The Reciprocal of a Real
Number
September 14, 2011

The Reciprocal of a Real
Number
Objective To simplify expressions involving reciprocals.

Reciprocals
Two numbers whose product is 1 are called reciprocals , or multiplicative inverses , of each other. For example:
5 and
1
5
are reciprocals because 5 ∙
1
5
= 1
5
4
and
4
5
are reciprocals because
5
4
∙
4
5
= 1
−1.25
and −0.8
are reciprocals because
− 1.25 ∙ −0.8 = 1

Reciprocals
1 is its own reciprocal because 1 ∙ 1 = 1
−1 is its own reciprocal because −1 ∙ −1 = 1
0 has no reciprocal because 0 times any number is 0 , not 1 .
The symbol for the reciprocal, or multiplicative
1 inverse, of a nonzero real number a is 𝑎 real number except 0 has a reciprocal.
. Every

Property of Reciprocals
For every nonzero real number a , there is a
1 unique real number 𝑎
such that 𝒂 ∙
𝟏 𝒂
= 𝟏 and
𝟏 𝒂
∙ 𝒂 = 𝟏

Reciprocals
Consider the following product:
−𝑎 ∙ −
1 𝑎
= −1𝑎 −1 ∙
1 𝑎
= −1 −1 𝑎 ∙
1 𝑎
= 1 ∙ 1 = 1
Therefore, −𝑎 and −
1 𝑎
are reciprocals.

Property of the Reciprocal of the Opposite of a Number
For every nonzero real number a ,
𝟏
−𝒂
= −
𝟏 𝒂
Read, “The reciprocal of −𝑎 is −
1 𝑎
.”

Reciprocals
Consider the following product: 𝑎𝑏
1 𝑎
∙
1 𝑏
= 𝑎 ∙
1 𝑎 𝑏 ∙
1 𝑏
= 1 ∙ 1 = 1
Therefore, 𝑎𝑏 and
1 𝑎
∙
1 𝑏
are reciprocals.

Property of the Reciprocal of a
Product
For all nonzero real numbers a and b ,
𝟏 𝒂𝒃
=
𝟏 𝒂
∙
𝟏 𝒃
The reciprocal of the product of two nonzero numbers is the product of their reciprocals.

Simplify
Example 1a
1
4
∙
1
−7
1
4
∙
1
−7
=
1
4 −7
=
1
−28
= −
1
28

Example 1b
Simplify 4𝑦 ∙
1
4
4𝑦 ∙
1
4
= 4 ∙
1
4 𝑦 = 1𝑦 = 𝑦

Example 1c
Simplify −6𝑎𝑏 −
1
3
−6𝑎𝑏 −
1
3
= −6 −
1
3 𝑎𝑏 = 2𝑎𝑏

Example 2
1
3
Simplify
1
3
42𝑚 − 3𝑣
42𝑚 − 3𝑣 =
1
3
42𝑚 −
1
3
3𝑣
=
1
3
∙ 42 𝑚 −
1
3
∙ 3 𝑣
= 14𝑚 − 𝑣

Class work
Oral Exercises p 80: 1-25
Homework p 81: 1-31 odd p 82: Mixed Review