## Zero in the Numerator

##### Think / Pair Share

Does the fraction \(\frac{0}{11}\) make sense?

- Write a “pies per child” story for the fraction \(\frac{0}{11}\). Does it make sense? How much pie does each individual child receive in your story?
- Think of \(\frac{0}{11}\) as the answer to a division problem. What is that division problem? Can you solve it?

It seems pretty clear that zero pies among eleven kids gives zero pies per child:

\[\frac{0}{11} = 0 \ldotp \nonumber \]

The same reasoning would lead us to say:

\[\frac{0}{b} = 0\; \text{for any positive number}\; b \ldotp \nonumber \]

The “Pies Per Child Model” offers one explanation: If there are no pies for us to share, no one gets any pie. It does not matter how many children there are. No pie is no pie is no pie.

We can also justify this claim by thinking about a missing factor multiplication problem:

\[\frac{0}{b}\; \text{is asking us to fill in the blank} :\; \_\_ \cdot \; b = 0 \ldotp \nonumber \]

The only way to fill that in and make a true statement is with 0, so \(\frac{0}{b} = 0\).

## Zero in the Denominator

What happens if things are flipped the other way round?

##### Think / Pair / Share

Does the fraction \(\frac{11}{0}\) make sense?

- Write a “pies per child” story for the fraction \(\frac{11}{0}\). Does it make sense? How much pie does each individual child receive in your story?
- Think of \(\frac{11}{0}\) as the answer to a division problem. What is that division problem? Can you solve it?

Students often learn in school that “dividing by 0 is undefined.” But they learn this as a rule, rather than thinking about why it makes sense or how it connects to other ideas in mathematics. In this case, the most natural connection is to a multiplication fact, the zero property for multiplication:

\[\text{any number} \cdot 0 = 0 \ldotp \nonumber \]

That says we can never find solutions to problems like

\[\_\_ \cdot 0 = 5, \qquad \_\_ \cdot 0 = 17, \qquad \_\_ \cdot 0 = 1 \ldotp \nonumber \]

Using the connection between fractions and division, and the connection between division and multiplication, that means there is no number \(\frac{5}{0}\). There is no number \(\frac{17}{0}\). And there is no number \(\frac{1}{0}\). They are all “undefined” because they are not equal to any number at all.

##### Think / Pair / Share

Can we give meaning to \(\frac{0}{0}\) at least? After all, a zero would appear on both sides of that equation!

- Cyril says that \(\frac{0}{0} = 2\) since \(0 \cdot 2 = 0\).
- Ethel says that \(\frac{0}{0} = 17\) since \(0 \cdot 17 = 0\).
- Wonhi says that \(\frac{0}{0} = 887231243\) since \(0 \cdot 887231243 = 0\).

Who is right? Can they all be correct? What do you think?

Cyril says that \(\frac{0}{0} = 2\), and he believes he is correct because it passes the check: \(0 \cdot 2 = 0\).

But 17 also passes the check, and so does 887231243. In fact, I can choose any number for x, and \(0 \cdot x = 0\) will pass the check!

The trouble with the expression \(\frac{a}{0}\) (with not zero) is that there is *no meaningful value* to assign to it. The trouble with \(\frac{0}{0}\) is different: There are *too many possible values* to give it!

Dividing by zero is simply too problematic to be done! It is best to avoid doing so and never will we allow zero as the denominator of a fraction. (But all is fine with 0 as a numerator.)