# What is 5 divided by 0?

Jul 30, 2016

Undefined.

#### Explanation:

Division by $0$ is (almost always) undefined.

Suppose we attempted to assign a value to $\frac{5}{0}$, say $a = \frac{5}{0}$

Then multiplying both sides by $0$ we get:

$0 = 5$

Since this is false, there is no such value $a$.

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Footnote

Why "almost always"? Are there any exceptions?

Suppose we try to add an object to our number system called $\infty$ which satisfies $\frac{1}{0} = \infty$ and $\frac{1}{\infty} = 0$. What can we do and not do with this object?

We can use it to provide the value of a function where it would otherwise have a vertical asymptote.

We can perform limited arithmetic on it:

If $a$ is any ordinary Real number, then $a + \infty = \infty + a = \infty$

If $a$ is any ordinary non-zero Real number, then $a \cdot \infty = \infty \cdot a = \infty$. So for example $\frac{5}{0} = 5 \cdot \frac{1}{0} = 5 \cdot \infty = \infty$

But expressions like $0 \cdot \infty$ have indeterminate value.

The expression $\frac{0}{0}$ is still indeterminate too.

This object $\infty$ does not fit in nicely with arithmetic as a whole, and should probably not be considered a number.