**Division by zero**is division where the denominator is zero such as 1/0. This is always undefined and has no meaning. The following are potential solutions to division by zero designed to illustrate why this is the case.

## Infinity Theories

One way to approximate answers for division by zero is to divide by smaller and smaller numbers to get closer to zero to see what happens:**1/0.1 = 10**

**1/0.0000001 = 10,000,000**

**1/0.0000000001 = 10,000,000,000**Looking at this you might assume that division by zero equals infinity as answers get larger and larger as the denominator approaches zero.

**1/0 = ∞ ????**However, if you approach zero from the other side by using small negative numbers the results are very different:

**1/-0.1 = -10**

**1/-0.0000001 = -10,000,000**

**1/-0.0000000001 = -10,000,000,000**These results imply that division by zero results in negative infinity as answers get smaller and smaller as the denominator approaches 0.

**1/0 = -∞ ????**Therefore, attempting to solve division by zero by approximation yields two equally valid arguments: the answer is infinity and negative infinity. As there are no two numbers further apart than infinity and negative infinity, this indicates the answer is indeterminate and undefined.

## A Proof of 1=2

If division by zero is possible, it allows for proofs of things we know not to be true. For example:Assumptions:**0 × 1 = 0**

**0 × 2 = 0**By deduction:

**0 × 1 = 0 × 2**If division by zero is allowed, this can be simplified:

**(0 x 1)/0 = (0 x 2)/0**As multiplication and division are inverse operations the zeros can be canceled out and this can be reduced to:

**1 = 2**Therefore, if division by zero is defined, 1=2.

## Inverse Operations

Multiplication and division are inverse operations, meaning that they can undo each other. For example:**(2/3) × 3 = 2**

**(33 / 4) × 4 = 33**

**(33,000 / 44) × 44 = 33,000**This doesn't work for division by zero:

**(1 / 0) × 0 ≠ 1**Any number multiplied by 0 is 0. As such, there is no possible value for (1/0) that can satisfy the principle that multiplication and division are inverse operations.

## What About 0/0?

There are two equally valid and conflicting theories for the answer to 0/0:**Zero divided by any number is 0. Therefore, 0/0 = 0**

**Any number divided by itself is 1. Therefore 0/0 = 1**As these are equally valid theories that contradict, the answer is indeterminate and undefined.

### Notes

Division by zero generates an exception on most computing platforms such that it can potentially crash software. As such, division by zero is often handled as a special case in code. Not handling division by zero in some intelligent way is a common type of bug.Overview: Division By Zero | ||

Type | ||

Definition | Division where the denominator is zero. | |

Related Concepts |