﻿ Fractions Involving Zero
FRACTIONS INVOLVING ZERO
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

Here, you will practice simplifying fractions involving zero.

FRACTIONS WITH ZERO IN THE NUMERATOR

Any fraction with zero in the numerator and a nonzero number in the denominator equals zero.

For example:   $\displaystyle\, \cssId{s10}{\frac{0}{5}} \cssId{s11}{= \frac{0}{-3}} \cssId{s12}{= \frac{0}{1.4}} \cssId{s13}{= 0}\,$

Why is this?
Here are two different ways you can think about it:

• Every fraction $\,\frac{N}{D}\,$ can be rewritten as:   $\,N\cdot \frac{1}{D}\,$

For example:   $\,\frac34 = 3\cdot\frac 14\,$

Thus:   $\cssId{s21}{\frac03} \cssId{s22}{= 0\cdot \frac13} \cssId{s23}{= 0}\,$
• The fraction $\,\frac{N}{D}\,$ answers both these questions:
• (the number of piles interpretation)
Given $\,N\,$ objects, if they are divided into equal piles of size $\,D\,$, how many piles are there? Answer: $\frac{N}{D}$
• (the size of piles interpretation)
Given $\,N\,$ objects, if they are divided into $\,D\,$ equal piles, what is the size of each pile? Answer: $\frac{N}{D}$
Now apply these interpretations to a fraction with zero in the numerator—say, the fraction $\,\frac03\,$:
• Given $\,0\,$ objects, if they are divided into equal piles of size $\,3\,$, how many piles are there?
• Given $\,0\,$ objects, if they are divided into $\,3\,$ equal piles, what is the size of each pile?
In both cases, the answer is zero. No objects, nothing to work with, you can't make any piles.

FRACTIONS WITH ZERO IN THE DENOMINATOR

Division by zero is not allowed, and we say that such a fraction is not defined.

For example:   $\displaystyle\,\frac{5}{0}\,$ is not defined;   $\displaystyle\,\frac{0}{0}\,$ is not defined

Why is this?
Consider, for example, the fraction $\frac50\,$.
You have $\,5\,$ objects. You want to divide them into piles of size $\,0\,$. How many piles?
Serious problem. With piles of size zero, you're going to have trouble getting rid of your five objects.
You can't just snap your finger and have matter disappear!

A more precise argument also covers the case $\,\frac00\,$, but uses material from later in this course.
Interested?

Master the ideas from this section