*“a”* Divided by Zero is Real

NOTE – This discussion assumes that number theory is related to reality. That is speculative to say the least, but it may still be a useful exercise.

I

In arithmetic we say that a number divided by zero is undefined; it has no meaning. What does that mean?

Arithmetic is a static system. *a*/*b *= *c* is about a stable, defined relationship between numbers. Division is the opposite operation from multiplication. Starting with a number *a* we can get a unique number *c* by dividing *a* by *b*. *a*/*b *= *c* means that *b***c *= *a*. If *b = 0*, that is the same as saying that *c* multiplied by *0* is equal to *a*. But there is no number *c* that will give any non-zero number *a* as a result of that operation. Starting in elementary school we are sometimes led to believe that there is a number “infinity” that is the inverse of *0*, and this somehow solves the problem.

A little thought shows it doesn’t. If *a* divided by *0* equals infinity, then infinity times *0* would have to equal *a*. Let’s give a value to *a*. Say *a *= 3,457. So 3,457 divided by 0 equals infinity, and infinity times *0* equals 3,457. But then that relation must be true for any value we give to *a*, so infinity times *0* equals 5, or 657, or 2,345,678, or ??. If someone just asks us what infinity times *0* equals, it could be anything. It’s now *a* that is undefined.

The problem is that infinity isn’t a number. It’s a process. A process takes time, which means it can’t be part of a static system, so infinity can’t be invoked in arithmetic.

And infinity isn’t just any kind of process; it’s a process that never ends. Here’s how it works.

In algebra, we use letters of the alphabet to represent numbers, but we reserve the last letters of the alphabet (*x* and *y* particularly) to represent place holders for numbers that we want to change according to some defined process while keeping the other numbers constant. So I’ll change the equation to *a*/*x *= *y*. In this equation, *a* is some number that stays the same while we change the value of *x* (which will change the value of *y*). If *x *= *a*, the left side of the equation becomes *a*/*a*, which equals 1 no matter what number *a* represents. If *x* is greater than *a*, *a*/*x* is less than 1. If *x* is less than *a*, *a*/*x* is greater than 1.

Starting with *x *= *a*, I’ll make *x* get smaller by dividing it in half over and over again. Each time I do this, *a*/*x* gets bigger. In theory, I can keep dividing *x* in half forever, so *y *= *a*/*x *means *y* will keep getting bigger forever. This is what we mean by infinity. You can see that since dividing a number in half over and over will never reach 0, we can never get a final answer to the equation *a*/*0 *= *y*. That’s why we say that *a*/*0* is undefined. (This is Zeno’s famous paradox in a different form.)

End of story? Maybe not.

II

In calculus we find a way to identify in some situations a value for a function *y *= *a*/*x* as *x* approaches *0*. That’s one way around the problem (still regarded as “handwaving” in some circles), and I think it indicates something important about our universe.

Here’s a step further.

Remember I said “in theory” I can divide *x* in half forever, so the approach of *x* to zero is effectively *0* and the increase of *a*/*x* is unbounded. But as Yogi Berra said, “In theory, theory and practice are the same. In practice, they’re not.”

Because *x* can’t get to zero in the real world, the progression of the value *a*/*x* as *x* approaches zero must effectively stop at some point, some number, below a certain upper limit. It’s an actual number. Its value may be taken to be real since the world has moved on and stopped calculating. At that point a value is fixed for *y *= *a*/*x*, with *x *~= *0* but not *0*.

What is that upper limit? It’s where the world stops calculating. In the real world there isn’t any difference between this very large number and infinity, no matter what the mathematicians say.

Note that this is different from the convergence of a series. It’s similar to the non-repeating, unending calculation of decimal places for an irrational number, which we calculate to the point where it matters to us. A value for *y* is determined only by the decision of the world to stop calculating.

That is useful if consciousness is involved in creating the world, or more accurately, in creating our perception of the world. To us, as conscious beings, they are the same.

This is another way into the worldview I have developed in other essays: reality is consensual.

When do we let go of one infinite progression (or regression) and move on to the next? Our mutual consciousness decides this. That is one definition of a moment. Mathematically, a moment is a point in time and has no duration, but in the real world it must. Its duration is small, and we can make it arbitrarily smaller if we want to, but we can never make it zero. We must all agree on duration if we are to share the same world.

Hugh Moffatt

Nashville

2017