1.1 Apply Properties of Real Numbers

 

Opposite

The opposite, or additive inverse, of any number b is -b.

 

Reciprocal

The reciprocal, or multiplicative inverse, of any nonzero number b is              .

 

SUBSETS OF REAL NUMBERS

The real numbers consist of the _rational_ numbers and the _irrational_ numbers. Two subsets of the rational numbers are the _whole numbers_ (0,1, 2, 3...) and the _integers_ ( -3, -2, -1, 0,1, 2, 3...).

 

Rational Numbers

Irrational Numbers

 

·         Can be written as quotients of integers

·         Cannot be written as quotients of integers

 

·         Can be written as decimals that terminate or repeat

·         Cannot be written as decimals that terminate or repeat

 

Example 1

 


Graph the real numbers              and             on a number line.

 

Solution

Note that           = _-2.6_. Use a calculator to approximate           to the nearest tenth:      » _2.4_. So, graph   between _–3_ and _–2_ and graph  between _2_ and _3_.

 

 



 

PROPERTIES OF ADDITION AND MULTIPLICATION

Let a, b, and c be real numbers.

Property

Addition

Multiplication

_Closure_

a + b is a real number.

 

ab is a real number.

 

Commutative

a + b = _b + a_

 

ab = ba

 

Associative

 

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

 

(ab)c = _a(bc)_

 

Identity

a + 0 = a, _0 + a_ = a

 

a · 1 = a, _1 · a_ = a

Inverse

a + (-a) = _0_

a ·      = 1, a ¹ 0

 

The following property involves both addition and multiplication.

 

Distributive                         a(b + c) = _ab_ + _ac_

 

Example 2

 

Identify the property that the statement illustrates.

  1. (6 · 3) · 2 = 6 · (3 · 2)  __ Associative_ property of _multiplication_
  2. 21 + (-21) = 0             _ Inverse_ property of _addition_

 

 

Complete the following exercises.

1.      Graph the numbers

 

 

2.      Identify the property that 10(6 + 8) = 10(6) + 10(8) illustrates.

Distributive property

 

DEFINING SUBTRACTION AND DIVISION

Subtraction is defined as _adding the opposite_. The opposite, or _additive inverse_, of any number b is -b. If b is positive, then -b is negative. If b is negative, then -b is positive.

a - b = a + (-b)                         Definition of subtraction

Division is defined as _multiplying by the reciprocal_. The reciprocal, or _multiplicative inverse_, of any nonzero number b is

a ¸ b = a ·        b ¹ 0                    Definition of division

 

Example 3

Use properties and definitions of operations

 

Show that 9 + (b - 9) = b.

9 + (b - 9)

= 9 + [b + ( -9 )]

Definition of subtraction

= 9 + [( -9 ) + b]

Commutative property of addition

= [9 + (-9)] + b

_Associative_ property of addition

= _0_ + b

Inverse property of addition

= _b_

Identity property of addition

 

Use properties and definitions of operations to show that the statement is true.

 

(a ¸ 5) = a

 

Definition of division

Commutative prop, of x

Associative prop, of x

=1 · a

Inverse prop, of x

= a

Identity prop, of x