Fractions Involving Zero

Need some basic practice with fractions first? Rewriting Fractions as a Whole Number plus a Fraction and Locating Fractions on a Number Line

Here, you will practice simplifying fractions involving zero.

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

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

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

First way

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

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

Thus:   $\cssId{s21}{\frac03} \cssId{s22}{= 0\cdot \frac13} \cssId{s23}{= 0}\,$

Second way

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? Read the text.

Practice