CHAPTER 1: REAL NUMBERS

LEARNING OUTCOME At the end of this topic, students are able to:

i. understand the terms involve in real number system.

ii. use the appropriate law in simplifying the exponent, radical and logarithms.

iii. perform calculation in solving the equation involving exponent, radical and logarithms.

INTRODUCTION:

- A real number is a value that represents a quantity along a continuous line.

- A symbol of the set of real numbers is ℝ.

- The real numbers include all the rational numbers and all the irrational numbers, either

algebraic or transcendental; either positive, negative or zero.

NOTES AND EXAMPLES

1.1 System of Real Numbers

Figure 1 is a concept map of the types of real numbers that we work in this module.

Figure 1 : Real Numbers Concept Map

Let’s review the types of numbers that make up the real number system.

1. Natural numbers - The numbers we use to count things

- Symbol : ℕ

- ℕ = {1,2,3,4,5, … }

2. Whole numbers - The numbers we use to count plus zero

- Symbol : 𝑊

- 𝑊 = {0,1,2,3,4, … }

3. Integers - The set of natural numbers and their opposite plus zero

- Symbol : ℤ

- ℤ = {… , −5, −4, −3, −2, −1,0,1,2,3,4,5, … }

4. Rational numbers - The numbers that can be expressed as the ratio of two integers

- Decimal representations of rational numbers either terminate or

Real Numbers

Irrational Numbers

Rational Numbers

Integer

Whole Numbers

Natural Numbers

repeat

- Symbol : ℚ

- ℚ = { 3

4, 6.747474747474, 4.5 }

5. Irrational numbers -The numbers that cannot be expressed as a ratio of two integers

- Their decimal representations neither terminate nor repeat

- Symbol : ℚ′

- ℚ′ = { 𝜋, √3, 𝑒2, √23

}

The set of all real numbers is usually denoted by the symbol ℝ. Figure 2 is a diagram of the

types of real numbers that shown in set.

Real numbers

Figure 2 : Real Number Set

Example A: True or False? If the statement is False, explain why.

i. −5 is a rational number?

ii. 0 is an integer?

iii. √16 is a natural number?

iv. All fractions are rational numbers.

v. All integers are whole numbers

vi. All irrational numbers are real numbers

Solution:

i. False because −5 is not a fraction.

ii. True.

iii. √16 = 4. So, True.

iv. True

v. False because whole numbers only include zero and natural numbers.

Rational numbers

5.33333 , -2.5 , 4/5

Integer

-3 , -2 , -1

Whole

0

Natural

1 , 2 , 3

Irrational numbers

𝝅

𝟑 , √𝟔 , 𝒆𝟐

vi. True

Example B: For the set {−4,2.3, √6, −𝜋, 0,3

4, 0. 36̅̅̅̅ , 7}, which of the elements are

a. Irrational numbers

b. Real numbers

c. Natural numbers

d. Rational numbers

e. Integers

Solution:

a. {√6, −𝜋}

b. {−4,2.3, √6, −𝜋, 0,3

4, 0. 36̅̅̅̅ , 7},

c. {7}

d. {−4,2.3,0,3

4, 0. 36̅̅̅̅ , 7}

e. {−4,0,7}

1.2 Properties of Real Numbers

There are several properties of the two primary operations, addition and multiplication, that

are very important in the study of algebra.

Properties of Real Number

1. Commutative

For any real number 𝑎 and 𝑏

(i) 𝑎 + 𝑏 = 𝑏 + 𝑎

When we add two numbers, the order doesn’t matter.

Example : 7 + 3 = 3 + 7

(ii) 𝑎𝑏 = 𝑏𝑎

When we multiply two numbers, the order doesn’t matter.

Example : 2 ∙ 5 = 5 ∙ 2

2. Associative

For any real number 𝑎, 𝑏 and 𝑐

(i) (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)

When we add three numbers, it doesn’t matter which

two we add first

Example : (3 + 4) + 5 = 3 + (4 + 5)

(ii) (𝑎𝑏)𝑐 = 𝑎(𝑏𝑐)

When we multiply three numbers, it doesn’t matter

which two we multiply first

Example : (2 ∙ 5) ∙ 3 = 2 ∙ (5 ∙ 3)

3. Distributive

For any real number 𝑎, 𝑏 and 𝑐

(i) 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐

Example : 5 ∙ (3 + 4) = 5 ∙ 3 + 5 ∙ 4

(ii) (𝑏 + 𝑐)𝑎 = 𝑎𝑏 + 𝑎𝑐 Example : (3 + 4) ∙ 5 = 5 ∙ 3 + 5 ∙ 4

When we multiply a number by a sum of two numbers, we get the same results as

multiplying the number by each of the terms and then adding the results.

4. Identity

For any real number 𝑎

(i) 𝑎 + 0 = 𝑎

Example : 3 + 0 = 3

(ii) 𝑎 ∙ 1 = 𝑎

Example : 4 ∙ 1 = 4

The identity property tells us that zero added to any number is the number itself. While for

multiplication, any number multiply with one, is the number itself.

Example C: State the property used to justify each statement.

a. 2 + (3 + 𝑏) = (2 + 3) + 𝑏

b. 0 + 𝑥 + 𝑦 = 𝑦 + 𝑥

c. 2 ∙ 5 = 5 ∙ 2

d. (1 ∙ 7) ∙ 5 = 1 ∙ (7 ∙ 5)

e. 7(𝑎 + 𝑏) = 7𝑎 + 7𝑏

Solution:

a. Associative

b. Identity and cummutative

c. Cummutative

d. Associative and identity

e. Distributive

1.3 Exponent

In this section, we discuss the rules for working with exponent notation. We also see how

exponents can be used to represent very large and very small numbers.

1.3.1 Exponential Notation

A product of identical numbers is usually written in exponential notation. For example 8 ∙ 8 ∙

8 = 83. Exponents are also called ‘powers’ or ‘indices’.

Exponential Notation

If 𝑎 is any real number and 𝑛 is a positive integer, then the 𝑛th power of 𝑎 is

𝑎𝑛 = 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ … ∙ 𝑎

𝑛 factors

The number 𝑎 is called the base and 𝑛 is called the exponent

Example D: Identify the base and exponent of the following.

a. (½)⁵ b. (−5)⁴

Solution:

a. Base: 1 2⁄ Exponent: 5

b. Base:−5 Exponent: 4

Example E: Expand and solve of the following.

a. −54 b. (−1

2)6

Solution:

a. – 5 ∙ 5 ∙ 5 ∙ 5 = −625

b. 64

1

2

1

2

1

2

1

2

1

2

1

2

1

Zero and Negative Exponents

If 𝑎 ≠ 0 is any real number and 𝑛 is a positive integer, then 𝑎0 = 1 and n

n

aa

1

Example F: Solve the following.

a. (⅜)⁰ b. 5⁻⁴ c. (−2)−6

Solution:

a. (⅜)0

Rational and Roots Exponents

If 𝑎 is any real number and 𝑛 and 𝑚 are positive integers, then n aa n 1

and n maa nm

Example G: Rewrite and solve the following.

a. 21

16 b. 53

)32( c. 23

4

Solution:

a. 41616 221

b. 832768)32()32( 55 353

1.3.2 Rules for Working with Exponents

Familiarity with the following rules is essential for our work with exponents and bases. In the

table below, the bases 𝑎 and 𝑏 are real numbers, and the exponents 𝑚 and 𝑛 are integers.

Laws of Exponents

1.Product Rule

nmnm aaa

2. Quotient Rule

nm

n

m

aa

a

3. Power Rule

mnnm aa )(

4. Product of Power Rule

nnn baab )(

5. Quotient of Power Rule

n

nn

ba

ba )(

Example H: Simplify each expression

a. 27 xx b.

27 )( x c. 3

2

a

a d.

4)3( x e. 3)

4(x

Solution:

a. 5)2(727 xxxx [Law 1]

Example I: Simplify each expression

a. 32 )3(

6

1xx b.

52

24

25

5

yx

yx c.

322 )6()3( bab

Solution:

a. 32332 )(3

6

1)3(

6

1xxxx

7

61

61

6

2

9

2

9

6

27

276

1

x

x

xx

xx

When simplifying an expression, you will find that many different methods will lead to the

same result; you should feel free to use any of the rules of exponents to arrive at your own

method.

We now give two additional laws that are useful in simplifying expressions with negative

exponents.

Laws of Exponents

6. nn

ab

ba )()(

7. m

n

n

m

a

b

b

a

Example J: Eliminate negative exponents and simplify each expression

a. 43

423

)(

)3()2(

x

xx b.

55

43

ba

ba

c.

33332 )3()2( vuvu

Solution:

a. 12

4232

43

423 3)(2

)(

)3()2(

x

xx

x

xx

2

2

1246

12

46

12

12

12

34

x

x

x

x

xx

1.3.3 Exponential Function

Exponent functions are functions whose defining equations involve the variables as an

exponent. The previous knowledge of these functions will allow us to consider many further

applications including population growth and radioactive decay.

Definition : Exponential Function

The exponential function with base a is defined for all real numbers 𝑥 by 𝑓(𝑥) =

𝑎𝑥 where 𝑎 > 0 and 𝑎 ≠ 1.

For instance; 𝑓(𝑥) = 2𝑥, 𝑔(𝑦) = (−6)𝑦 and ℎ(𝑧) = (𝜋

3)

𝑧

Example K: Let 𝑓(𝑥) = 3𝑥 and evaluate the following

a. 𝑓(2) b. 𝑓(−2/3) c. 𝑓(𝜋) d. 𝑓(√2)

Solution:

a. 𝑓(2) = 32 = 9

This yields an important property that can be used to solve certain types of equations

involving exponents.

Exponential Property

If 𝑎 > 0 and 𝑎 ≠ 1, then

𝑎𝑚 = 𝑎𝑛 if and only if 𝑚 = 𝑛 in which 𝑚 and 𝑛 are any real numbers.

Example L: Solve each of the following equations for 𝑥.

a. 82 x b. 8132 x

c. 16

12 1 x

Solution:

a. 82 x 322 x

3x

Example M: Solve each of the following equations for 𝑥.

a. 22164 xx b. 0273632 xx

c. 81363 13 xx

Solution:

a. 22164 xx

222 )4(4 xx

4444 xx

44 xx

43 x

3

4x

b. 0273632 xx

02736)3( 2 xx , let

xa 3

02762 aa

Solve by using factorize;

0)3)(9( aa

09 a

9a

Recall xa 3

93 x

233 x

2x

03 a

3a

Recall xa 3

33 x (Ignored)

In this section we learn the meaning of expressions such as nm

a in which exponent nm is a

rational number.

1.4.1 Radical Definition

An irrational number is a number that is not rational, so it cannot be expressed in the form b

a

where 0b and 𝑎 and 𝑏 are integers. Examples of irrational numbers include ,5,2 𝜋. An

irrational number whose exact value can only expressed using ‘radical’ or ‘roots’ symbol

For example, 53 8,5,7 .

Definition

If 𝑛 is any positive integer, then the 𝑛th radical of 𝑎 is defined as

√𝑎𝑛

= 𝑏 means 𝑎 = 𝑏𝑛

If 𝑛 is even, we must have 𝑎, 𝑏 ≥ 0.

For instance;

If 𝑛 is even,√814

= 3

81 = 34

If 𝑛 is odd,√−1253

= −5

−125 = (−5)3

Example N: Solve the following expression by using the definition of radical.

a. √𝑥 = 3 b. √−𝑦3 = −10 c. √𝑎6

= 2

Solution:

a. √𝑥 = 3

𝑥 = 33

= 9

Radical is sometimes written in exponential notation. Let’s recall back the idea that we have

discussed in Section 1.2.

Rational and Roots Exponents

If 𝑎 is any real number and 𝑛 and 𝑚 are positive integers, then n aa n 1

and n maa nm

Example O: Write each radical expression using exponent, and each exponential expression

using radicals.

a. √16 b. √723 c. 4

23 d. 𝑎−

32

Solution:

a. √16 = 1612

1.4.2 Rules for Working with Radicals

In this section, we discuss on how to write radicals in simplified form. To accomplish this,

we will need to use the law of radicals.

Laws of Radicals

1. bcabcba )(

2. bcabcba )(

3. bdacdcba )()()(

4. d

b

c

a

dc

ba

5. abba

6. b

a

b

a

7. nn aa )(

8. aaan nnn )(

9. m

m

aa

1

10. m

ma

a

1

Example P: Simplified each of the following expressions.

a) 2624

b) 2372316

c) 3437310

d) 3 27

32

62235

Solution:

a) 2)24(2624

26

1.4.3 Simplifying Radical

When working with the simplification of radicals you must remember some basic information

about perfect square numbers. A perfect square is a number who have been square

produces a natural number as shown in table below.

List of Perfect Squares

22 4

32 9

42 16

52 25

62 36

72 49

⋮ ⋮

Let’s take a look at the example below to help us understand the steps involving in

simplifying radicals.

Example Q: Simplify √12

Step Solution

Step 1: Find the largest perfect square that is a factor of the radicand

12 is the product of 4 × 3

(4 is the largest perfect square, 3 is

its matching factor)

Step 2: Rewrite the radical as a product of the perfect square and its matching factor found in Step 1

√12 = √4 × 3

Step 3: Rewrite as Law 5 √12 = √4 ∙ √3

Step 4: Simplify √12 = 2√3

For more examples;

Example R: Simplify the following radicals.

a. √45

b. √32

c. √72

d. √20

e. √27

f. √48

g. √75

h. √4500

i. √3200

Solution:

a. √45 = √9 × 5

= √9 × √5

= 3√5

b. √32 = √16 × 2

= √16 × √2

= 4√2

1.4.4 Rationalization of Denominator

It is often useful to eliminate the radical in a denominator by multiplying both numerator and

denominator by an appropriate expression. This procedure is called rationalizing the

denominator.

Example:

Remember fractions

1

2

↔↔

numerator

denominator

Fractions can contain radicals in the

numerator, denominator or both:

5

3

34

5

53

23

Removing the radical from numerator or denominator

Remember the rules:

i) Single term with radical, a

To remove radical : Multiply by itself, √𝒂

√𝑎 × √𝑎 = 𝑎

ii) Two term containing both radical,√𝑎 + √𝑏 or √𝑎 − √𝑏

To remove radical : Multiply by the conjugate √𝑎 − √𝑏 or the conjugate √𝑎 + √𝑏

Resulting; (√𝑎 + √𝑏)(√𝑎 − √𝑏) = 𝑎 − 𝑏

iii) One of two term containing radical,𝑎 + √𝑏 or 𝑎 − √𝑏

To remove radical : Multiply by the conjugate 𝑎 − √𝑏 or the conjugate 𝑎 + √𝑏

Resulting; (𝑎 + √𝑏)(𝑎 − √𝑏) = 𝑎2 − 𝑏

iv) Two term containing both radical and constant,𝑥√𝑎 + 𝑦√𝑏 or 𝑥√𝑎 − 𝑦√𝑏

To remove radical :

Multiply by the conjugate 𝑥√𝑎 − 𝑦√𝑏 or the conjugate 𝑥√𝑎 + 𝑥√𝑏

Resulting; (𝑥√𝑎 + 𝑦√𝑏)(𝑥√𝑎 − 𝑦√𝑏) = 𝑥2𝑎 − 𝑦2𝑏

v) One of two term containing radical and constant, 𝑥𝑎 + 𝑦√𝑏 or 𝑥𝑎 − 𝑦√𝑏

To remove radical :

Multiply by the conjugate 𝑥𝑎 − 𝑦√𝑏 or the conjugate 𝑥𝑎 + 𝑦√𝑏

Resulting; (𝑥𝑎 + 𝑦√𝑏)(𝑥𝑎 − 𝑦√𝑏) = 𝑥2𝑎2 − 𝑦2𝑏

Summarize

i) √𝑎 × √𝑎 = 𝑎

ii) (√𝑎 + √𝑏)(√𝑎 − √𝑏) = 𝑎 − 𝑏

iii) (𝑎 + √𝑏)(𝑎 − √𝑏) = 𝑎2 − 𝑏

iv) (𝑥√𝑎 + 𝑦√𝑏)(𝑥√𝑎 − 𝑦√𝑏) = 𝑥2𝑎 − 𝑦2𝑏

v) (𝑥𝑎 + 𝑦√𝑏)(𝑥𝑎 − 𝑦√𝑏) = 𝑥2𝑎2 − 𝑦2𝑏

Let’s take a look for the following example.

Example S: Rationalize the denominator in the expressions below:

a) 5

3

b) 73

9

c) 15

2

d) 1132

3

e) 7253

6

Solution:

a) 5

3

Multiply top and bottom by 5

5

53

5

5

5

3

b) 74

9

Multiply top and bottom by 7

28

79

)7(4

79

7

7

74

9

1.5 Logarithm

In this section, we study the inverse of exponential functions.

1.5.1 Logarithm Definition

The idea of logarithms is to reverse the operation of exponential, that is raising a number to a

power.

Definition

Let 𝑏 be a positive number with 𝑎 ≠ 1 and 𝑥 > 0. Then the logarithmic

function with base a denote by 𝑙𝑜𝑔𝑏 is defined by

𝑙𝑜𝑔𝑏𝑥 = 𝑦 is equivalent to 𝑥 = 𝑏𝑦

Read this as ‘’log base 𝑏 of 𝑥’’

In this definition 𝑙𝑜𝑔𝑏𝑥 = 𝑦 is called the logarithm form and 𝑥 = 𝑏𝑦 is called

the exponential form.

For instance;

log4𝑥 = 2

𝑥 = 42

log9𝑦 = 1/2

𝑦 = 91/2

log2(𝑧/3) = 4

𝑧/3 = 24

Example T: Write each of the following in logarithmic form.

a) 𝑏𝑛 = 𝑥

b) 23 = 8

c) 102 = 100

d) 5−2 =1

25

Solution:

a) 𝑏𝑛 = 𝑥

nxb log

b) 23 = 8

38log2

1.5.2 Properties of Logarithms

Logs have some very useful properties which follow from their definition and the

equivalence of the logarithmic form and exponential form. For the below properties, we

require that 𝑏 > 0, 𝑏 ≠ 1 and 𝑚, 𝑛 > 0.

Properties of Logarithms

1. log𝑏(𝑚𝑛) = log𝑏𝑚 + log𝑏𝑛

2. log𝑏(𝑚/𝑛) = log𝑏𝑚 − log𝑏𝑛

3. log𝑏𝑚𝑎 = 𝑎 log𝑏𝑚

4. log𝑏𝑚 = log𝑏𝑛 if and only if 𝑚 = 𝑛

5. b

mm

a

ab

log

loglog

6. log𝑏1 = 0 for any 𝑏 ≠ 0

7. log𝑏𝑏 = 1

8. log𝑏𝑏𝑥 = 𝑥 for any real number 𝑥

9. 𝑏log𝑏𝑥 = 𝑥 for 𝑥 > 0

Example U: Expand and solve the following expression.

a) )(logz

xyb

b) p5log 5

c) 3

1

)8(log 2 x

d) Find 𝑥 if xbbbb log3log9log2

15log2

e) y

x

2

8log

3

2

Solution:

a) zxyz

xybbb log)(log)(log

zyx bbb logloglog

b) 5log5log 55 pp

p

p

)1(

1.5.3 Natural Logarithm

The natural logarithm is often written as ‘ln’ which you may notices on your calculator

xx elogln

The symbol 𝑒 symbolizes a special mathematical constant. It has importance in growth and

decay problems. The logarithmic properties listed above hold for all bases of logs. In

particular,

xe y ↔ yx ln are equivalent statements. We also have 𝑒0 = 1 and ln 1 = 0

Example V:

a) 𝑒log𝑒𝑥

b) 𝑒𝑎 log𝑒𝑥

c) log𝑒𝑒2𝑦

d) 5

log2x

e

Solution:

a) 𝑒log𝑒𝑥

EXERCISE

1. True or False? If the statement is False, explain why.

(a) −3. 25̅̅̅̅ is an integer.

(b) √8 is a rational number.

(c) √7 is a real number.

(d) All negative numbers are integers.

(e) All integers are natural numbers.

2. List the elements of the given set that are

(i) natural numbers

(ii) integers

(iii) rational numbers

(iv) irrational numbers

(a) {0, −4, 16,22

7, 0.25, √8, 1. 23̅̅̅̅ , −1/2, √3

3}

(b){1.01,0.333 … , −𝜋, −11, 11,13

14, √25, 3.14,

16

4}

3. State the property of real numbers being used

(a) 2(3 + 5) = (3 + 5)2

(b) 2(𝐴 + 𝐵) = 2𝐴 + 2𝐵

(c) (2 + 3) + 1 = 2 + (3 + 1)

(d) (0 + 7) + 5 = 0 + (7 + 5)

(e) 3𝑥(𝑦 + 𝑧) = (𝑦 + 𝑧)3𝑥

4. Evaluate each expression (Show your working)

(a) 32 ∙ 33

(b) (24)0

(c) −60

= 𝑥

(d) 6

4

33

1

(e) 8

3

2

4

(f)

24

2

5

2

1

5. Solve

(a) 3𝑥 = 9

(b) 3𝑦 = 27

(c) 5𝑧 = 125

(d) 5𝑛 = 625

(e) 6𝑚 = 216

6. Simplify each expression.

(a)

2

432

25

1)5(

xx

(b) 1052 )2( yy

(c)

2443

xx

(d) )12(2

1 4324

yyxx

(e)

5

2

23

c

ba

bc

a

7. Simplify the expression and eliminate any negative exponent.

(a) 49aa

(b) )16(4

1)2( 4613 tsts

(c)

3

3

2234 )(

c

ddc

(d)

3

432

32

zyx

zxy

(e)

2

3

23 2

)3(

c

bacab

8. Solve each of the following equations for 𝑥.

(a) 03221222 xx

(b) 01083153 21 xx

(c) 16232 2 xx

(d) )4(22 134 xx

(e) 419)4(8 26 x

9. Simplify each expression.

(a) 123 xe

(b) 300500 xe

(c) 0542 xx ee

(d) 0432 xx ee

(e) 22

30002

xe

10. Simplify each expression.

(a) 24

(b) 50

(c) 96

(d) 150

1

(e) 312

11. Rationalize the denominator

(a) 6

1

(b) 73

2

(c) 71

6

(d) 732

3

(e) 532

8

12. Rewrite exponential form in logarithmic form

(a) 25 = 32

(b) 10−3 =1

1000

(c) 50 = 1

(d) 32 = 9

13. Evaluate

(a) log1010000

(b) log5(1

125)

(c) log(

1

2)(

1

8)

(d) logb𝑏

(e) log𝑐𝑐7

(f) log3(√34

)

14. Solve

(a) 3)25(log2 x

(b) 16)2(log34 10 x

(c) 4)2(log2 10 x

(d) 2)124(log)18(log 2

3

3

3 xxx

(e) 2

1)425(loglog2 2525 xx

15. Assume that 3.5log3 x and 1.2log3 y . Evaluate the given quantities.

(a) 2

3

3logy

x

(b)

x3log

(c)

3

3log xy

PROBLEM

1. Period of Pendulum

The period 𝑇 (in seconds) of a pendulum is 3T 3

128

L where 𝐿 is the length of the

pendulum (in feet). Find the period of pendulum whose length is 2 feet.

2. List all possible digits that occur in the units place of the square of a positive integer. Use

that list to determine whether √168 is an integer.

3.Computer Virus

The number 𝑉 of computer s infected by a computer virus increases according to the

model 𝑉(𝑡) = 1 + 100𝑒4.6052𝑡, where 𝑡 represents the time in hours. Find

(a) 𝑉(1)

(b) 𝑉(1.5)

(c) 𝑉(2)

4. Human Memory Model

Students in mathematics class were given an exam and then retested monthly with an

equivalent exam. The average scores for the class are given by the human memory model

ℎ(𝑡) = 18 log(𝑡 + 1) + 65, 0 ≤ 𝑡 ≤ 12 where 𝑡 is the time in months.

(a) what was the average score on the original exam?

(b) what was the average score after 5 months?

(c) what was the average score after 10 months?

(d) what was the average score after 12 months?