Why Division By Zero is Undefined

John Spacey, June 08, 2018

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.