How to divide by 0

In elementary school, we learn how easy it is to multiply by zero and one. Anything times one is itself, and anything times zero is zero. Things seem to be going well when we next learn that anything divided by one is itself. By analogy, we guess that anything divided by zero is zero.

But when we test this theory and divide by zero, all we get is an error message from on high. The teacher-prophet comes down from the mountain and delivers the First Commandment of Mathematics: “Thou shalt not divide by zero.”

Unfortunately, the teacher gives little more explanation for this immutable prohibition than “because it’s undefined.”

To see why we can’t divide by zero, we should back up and ask what it means to divide.

Division is just a name for multiplication by the reciprocal. The reciprocal is the “multiplicative inverse” or “one over” a number.

So to divide by zero, we’d need to know what “1/0” means.

Except that if we assume the usual properties of numbers (precisely speaking, the ring axioms), it becomes difficult to define “1/0.”

Let’s pretend “1/0” is a number, called n (in honor of Chuck Norris, who supposedly can divide by zero). Then what’s 0*n?

Well, on the one hand, 0*n = 1, because that’s what we wanted n to be: the reciprocal of 0. And a number times its reciprocal is 1 by definition. But on the other hand, 0*n = 0, since zero times anything is zero.

So 1=0. Yikes!

This is not good: multiplying the last equation by any number shows it is equal to zero. Thus, using the normal rules of numbers, we can divide by zero only if we are willing to live in a universe where the only number is zero.

However, there are ways of changing our definitions to allow division by zero.

After learning about limits in calculus, students often think that “1/0” should be infinity. But even in calculus, this limit does not exist.

Approaching zero from one side, we get positive infinity, and from the other we get negative infinity.

The other problem is that neither “infinity” is a real number. One can, however, add an unsigned “point at infinity” to the real numbers to get the “real projective line.” Here, anything nonzero divided by zero is infinity. However, there are still problems with defining 0/0.

Some work from the past 10 years has created algebraic systems which allow division by zero. Called “wheels,” these extensions of the usual numbers have somewhat aesthetically perverse axioms.

Division must be defined not as multiplication by the reciprocal, but as a separate operation. Also, anything times zero is not necessarily zero in a wheel.

The problem with discussing division by zero (and other tricky issues like why 0.999...=1) is a lack of clarity in definitions. Internet arguments on such things often devolve until nobody knows what definitions are in force.

Thus, except in some exotic or trivial settings, division by zero is not defined. Usually, it’s just not allowed.

In other words: you can’t divide by zero, except when you can.  
 

E-mail: brownjoh@indiana.edu