#### Why are you not allowed to divide by zero?

Despite calculators and computers often describing it as an "illegal operation", there is no law against it. The simple answer is that the result is unknown.

Suppose you attempt to divide one by zero. Let the quotient (the result after dividing) be q.

As 1/0 = q, it follows that 1 = 0×q = 0, which is absurd. In fact it can be seen that any non-zero quantity, r, divided by zero would lead to the same contradiction: r, a non-zero value, is equal to zero.

This would suggest that calculators and computers got it right. However, if we consider the special case, 0/0 = q, we get, 0 = 0×q = 0, and this does not seem to lead to a contradiction. So is it okay to divide zero by zero?

Consider the well known algebraic identity,

xn–1 = (x–1)(xn–1+xn–2+...+x2+x+1)

Rearranging, (xn–1)/(x–1) = xn–1+xn–2+...+x2+x+1.

If we let x = 1, this leads to, 0/0 = 1 + 1 + ... + 1 = n; that is, we can show that 0/0 can be equal to any natural number you choose and so the value of 0/0 is unknown.

Previously we showed that, r/0 = q, where r is not equal to zero, leads to the contradiction, r = 0. This supports the fact that any non-zero value divided by zero is not equal to a finite value. However, there remains one other possibility: the result of dividing by zero is not finite, but infinite.

Consider the ratio, 1/x as x gets closer to zero.

1/0.1 = 10
1/0.01 = 100
1/0.001 = 1000
and so on.

This suggests that as x tends towards zero, 1/x tends towards infinity. However...

1/-0.1 = -10
1/-0.01 = -100
1/-0.001 = -1000
and so on.

That is, as x tends towards zero (from a negative direction), 1/x tends towards negative infinity.

So which is it? Does 1/0 = ∞ or does 1/0 = -∞?

To resolve this quandary, we say the anything divided by zero is unknown, or indeterminate: there is no reason why it should be equal to any one particular value over the other possible choices.