Division by zero

In mathematics, the result of division by zero, such as a ÷ 0, is undefined and not allowed in arithmetic. The reason is the following: division ought to be the inverse operation of multiplication, which means that a ÷ b should be the solution x of bx = a, but for b=0 this has no solution if a≠0, and any x as solution if also a=0. In both cases a ÷ b can not be defined meaningfully.

In particular, it is incorrect to say that a ÷ 0 is infinity. The argument that any number a, divided by a very small one, becomes extremely large is unconvincing: a negative number a divided by a small positive number does not become large, and neither does a positive number a divided by a small negative number.

Another way to see why division by zero does not work is to work backwards from multiplication, remembering that anything multiplied by zero is zero. So

2 × 0 = 0,

which, if we are allowed to divide by zero, means that

0 ÷ 0 = 2.

But

4 × 0 = 0,

so

0 ÷ 0 = 4,

suggesting that 2 = 4, which is nonsense.

It is possible to disguise a division by zero in a long algebraic argument, leading to such things as a spurious proof that 2 equals 1.

It is both possible and meaningful to find the limit as x approaches 0 of some divisions by x; see l'Hopital's rule for some examples.

Extension to complex numbers

For the complex plane, see also Pole (complex analysis).

Computers

Many computer architectures produce a runtime exception[?] when an attempt is made to divide by zero.