Division by zero

In mathematics, the result of division by zero, such as a ÷ 0, is undefined and not allowed in real numbers and integers. 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, within real numbers, it is incorrect to say that a ÷ 0 is infinity because infinity is not a real number and does not follow the rules for real numbers.

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; see also indeterminate form.

Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other number systems. Zero divisors are frequently found in group theory and in hyperreal numbers and surreal numbers.

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 an integer by zero. However, most programss that use user input for calculations perform checks to make sure a divide by zero operation is not attempted. The IEEE standard for computer floating-point numbers states that x ÷ 0 is Infinity when x is positive, negative Infinity when x is negative, and NaN (Not a Number) when x is zero.