Why Can't I Divide by Zero ?

Many people get rebellious when they are told they cannot divide by zero.Why can't I divide by zero?", they ask. Nobody likes being told they can't do something. But the truth of the matter is, they can! Anyone can defi ne things like 1/0 to be any number they like. It's a free country. The same goes for dividing by zero. You can do it, if you wish, but you will give something up in the bargain, just as the lawmakers and they would make some law. In the case of division by zero, what must be given up are the familiar laws of arithmetic. So I'm not going to try to convince you in what follows that you cannot divide by zero. I shall merely try to convince you that you should not divide by zero.

All the familiar rules of arithmetic can be summarized in about 11 rules called the Field Axioms. An axiom is an assumption that everyone agrees to believe without proof, and from which all other mathematical facts are derived. The field axioms include the familiar commutative and associative laws of addition and multiplication,

the distributive law, as well as the seemingly innocuous requirement that 

                                                             1 not= 0 -  (*)

I should point out that the 1 and 0 in this statement do not necessarily refer to the familiar numbers from arithmetic. It is one of the remarkable discoveries of modern mathematics that there are alternative number "systems" which equally well satisfy the eld axioms. The simplest example is a number system with only two "numbers" in it, denoted 1 and 0. All the usual rules of arithmetic hold except that 1 + 1 = 0.

One of the cardinal principles of mathematical reasoning is that facts derived from axioms cannot contradict any of the axioms themselves. If this ever happened, the set of axioms would have to be modified or abandoned. Unfortunately, this is just what happens if one attempts to introduce additional axioms that defi ne values for expressions like 1/0.

For example, suppose we attempted to defi ne 1/0 = 1. Then multiplying both sides by zero would give

   0 x 1 / 0 = 1 x  0.

Cancelling zero on the left and using 1 0 = 0 on the right, we obtain 1 = 0. Oops! We just violated axiom (*).

What about 0/0 ? This expression arises in connection with the study of limits in calculus, and such limits would be much easier to handle if we had a value for 0/0. Let's try defi ning 0 / 0 = 1: If we then multiply both sides by -1 we would have

-1 x ( 0 / 0 ) = -1 x 1 = -1;

or

-1 x 0 / 0 = -1.

Since -1 x 0 = 0; we would conclude that 0 /0 = -1: But we defi ned this fraction to have the value 1, so we must (reluctantly) conclude that -1 = 1: Trouble is looming on the horizon. Indeed, if we add 1 to both sides and then divide by 2 we're right back to 1 = 0.

A similar diffi culty arises if 0 /0 is de ned to be any nonzero number. As a last  resort, let's try defi ning

0 / 0 = 0.

This doesn't work either, but the trouble lies a bit deeper this time. First, we must understand that division isn't really an independent operation at all. You won't nd division mentioned anywhere in the 11 fi eld axioms that summarize all the important properties of arithmetic. Indeed, the expression a / b is merely shorthand for the expression a x b-1, where the fi eld axioms guarantee the existence of a multiplicative inverse b-1 for any b . It is unique for a given b and satisfi es b  x b-1 = 1.

The field axioms don't directly forbid 0 to have a multiplicative inverse, but it turns out they do so indirectly. When we say 0 / 0 = 0; we're really saying two things: fi rst, that 0 does have a multiplicative inverse after all; and secondly, that it satis es 0 x 0-1 = 0: But also 0 x 0-1 = 1; since that's what it means to be a multiplicative

inverse. Thus, 1 = 0. Another way to reach the same conclusion is using the usual rules for adding fractions:

                                            1/1 + 0/0 = 0 x 1 + 1 x 0 / 0 x  1  = 0 / 0 = 0 .

On the other hand,

1 / 1 + 0 / 0 = 1 + 0 = 1.

Once again, we have 0 = 1: (Thanks to Andy Vogel and Dan Zacharia for this last example.)

In conclusion I should remark that axiom (*) really has only one purpose: to rule out the trivial example of a eld with only 1 element, 0. Why not then simply allow this as an example of a eld? Little would be gained by doing this, and much would be lost. Virtually every theorem proved about fi elds would have have to include a special proviso to handle the exceptional example.