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 ))
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.
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: AddisonWesley, 2001.
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