Why isn't zero divided by zero either zero or one?

Consider the equation 0 times x = 0.

This is true for every number x.

Divide both sides by 0; we get x = 0/0.

So zero divided by zero could be any number at all; it could be -42, or 273.15, or anything else.

If we try to pick one value for 0/0, we will eventually get into trouble.

Examples:

Say 0/0 = 1 = 1/1.

Multiply the numerator of both sides by 3. Then

(3 times 0)/0 = (3 times 1)/1.

Therefore 0/0 = 3.

Since 0/0 = 1, we get 1 = 3, which we really don't want, as all of our mathematics will become useless.

Say 0/0 = 0.

Then 0/0 = 0/1.

Turn both fractions upside down. We get

0/0 = 1/0, but since 0/0 = 0, we get

0 = 1/0.

Multiplying both sides by 0 gives

0 times 0 = 1,

so 0 = 1, which we don't want either.

The best thing to do is not to give 0/0 any value; we say 0/0 is undefined. Also we take x/0 to be undefined for every number x.