## Fractions are Hard!

### 3.3 Dividing by zero

As a consequence of our beliefs we saw that $$b\times \dfrac{a}{b}=a$$.  This provides a check to see if a fraction calculation is correct.

For example, suppose I am having trouble computing $$\dfrac{20}{4}$$. I think the answer is $$3$$.

We can check.

We should have $$4\times \dfrac{20}{4}=20$$  .

Does 3 fit this bill? No! $$4\times 3$$  is not $$20$$.

In general, one checks whether or not a division problem is correct by performing multiplication. For example

$$\dfrac{6}{2}=3$$ is correct because $$2$$ times $$3$$ is indeed $$6$$.

$$\dfrac{20}{4}=5$$  is correct because $$4$$ times $$5$$ is indeed $$20$$.

$$\dfrac{83}{9}=11$$ is not correct because $$9$$ times $$11$$ is not $$83$$.

$$\dfrac{18}{0.1}=180$$ is correct because $$0.1$$ times $$180$$ is indeed $$18$$.

 Thinking Question: a)       Cyril says that $$\dfrac{5}{0}$$ equals $$2$$. Why is he not correct? b)      Ethel says that $$\dfrac{5}{0}$$ equals $$17$$. Why is she not correct? c)       Wonhi says that $$\dfrac{5}{0}$$ equals $$887231243$$. Why is he not correct? d)      Duane says that there is no answer to $$\dfrac{5}{0}$$. Explain why he is correct.

 Thinking Question: Cyril says that $$\dfrac{0}{0}$$ equals $$2$$. Ethel says that $$\dfrac{0}{0}$$  equals $$17\frac{1}{2}$$. Wonhi says that $$\dfrac{0}{0}$$ equals $$887231243$$. Why do they each believe that they are correct? What might Duane say here?

To answer these questions …

Notice that if $$\dfrac{5}{0}=2$$, as Cyril says, then we should have that $$2$$ times $$0$$ is $$5$$, according to the check. This is not correct. In fact, the check shows that there is no number $$x$$ for which $$\dfrac{5}{0}=x$$ is correct.

On the other hand, Cyril says that $$\dfrac{0}{0}=2$$ and he believes he is correct because it passes the check: $$2$$ times $$0$$ is indeed zero.  But so too do $$\dfrac{0}{0}=17\frac{1}{}2$$ and $$\dfrac{0}{0}=887231243$$ pass the check!  In fact, $$\dfrac{0}{0}=x$$ passes the check for any number $$x$$ .

The trouble with $$\dfrac{a}{0}$$ (with $$a$$ not zero) is that there is no meaningful value to assign to it, and the trouble with $$\dfrac{0}{0}$$ is that there are too many possible values to give it!

In general, most people would say that dividing by zero is “undefined.” There is no means to give either an answer that is consistent with the arithmetic. The Basic Multiplication Belief suggests then that we should never allow the denominator of a fraction to be zero.

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