Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Why can't we divide by zero?

The reason that the result of a division by zero is undefined is the fact that any attempt at a definition leads to a contradiction.

To begin with, how do we define division? The ratio r of two numbers a and b:


is that number r that satisfies


Well, if b=0, i.e., we are trying to divide by zero, we have to find a number r such that

r*0=a. (1)
for all numbers r, and so unless a=0 there is no solution of equation (1).

Now you could say that r=infinity satisfies (1). That's a common way of putting things, but what's infinity? It is not a number! Why not? Because if we treated it like a number we'd run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that's so, then

infinity = infinity+1 = infinity + 2

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

What about 0/0?

I said above that we can't solve the equation (1) unless a=0. So, in that case, what does it mean to divide by zero?

Again, we run into contradictions if we attempt to assign any number to 0/0.

Let's call the result of 0/0, z, if it made sense. z would have to satisfy

z*0=0. (2)

That's OK as far as it goes, any number z satisfies that equation. But it means that the result of 0/0 could be anything. We could argue that it's 1, or 2, and again we have a contradiction since 1 does not equal 2.

But perhaps there is a number z satisfying (2) that's somehow special and we just have not identified it? So here is a slightly more subtle approach. Division is a continuous process. Suppose b and c are both non-zero. Then, in a sense that can be made precise. the ratios a/b and a/c will be close if b and c are close. A similar statement applies to the numerator of a ratio (except that it may be zero.)

So now assume that 0/0 has some meaningful numerical value (whatever it may be - we don't know yet), and consider a situation where both a and b in the ratio a/b become smaller and smaller. As they do the ratio should become closer and closer to the unknown value of 0/0.

There are many ways in which we can choose a and b and let them become smaller. For example, suppose that a=b throughout the process. For example, we might pick

a=b = 1, 1/2, 1/3, 1/4, ....



for all choices of a we get the ratio 1 every time! This suggests that 0/0 should equal 1. But we could just as well pick

b = 1, 1/2, 1/3, 1/4, ....

and let a be twice as large as b. Then the ratio is always 2! So 0/0 should equal 2. But we just said it should equal 1! In fact, by letting a be r times as large as b we could get any ratio r we please!

So again we run into contradictions, and therefore we are compelled to

let 0/0 be undefined.

It's a common strategy in teaching to simplify concepts when they are first encountered. In other words, it's common for your teacher to lie to you. I just did! Actually, there is a way to make sense of the expression 0/0. The basic idea is to let both the numerator and the denominator become smaller and smaller, and to make the value of 0/0 dependent upon the way in which numerator and denominator approach 0. This is explained more thoroughly here.

Fine print, your comments, more links, Peter Alfeld, PA1UM