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

x² = xy (after multiplying both sides by x)

x² - y² = xy - y² (after subtracting y² from both sides)

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

(x + y) = y (after dividing both sides by (x - y))

2y = 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.

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.