Question

What is 5 divided by 0?

Answer 1

Undefined.

Explanation:

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

Suppose we attempted to assign a value to `5/0`, say `a=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 `oo` which satisfies `1/0 = oo` and `1/oo = 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 + oo = oo + a = oo`

If `a` is any ordinary non-zero Real number, then `a*oo = oo*a = oo`. So for example `5/0 = 5*1/0 = 5*oo = oo`

But expressions like `0*oo` have indeterminate value.

The expression `0/0` is still indeterminate too.

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