College AlgebraFifth EditionJames Stewart Lothar Redlin Saleem Watson

PrerequisitesP

Real Numbersand Their PropertiesP.2

Types of Real Numbers

Introduction

Let’s review the types

of numbers that make up

the real number system.

Natural Numbers

We start with the natural

numbers:

1, 2, 3, 4, …

The integers consist of the natural

numbers together with their negatives

and 0:

. . . , –3, –2, –1, 0, 1, 2, 3, 4, . . .

Integers

Rational Numbers

We construct the rational numbers

by taking ratios of integers.

• Thus, any rational number r can be expressed as:

where m and n are integers and n ≠ 0.

m

rn

Rational Numbers

Examples are:

• Recall that division by 0 is always ruled out.

• So, expressions like 3/0 and 0/0 are undefined.

3 461 172 7 1 10046 0.17

Irrational Numbers

There are also real numbers, such as ,

that can’t be expressed as a ratio of integers.

Hence, they are called irrational numbers.

• It can be shown, with varying degrees of difficulty, that these numbers are also irrational:

2

32

33 5 2

Set of All Real Numbers

The set of all real numbers is

usually denoted by:

• The symbol

Real Numbers

When we use the word ‘number’

without qualification, we will mean:

• “Real number”

Real Numbers

Figure 1 is a diagram of the types

of real numbers that we work with

in this book.

Repeating Decimals

Every real number has a decimal

representation.

If the number is rational, then its

corresponding decimal is repeating.

Repeating Decimals

For example,

• The bar indicates that the sequence of digits repeats forever.

12

23

157495

97

0.5000... 0.50

0.66666... 0.6

0.3171717... 0.317

1.285714285714... 1.285714

Non-Repeating Decimals

If the number is irrational, the decimal

representation is non-repeating:

2 1.414213562373095...

3.141592653589793...

Approximation

If we stop the decimal expansion of

any number at a certain place, we get

an approximation to the number.

• For instance, we can write π ≈ 3.14159265

where the symbol ≈ is read “is approximately equal to.”

• The more decimal places we retain, the better our approximation.

E.g. 1—Classifying Real Numbers

Determine whether

a) 999 d)

b) –6/5 e)

c) –6/3

is a natural number, an integer, a rational

number, or an irrational number.

25

3

E.g. 1—Classifying Real Numbers

a) 999 is a positive whole number, so it is a

natural number.

b) –6/5 is a ratio of two integers, so it is a

rational number.

c) –6/3 equals –2, so it is an integer.

E.g. 1—Classifying Real Numbers

d) equals 5, so it is a natural number.

e) is a nonrepeating decimal

(approximately 1.7320508075689),

so it is an irrational number.

3

25

Operations on Real Numbers

Operations on Real Numbers

Real numbers can be combined using the

familiar operations:

• Addition

• Subtraction

• Multiplication

• Division

Order of Operations on Real Numbers

When evaluating arithmetic expressions that

contain several of these operations, we use

the following convention to determine the

order in which operations are performed:

Order of Operations on Real Numbers

1. Perform operations inside parenthesis

first, beginning with the innermost pair.• In dividing two expressions, the numerator and

denominator of the quotient are treated as if they are within parentheses.

2. Perform all multiplication and division• Working from left to right

3. Perform all addition and subtraction• Working from left to right

E.g. 2—Evaluating an Arithmetic Expression

Find the value of the expression

8 10

3 4 2 5 92 3

E.g. 2—Evaluating an Arithmetic Expression

First we evaluate the numerator and

denominator of the quotient.• Recall, these are treated as if they are inside

parentheses.

8 10 183 4 2 5 9 3 4 2 5 9

2 3 6

3 3 4 2 5 9

3 7 2 14

21 28

7

Properties of Real Numbers

Introduction

We all know that: 2 + 3 = 3 + 2

5 + 7 = 7 + 5

513 + 87 = 87 + 513

and so on.

• In algebra, we express all these (infinitely many) facts by writing:

a + b = b + a where a and b stand for any two numbers.

Commutative Property

In other words, “a + b = b + a” is a concise

way of saying that:

“when we add two numbers, the order

of addition doesn’t matter.”

• This is called the Commutative Property for Addition.

Properties of Real Numbers

From our experience with numbers, we

know that these properties are also valid.

Distributive Property

The Distributive Property

applies:

• Whenever we multiply a number by a sum.

Distributive Property

Figure 2 explains why this property works

for the case in which all the numbers are

positive integers.

• However, it is true for any real numbers a, b, and c.

E.g. 3—Using the Properties

2 + (3 + 7)

= 2 + (7 + 3) (Commutative Property of Addition)

= (2 + 7) + 3 (Associative Property of Addition)

Example (a)

E.g. 3—Using the Properties

2(x + 3)

= 2 . x + 2 . 3 (Distributive Property)

= 2x + 6 (Simplify)

Example (b)

E.g. 3—Using the Properties

(a + b)(x + y)

= (a + b)x + (a + b)y (Distributive Property)

= (ax + bx) + (ay + by) (Distributive Property)

= ax + bx + ay + by (Associative Property

of Addition)

• In the last step, we removed the parentheses.• According to the Associative Property, the order

of addition doesn’t matter.

Example (c)

Addition and Subtraction

Additive Identity

The number 0 is special for addition.

It is called the additive identity.

• This is because a + 0 = a for a real number a.

Subtraction

Every real number a has a negative, –a,

that satisfies a + (–a) = 0.

Subtraction undoes addition.

• To subtract a number from another, we simply add the negative of that number.

• By definition, a – b = a + (–b)

Note on “–a”

Don’t assume that –a is a negative number.• Whether –a is a negative or positive number

depends on the value of a.

• For example, if a = 5, then –a = –5.– A negative number

• However, if a = –5, then –a = –(–5) = 5.– A positive number

Properties of Negatives

To combine real numbers involving

negatives, we use these properties.

Property 5 & 6 of Negatives

Property 5 is often used with more than

two terms:• –(a + b + c) = –a – b – c

Property 6 states the intuitive fact

that: • a – b and b – a are negatives of each other.

E.g. 4—Using Properties of Negatives

Let x, y, and z be real numbers.

a) –(3 + 2) = –3 – 2 (Property 5: –(a + b) = –a – b)

b) –(x + 2) = –x – 2 (Property 5: –(a + b) = –a – b)

c) –(x + y – z) = –x – y – (–z) (Property 5)

= –x – y + z (Property 2:

–(– a) = a)

Multiplication and Division

Multiplicative Identity

The number 1 is special for multiplication.

It is called the multiplicative identity.

• This is because a . 1 = a for any real number a.

Division

Every nonzero real number a has an inverse,

1/a, that satisfies a . (1/a).

Division undoes multiplication.• To divide by a number, we multiply by

the inverse of that number.

• If b ≠ 0, then, by definition, a ÷ b = a . 1/b

• We write a . (1/b) as simply a/b.

Division

We refer to a/b as:

The quotient of a and b or as

the fraction a over b.

• a is the numerator. • b is the denominator (or divisor).

Division

To combine real numbers using division,

we use these properties.

Property 3 & 4

When adding fractions with different

denominators, we don’t usually use

Property 4.

• Instead, we rewrite the fractions so that they have the smallest common denominator (often smaller than the product of the denominators).

• Then, we use Property 3.

LCD

This denominator is the Least

Common Denominator (LCD).

• It is described in the next example.

E.g. 5—Using LCD to Add Fractions

Evaluate:

• Factoring each denominator into prime factors gives:

36 = 22 . 32

120 = 23 . 3 . 5

5 7

36 120

E.g. 5—Using LCD to Add Fractions

We find the LCD by forming the product of all

the factors that occur in these factorizations,

using the highest power of each factor.

• Thus, the LCD is:

23 . 32 . 5 = 360

E.g. 3—Using LCD to Add Fractions

So, we have:

(Use common denominator)

(Property 3: Adding fractions

with the same denominator)

5 7

36 1205 10 7 3

36 10 120 350 21

360 36071

360