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3.10 Division of Fractions

Our intuition is not very good when we think of dividing two fractions. How many times does 2 3 “go
3
into” , for example? Equivalently, what fraction results from the division problem
4
3 2
÷ ?
4 3
It is not at all obvious. We will derive a simple rule, but we’ll need an auxiliary concept.

3.10.1 Reciprocals

Two non-zero numbers are reciprocal if their product is 1. Thus, if x and y are nonzero numbers, and
if
x · y =1,

then x is the reciprocal of y,and, also, y is the reciprocal of x.
The rule for multiplying fractions, together with obvious cancellations, shows that
1 1
a b '✒ a · b 1 · ✚ b ✚❃ 1
· = = = =1.
b a 1 ✚❃ 1 1
b · '✒ a ✚ b · 1
This means that

a b
The reciprocal of the fraction is the fraction .
b a
(both a and b nonzero)

n
Since every whole number n can be written as the fraction ,we have the following special case:
1
The reciprocal of the whole number n (nonzero)
1
is the fraction .
n

8
1
Example 116. The reciprocal of 5 is . The reciprocal of 1 9 is 9. The reciprocal of 3 8 is , or, expressed
3
5
2
as a mixed number, 2 .
3
We note some important facts and special cases:
• the reciprocal of the reciprocal of a number is the number itself. For example, the reciprocal of
2/3is 3/2, and in turn, the reciprocal of 3/2is 2/3.
1
• 1 is the only (positive) number that is its own reciprocal (since1 = .)
1
• the reciprocal of a number less than 1 is a number greater than 1, and vice versa (since the
reciprocal of a proper fraction is an improper fraction).

• 0 has no reciprocal (since division by 0 is undefined).

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