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3.2.1 Zero as Numerator and Denominator

The whole number 0 can be written as a fraction in infinitely many ways:
0
0= , for any non-zero b.
b
It is easy to understand why this is true: you can divide something into any number of equal pieces
(say, b of them), but if you take none of them,you have taken an amount equal to 0 – no matter the
value of b.Thus,
0 0 0
0= = = =...
1 2 3
So 0 can certainly be the numerator of a fraction. Can it be the denominator? The answer is no.
Here is one way to think about it: does it make sense to divide something into 0 pieces? (1 piece, yes,
but 0 pieces?) This is closely related to the fact that 0 cannot be the divisor in a division problem (see
Section 1.6.) There is another reason why 0 cannot be the denominator of a fraction. Multiplication
and division are mutually inverse operations, meaning that the equation a = c is equivalent to the
b
equation a = b · c whenever b ̸= 0. Suppose we could assign a numerical value to the fraction 1/0, say,
1 =1. That would mean that 1 =0 · 1. But of course, 0 · 1 = 0, so we arrive at 1 = 0, an obvious
0
contradiction.
For these reasons, we say that a fraction with denominator 0 is undefined.

n
For any whole number n,including 0, the fraction is un-
0
defined.

3.2.2 Exercises

1. What improper fraction does the following picture represent?

+ +

Use rectangles, circles, or squares to represent the following fractions.
2. 3
2
3. 3
4
4. 5
8
5. 11
6
6. 4 3

7. 6 2

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