Math CentralQuandaries & Queries


Question from Bill:

Why does multiplication have a favored status so as to make division by zero undefined?

Hi Bill,

The number 1 plays a special role in the arithmetic of the real numbers. For any real number a,

a × 1 = 1 × a = a.

There are times when you have a real number and you somehow want to use this number to generate the number 1. For example you might ask

"Given a real number b is there a real number c so that b × c = 1?"

If such a number c exists we call it the reciprocal of b and write it b-1 or sometimes 1/b. The difficulty that results is that if the number b that is given is actually 0 then we are asking "Given a real number 0 is there a real number c so that 0 × c = 1?" Clearly the answer is no since for any real number c, 0 × c = 0. Thus the number we might want to call the reciprocal of 0 and write 0-1 or 1/0 doesn't exist.

I hope this helps,

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