Getting Philosophical

 Getting Philosophical

By Bobby Neal Winters

I’ve been getting more philosophical in my old age.  People mean a variety of things when they say that.  Some mean they are getting more stoical, i.e. more willing to take what comes as it is and to endure it.  That may be true of me as well--I’ll think about that--but it’s not what I mean when I say philosophical.

What I mean is that I tend to think more about the true nature of reality.  This, and because I am a mathematician, means that I find myself spending more time thinking about what numbers are and the nature of their existence.

One can ask, do numbers exist?

The quick answer is of course they do.  You could ask a random person on the street this.  They would say sure.  If you asked them to show you three and they had been out buying fruit, they could show you three apples.

The response to this would be, “You haven’t shown me three; you’ve shown me some apples.” You could then take three of their oranges and say, “You’ve shown me three apples.  Here are three oranges.  They are manifestations of “three,” but they are not “three.”  “Three” is an idea.”

If the person you’d stopped on the street were a construction worker, you might find yourself with six pieces of fruit rammed up your backside. (Not the number six itself, which is also an idea, but the actual, physical fruit.)

The fact that numbers are ideas only gets more interesting when we push past whole numbers.

Because of the way we learn math, we’ve been conditioned to think of numbers as decimals.  Numbers are not decimals.  As has been said, numbers are ideas; numbers are words; but numbers are not decimals.  Decimals are ways in which we attempt to write numbers.

The decimal system of writing numbers is a means of doing so which is very convenient when we need to do arithmetic.  We can use them in hand-written work, sure, but they are very handy to just tap into a calculator.  I think that this has contributed to the belief that numbers are decimals.

Here’s the thing: Not every number can be represented exactly as a decimal.  The original example of this was the square root of two.  This was discovered by Pythagoras. He didn’t put it that way because decimals had not been invented yet.  More accurately, we would say that he discovered that the square root of two couldn’t be expressed as a fraction.  This isn’t exactly true either, but I am too far off in the weeds already.

It is not difficult to show this is true, but that isn’t my point today.  Let’s push on.

We refer to numbers like the square root of two as irrational numbers.

There are a lot of other irrational numbers, but the one that most people are the most familiar with is Pi.  Pi is the ratio of the circumference of a circle to its diameter.  Its decimal representation begins as 3.14159... and just goes on forever without going into a cyclic representation like 1/3 as 0.333... or 1/9 as 0.111111... .

Pi is also an example of a transcendental number, but let’s put a pin in that.  We might come back to it if I get people asking for it, but it being irrational is enough for today.

While I could sit down with you at a table at Signet Coffee Roasters with a napkin and a pen and--for the price of a chocolate chip cookie--show you that the square root of two is irrational, to show you pi is irrational is more work.

It can be done, but you have to bring a little more along with you.

Over the past few weeks, I’ve been working with a student on irrational numbers and transcendental numbers.  As a part of this, I’ve been doing some research on them and have found a nice proof that pi is irrational.

This proof is due to a man called Ivan Niven.  Niven was a Canadian-American number theorist.  His proof that pi is irrational is delightfully simple in the fact that it uses no mathematics beyond Calculus II, which is a Freshman course for math majors.

It is a beautiful proof. In its structure and because it uses such simple tools.  In its original form, it fits easily on one page of a journal with some white space to spare. The version I worked with was a little longer than that, but not much.

The notes I’ve made from it to use to explain it to my student go six pages from a yellow pad.  I don’t know what that will type up to, but it’s still not too bad.

To understand this by analogy, it’s like someone has made a chest of drawers with a hand saw, a plane, and a chisel.  All of the joins are perfect.  You can use a hand saw; you can use a plane; you can use a chisel.  Making that perfect chest of drawers is a different thing, though.

As a part of the proof, Niven uses a particular polynomial in a particular way.  At the same time I was thinking of the question on my own, the author reproducing the proof asks, “Where did Niven get that polynomial from?” I was thinking the same thing myself.

It made me feel good to know I wasn’t the only one with the question. But it’s also good to know there are minds sharper than mine.

It makes me philosophical in another way.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.