Division by zero

Division by zero -- A division, such as a &divide b is defined only when b is nonzero. The smaller the divisor in a division, the larger the quotient (answer), and as the divisor approaches zero, the quotient approaches infinity. It could be said that the answer to ?anything divided by zero? is infinity; however, allowing something to equal infinity causes the rules of arithmetic to break down, so instead the division itself is not allowed within the rules of arithmetic.

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 &times 0 = 0,

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

0 &divide 0 = 2.

Yet any number divided by itself is 1, so this cannot be right. Furthermore,

4 &times 0 = 0,

so

0 &divide 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 the 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 derivative (calculus) for more information.

Extension to complex numbers

In the complex plane. division by zero is an example of a simple pole.

Computers

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

Non-standard analysis

In some non-standard formulations of mathematics, such as Conway numbers, division by zero is permitted, with a defined result.