# Division by Zero

# Division by Zero

The number 0 has unique properties, including when a number is multiplied or divided by 0. Multiplying a number by 0 equals 0. For example, 256 × 0 = 0. Dividing a number by 0, however, is undefined.

Why is dividing a number by 0 undefined? Suppose dividing 5 by 0 produces a number *x* :

From it follows that 0 × *x* must be 5. But the product of 0 and any number is always 0. Therefore, there is no number *x* that works, and division by 0 is undefined.

## A False Proof

If division by 0 were allowed, it could be proved—falsely—that 1 = 2. Suppose *x = y*. Using valid properties of equations, the above equation is rewritten

*x* ^{2} = *xy* (after multiplying both sides by *x* )

*x* ^{2} − *y* ^{2} = *xy − y* ^{2} (after subtracting *y* ^{2} from both sides)

(*x - y* )(*x + y* ) = *y* (*x − y* ) (after factoring both sides)

(*x + y* ) = *y* (after dividing both sides by (*x − y* ))

2*y = y* (*x = y*, based on the original supposition)

2 = 1 (after dividing both sides by *y* )

This absurd result (2 = 1) comes from division by 0. If *x = y*, dividing by (*x − y* ) is essentially dividing by 0 because *x − y* = 0.

## Approaching Limits

It is interesting to note that dividing a number such as 5 by a series of increasingly small numbers (0.1, 0.01, 0.001, and so on) produces increasingly large numbers (50, 500, 5000, and so on). This division sequence can be written as where *x* approaches but never equals 0. In mathematical language, as *x* approaches 0, increases without limit or that **infinity** .

see also Consistency; Infinity; Limit.

*Frederick Landwehr*

## Bibliography

Amdahl, Kenn, and Jim Loats. *Algebra Unplugged.* Broomfield, CO: Clearwater Publishing Co., 1995.

Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. *Mathematical Ideas,* 9th ed. Boston: Addison-Wesley, 2001.

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