set B = {3} is a subset of set C = {2, 3, 5}, that is, we can write B C.

Now, consider a set D = {1, 2}. Is set D a subset of set C? No, set D is not a subset of set C because there exists an element, 1, of set D which does not belong to set C. Therefore, we have, D C.

Now we shall use the concept of subsets of a set to recognize the subsets of real numbers.

2. Real Number SystemThe real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count. For example, The number of sheep a farmer owns. These are called natural numbers or sometimes counting numbers.Natural Numbers: These numbers are also known as counting numbers.

Example: 1, 2, 3,…, 10,…, 90, …, 112,…, 1100,… are all natural numbers.NOTE: The use of three dots at the end of the list is a common mathematical notation to indicate that the list continues till infinity.

The number of days in a week is a natural number. The number of schools in a city is a natural number. Decimal numbers, such as 3.9 or 6.75 are not natural numbers. Similarly fractions, such as ½, 3/9 and 109/21 are also not

natural numbers. The set of natural number is denoted by . Some common facts about natural numbers are summarized below:

The first and the smallest natural number is 1.A natural number (except 1) can be obtained by adding 1 to the previous natural number.For number 1, there is no previous number. (0 is not a natural number. )There is no last or greatest natural number.We cannot complete the counting of all natural numbers. We express this fact by saying that there are infinite natural numbers.

Natural Numbers

{1, 2, 3,…}

2. Real Number System (Cont…)Whole Numbers: The natural numbers along with the number zero form the set of whole numbers. That is, numbers 0, 1, 2, 3,…are whole numbers.

Therefore, a whole number is either the number zero or a natural number. The set of whole numbers is usually denoted by . Some common facts about whole numbers are summarized below:

The number 0 is the first and the smallest whole number.The system of natural numbers is a subset of the set of whole numbers.There is no last or greatest whole number.There are infinite whole numbers.

Negative Numbers: Till now we have studied two system of numbers – the system of natural numbers and the system of whole numbers. However, these numbers do not meet all our requirements. There is a strong need for numbers beyond the system of whole numbers.

Observe that if a whole number is subtracted from a whole number, the result may not be a whole number. For example: 5 – 7 is not a whole number. Therefore, we need to extend the whole number system to include the so called negative numbers. We use both positive and negative numbers in real life. Profit and loss are the two opposite situations for a trader. If profit is denoted by a positive number, the loss is denoted by a negative number.

Similarly. the temperature in the cities around the equator is above 0° C and is expressed using positive numbers, while the temperature near the poles is below 0° C and is expressed using negative numbers. -1, -3, -5,… are all examples of negative numbers.

Natural Numbers1, 2, 3,…

0

Whole Numbers

2. Real Number System (Cont…)Integers: The set of integers is formed by adding negatives of the natural numbers to the set of whole numbers. In other words, integers include natural numbers, their negatives and the number 0.

Therefore, the numbers –5, -3, -1, 0, 1, 4, 6,… are all called integers. Natural numbers 1, 2, … as integers are called positive integers,

whereas numbers –1, -2, -3,… are all called negative integers. The integers 0 is neither positive integer nor a negative integer.

The set of all integers is denoted by .Rational Numbers: All numbers of the form a/b, where a and b are integers and b 0 are called rational numbers.

Rational numbers include what we usually call fractions.Notice that the word “rational” contains the word “ratio,” which should remind you of fractions.

Rational numbers, such as all fractions, can be expressed as a decimal.

Rational numbers are either terminating or non-terminating but recurring decimals. Example: 5/16 = 0.3125, ¾ = 0.75, 5/11 = 0.454545…

Rational numbers are denoted by Q. All integers can also be thought of as rational numbers, with a

denominator of 1.Example:

This indicates that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of rational numbers.

Natural Numbers1, 2, 3,…

Integers-1

-2

-30

Whole Numbers

RationalNumbers

-1/2

3/4

10.25

33

1

2. Real Number System (Cont…)Irrational Numbers: There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.

An irrational number is a non-terminating and non-recurring decimal, that is, it can not be written in the form a/b, b 0.

Example: 2 = 1.41421356…(never repeats or terminates)

pi() = 3.1415…NOTE: Both rational and irrational numbers are separate

number systems and there is no overlap. Real Numbers: The set of real number is formed by

combining the set of rational numbers and the set of irrational numbers.

That is, every real number is either a rational number or an irrational number.

Natural Numbers1, 2, 3,…

Integers-1

-2

-30

Whole Numbers

RationalNumbers

-1/23/4

10.25

IrrationalNumbers

2

Real Numbers

3. Real Number LineNow we will see how to represent real numbers on a number line:

The real number line is graph used to represent the set of real numbers.

We draw a line and designate an arbitrary point on the number line to be the origin, zero, and choose a scale, then every point on the line corresponds uniquely to a real number.

The number line is divided into two symmetric halves by the origin, i.e. the number zero. Customarily, positive numbers lie on the right side of zero, and negative numbers lie on the left side of zero.

Every real number corresponds to a distance on the number line, starting at the origin (zero).

Negative numbers represent distances to the left of zero, and positive numbers are distances to the right.

The arrows on the end indicate that it continues till infinity in both directions.

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

Negative Numbers Positive Numbers

OriginNegative Direction Positive Direction

3. Real Number Line (Cont…) Real numbers are plotted on a number line by placing a dot at the correct location

for each number. Example: We plot the real numbers –3, -1, 0, ½, 5 on the number line.

Every real number corresponds to a point on the number line and every point on the number line corresponds to a real number. Therefore, there is one-to-one correspondence between all the real numbers and all the points on the real number line.

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Negative Direction Positive Direction

1/2

4. Ordering the Real NumbersReal numbers have the property that they are ordered, which indicates that given any two different numbers we can always say that one is greater or less than the other. For any two real numbers a and b, one and only one of the following three statements is true:

1. a is less than b, (expressed as a < b)2. a is equal to b, (expressed as a = b)3. a is greater than b, (expressed as a > b)

The symbols ‘<‘ and ‘>’ are called inequality symbols.On the real number line, the real numbers increase from left to right.

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Numbers increase from left to right

Symbols Meaning Example

a < b a is less than b. 4 < 8

a is less than or equal to b.

a > b a is greater than b.

8 > -2

a is greater than or equal to b.

a b 5 5

a b 1 1

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function. The absolute value of a number is the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number.The absolute value of a real number a, is denoted by |a|. It is read as “absolute value of a”.

If a number is positive (or zero), the absolute value function does nothing to it: |3| = 3

The absolute value of 3 is 3 because 3 is 3 units from the origin, 0.

The absolute value of a positive real number or 0 is the number itself. If a number is negative, the absolute value function makes it positive: |-3| = 3

The absolute value of -3 is also 3 because -3 is also 3 units from the origin, 0.

The absolute value of a negative real number is the number without the negative sign.

NOTE: The absolute value notation is bars, not parentheses or brackets. Remember to use the proper notation because other notations do not indicate the same thing.

5. Absolute Value

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|3| = 3

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

| -3 | = 3

It is also important to note that the absolute value bars do not work in the same way as the parentheses.

Example: We already know that -(-3) = +3, now let us find -|-3|.-|-3| = -(+3) (Absolute value |- a| = a) = -3

We can define the absolute value of the real number x without referring to the number line as well. The algebraic definition of absolute value is given as:

Absolute Value:

|x| =

In other words we can say that if x is non negative, that is, , the absolute value of x is the number itself.

Example: |6| = 6, || = , |0| = 0 If x is negative, that is, x < 0, the absolute value of x is the opposite of the number x.

Example: |-5| = -(-5) = 5 (Minus. Minus = Positive)

5. Absolute Value (Cont…)

x 0

x, if x 0

-x, if x < 0

5. Absolute Value (Cont…)Let us now state some properties of absolute value which can be derived from the definition of absolute value:

Properties of Absolute Values

Examples

For all real number a and b,1.

2.

3.

4.

5.

6. (This is called triangle

inequality)

| a | 0

| a | = |a|

a |a|

| a.b | = |a|.|b|

a |a| = ,b 0

b |b|

| a b | | a | | b |

|1| = 1 0

|-1| = 1 0

| 2 | 2 | 2 |

4 | 4 |

| 4 . 5 | | 20 | 20 4 . 5 | 4 | . | 5 |

3 3 | 3 |

2 2 | 2 |

| 4 5 | | 9 | 9

| 4 | | 5 | 4 5 9

Therefore, |4 + 5| |4| + |5|

5. Absolute Value (Cont…)Evaluating Absolute Value:

Example 1: Simplify |5 - 2||5 - 2| = |3| = 3

Example 2: Simplify |2 - 5||2 - 5| = |-3| = 3

NOTE: If there are some operations to be performed inside the absolute value sign, then we perform these operations first. The absolute value is taken of the result that is obtained after performing the operations.

|5 + (- 2)| = 5 + 2 = 7 wrong!The correct result is |5 + (- 2)| = |5 - 2| = |3| = 3

Example 3: Rewrite each expression without the absolute value bars:

Solving the expression inside the absolute value sign, we get

As the expression inside the absolute value bars is negative, we get

| 2 5 |

| 2 5 | |1.41 5 | | 3.59 |

| 2 5 | ( 2 5) (|x| = -x when x < 0)

2 5 (minus. minus = plus)

= 5 - 2

6. Distance Between Points on a Real Number Line

The absolute value of a number is the distance from that number to the origin (zero) on the number line. Therefore, the absolute value function is used to determine the distance between two points on a real number line.

If a and b are any real numbers, then the distance between a and b is the absolute value of their difference, that is, |a - b|.Therefore, we can state the following generalization:

Distance between two points on the real number line:If a and b are any two points on a real number line, then the distance between a and

b is givenby |a - b| or |b - a|.

Example: Let us find the distance between –4 and 5 on the real number lineAs the distance between two points a and b is given by |a - b|, the distance between –4 and 5 is |-4 - 5| = |-9| = 9

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9

7. Properties of Real NumbersNow let us study some important properties of real numbers:

Property Definition Example Remarks

Commutative Property (Addition).

a + b = b + a 3 + 5 = 5 + 3 = 8 Two real numbers can be added in any order.

Commutative Property (Multiplication).

a . b = b. a 4 . 5 = 5. 4 = 20 The order in which any two numbers are multiplied does not change its value.

Associative Property (Addition).

(a + b) + c = a + (b + c)

2 + (4 + 6) = (2 + 4) + 6 = 12

If three real numbers are added, it makes no difference which two are added first.

Associative Property (Multiplication).It is important to note here that both commutative and associative properties do not work for subtraction or division

(a . b) . c = a . (b . c) (2. 3) . 4 = 2 . (3 . 4) = 24 If three real numbers are multiplied, it makes no difference which two are multiplied first.

7. Properties of Real Numbers (Cont…)

Property Definition Example Remarks

Distributive Property a. (b + c) = a. b + a. c 5 (2 + 3) = 5 . 2 + 5 . 3 Multiplication distributes over addition.

Identity Property (Addition) a + 0 = a0 + a = a

3 + 0 = 30 + 3 = 3

When you add 0 to any number, you obtain the same number as a result.

Identity Property (Multiplication)

a . 1 = a1 . a = a

4 . 1 = 41 . 4 = 4

When you multiply any number by 1, you obtain the same number as a result.

Inverse Property (Addition) For each real number a, there is a unique real number, denoted -a, such that  a + (-a) = 0(-a) + a = 0

5 + (-5) = 0(-4) + 4 = 0

When you add a number to its additive inverse, the result is 0. Other terms used for additive inverse is opposites or negatives.

Inverse Property (Multiplication)

For each real number a, except 0, there is a unique real number 1/a such that a . (1/a) = 1; a 0(1/a) . a = 1; a 0

6 . 1/6 = 1½ . 2 = 1

When you multiply a number by its multiplicative inverse the result is 1.  A more common term used to indicate a  multiplicative inverse is reciprocal. 

8. Algebraic ExpressionsThe diameter d of a circle of radius r is given by the formula d = 2r.

In this formula, 2 is a fixed number whereas d and r are not fixed because they depend upon the size of the circle.

For circles of different sizes, the values of d and r will be different.

Similarly, the perimeter P of a square of side s is given by the formula P = 4s. In this formula, 4 is a fixed number, but the values of s and P vary because they depend

on the size of the square.

In Algebra we come across two types of symbols, namely constants and variables as defined below:

Constant: A symbol which has a fixed numerical value is called a constant. Variables: A symbol which takes various numerical values is called a variable.

In the formulae d = 2r and P = 4s discussed above, 2 and 4 are constants where as r, P and s are variables.

In the formula d = 2r, observe that 2r combines the number 2 and the variable r using the operation of multiplication. Such a combination of constant and variable connected by the signs of fundamental operations of addition, subtraction, multiplication and division is called an algebraic expression.

Example:

3x + 2y – 4z, 4, 7, ¾, 3x – 5, a2 + b2 + c2, xy + yz + zx etc are all examples of algebraic expressions.

8. Algebraic Expressions (Cont…) Terms: The terms of an algebraic expression are those parts that are

separated by addition. Example: Algebraic expression 7x2 - 3xy + 8 can be rewritten as 7x2 + (-3xy) + 8 is an algebraic expression consisting of three terms, namely 7x2, 3xy, and 8.Numerical Coefficients: Coefficients are the number part of the terms with variables. In 7x2 - 3xy + 8, the coefficient of the first term is 7. The coefficient of the second term is 3.

If a term consists of only variables, its coefficient is 1. Constant Term: Constants are the terms in the algebraic expression that contain only numbers. That is, they are the terms without variables.

We call them constants because their value never changes, since there are no variables in the term that can change its value. In the expression 7x2 - 3xy + 8 the constant term is "8“.

Factors: Each term in an algebraic expression is a product of one or more number(s) and/ or variable(s). These numbers or variables are known as the factors of that term.

For example: Algebraic expression 7x is a product of number 7 and the variable x. Therefore, 7 and x are factors of the expression 7x.

7x2 + 3xy + 8

Numerical coefficients

Constants

Like terms: The terms having the same variable factors with the same exponents on the variables are called like terms or similar terms.Unlike terms: The terms not having same variable factors are called unlike terms.

For example, in the algebraic expression 5x2y + 7xy – 3xy + 4xy2, 7xy and 3xy are like terms whereas 5x2y and 4xy2 are unlike terms.

Evaluating Algebraic expressions: Evaluating an algebraic expression means to find the value of the expression for a

particular value of the variable. For Example: If we have to evaluate an expression 7x + 3 when x = 2, we

substitute 2 for x in the algebraic expression. Many algebraic expressions involve more than one operation. Evaluating such

expressions can be a simple process but we need to follow an order of operations to get the right answer.Order of Operations: Rule 1: First perform any calculations inside parentheses.

If the algebraic expression involves a fraction, treat the numerator and the denominator as if they were enclosed in parentheses.Rule 2: Evaluate all exponential expressions.Rule 3: Next perform all multiplications and divisions, working from left to right.Rule 4: Lastly, perform all additions and subtractions, working from left to right.

8. Algebraic Expressions (Cont…)

8. Algebraic Expressions (Cont…)Remark: The sequence details the order we follow:

Parenthesis | Exponents | Multiplication | Division | Addition | Subtraction

A popular way of remembering the order is by putting together the first letter of each word:

P.E.M.D.A.S.

We can also create a little phrase to go along with this, like “Please Excuse My Dear Aunt Sally”.

Example: Let us evaluate the following algebraic expression for the given value of the variable:

5(x 2);x 10

2x 145(10 2)

(Substituting the value of x)2(10) 14

5(12) (Parentheses)

20 1460

(Multi6

plication)

=10 (Division)

8. Algebraic Expressions (Cont…)Simplifying Algebraic Expression:

Algebraic expressions are simplified when parentheses have been removed and like terms are combined.

To add like terms we add their numerical coefficients. Example 1: Simplify 7(3y - 5) + 2(4y + 3)

7(3y - 5) + 2(4y + 3) = 7. 3y – 7. 5 + 2. 4y + 2. 3 (Using distributive property)

= 21y – 35 + 8y + 6 (Multiply) = (21y + 8y) + (– 35 + 6) (Grouping like terms) = 29y – 29 (Combining like terms)

Example 2: Simplify 7 – 4[3 – (4y - 5)] 7 – 4[3 – (4y - 5)] = 7 – 4[3 – 4y + 5] (minus.minus = plus) = 7 – 4[8 – 4y] (Combining like terms) = 7 – 4. 8 + 4. 4y (Using distributive property) = 7 – 32 + 16y (Multiply) = -25 + 16y (Combining like terms) = 16y - 25

9. Properties of NegativesNow let us remember some important properties regarding negatives which we come across frequently:Let a and b represent real numbers, variables, or algebraic expressions.

Property Example

(-1)a = -a (-1)3x = -3x

-(-a) = a -(-4y) = 4y

(-a)b = -ab (-2)3x = -6x

a(-b) = -ab 3x(-4) = -12x

-(a + b) = -a – b -(3x + 5y) = -3x – 5y

-(a - b) = -a + b = b – a

-(6x – 7y) = -6x + 7y = 7y – 6x

SummaryLet us recall what we have learnt so far:

Set: A well defined collection of objects is called a set.

Subsets: If A and B are two sets such that every element in set A is also in set B.Real Numbers: The set of real number is formed by combining the set of rational numbers and the set of irrational numbers.

The real number line is graph used to represent the set of real numbers.The absolute value of a number is the distance from that number to the origin (zero) on the number line.

|x| =

Distance between two points on the real number line:If a and b are any two points on a real number line, then the distance between a and

b is givenby |a - b| or |b - a|.

Properties of Real Numbers:

Commutative Property (Addition). a + b = b + a Commutative Property (Multiplication). a . b = b. aAssociative Property (Addition). (a + b) + c = a + (b + c)Associative Property (Multiplication). (a . b) . c = a . (b . c)

x, if x 0

-x, if x < 0

Summary (Cont…)Distributive Property a. (b + c) = a. b + a. cIdentity Property (Addition) a + 0 = a

0 + a = aIdentity Property (Multiplication) a . 1 = a

1 . a = aInverse Property (Addition) a + (-a) = 0

(-a) + a = 0Inverse Property (Multiplication) a . (1/a) = 1; a 0

(1/a) . a = 1; a 0

Algebraic Expressions: A combination of constant and variable connected by the signs of fundamental operations of addition, subtraction, multiplication and division is called an algebraic expression.

Order of Operations:

Rule 1: First perform any calculations inside parentheses.

Rule 2: Evaluate all exponential expressions.

Rule 3: Next perform all multiplications and divisions, working from left to right.

Rule 4: Lastly, perform all additions and subtractions, working from left to right.