The Justifiability of Irrational Numbers

I have been doing Algebra 1 this year, and I really enjoyed it - until I came to the chapter on irrational numbers. Now don’t be intimidated by my mentioning algebra/math. This is really going to be a very simple post. But I will warn you ahead of time, that you can’t get much out of it by skimming it. Get in, or stay out.

This post was inspired by a discussion I had with a clever young man, at a chess table at the 2009 Home school Book Fair. (Though I think I have had this conversation more than once, with more than one person.) Being multitaskers, we tried to have the discussion, with inserts from several other bystanders, and play chess at the same time. It was quite educational. I think it started with him trying to explain to me why one cannot know both the velocity of an object and the position of that object, and from there we got onto irrational numbers and the Theory of Relativity. (This post only covers irrational numbers.)

Irrational numbers are introduced with the Pythagorean theorem:

In other words: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

This is true for any right triangle. Example:

hypotenuse = c

1 leg = a

2 leg = b

So we translate that as follows: 4² + 3² = ?²

4 squared really means 4 times itself, which comes out to be 4 x 4 = 16

3 squared comes out to be 3 x 3 = 9

16 + 9 = 25

25 now equals the hypotenuse, squared. In order to undo the square, we must find the square root of 25. If you have any algebra, it is not hard to figure out that the square root of 25 = 5. (5 x 5 = 25) All very well and fine. But it doesn’t always work. What if you do this?

Should be pretty easy, right?

1² + 1² = ? ²

1² = 1 (1 x 1 =1) so 1 + 1 = 2²

? ² = 2²

Thus, the square root of two is the length of the hypotenuse.

How in the wide and wild world does one ever find the square root of 2?

It doesn’t seem so complicated at first, unless you have already been through it before, and are already thoroughly aware of the scandalous problem. But if not, it won’t take long to become so. There can only be so many numbers, right? 1x1 = 1 which is too low. 2x2 = 4 which is two high. It must be a fraction/decimal. But you could spend weeks trying to find what fraction works, and never succeed. In other words, the square root of 2 cannot be represented by a whole number or a fraction.

1.4142135623730950488016887242097 etc. is the closest you can get. But it is still too low. What is more, that number does not end with the 7, but goes on into eternity, in an endless string of numbers that NEVER repeats itself. There is no pattern. It is absolutely random.

1.4142135623730950488016887242097 x 1.4142135623730950488016887242097 = 1.9, with a repeating 9 that disappears into eternity. Not quite 2. The square root of two is called an ‘irrational number.’ What’s more, there are hundreds of other ‘irrational numbers,’ and unfortunate high-school students all over the world are expected to put up with the infamous incongruity and use irrational numbers in their calculations like any other number.

But in spite of the best algebra teachers available, the theory still does not make sense. It defies reason. How can a finite length, in this case the hypotenuse of the above triangle, be nonexistent? Because if it is not a whole number or a fraction, how can it exist? How can that simple black line have no definite length? It simply cannot be.

You can take a ruler and measure that mysterious black line. No, you won’t get an exact number, the ruler isn’t precise enough for that, but surely an exact number must exist. To say that it does not defies the imagination. How can that line have NO exact length?

I don’t know quite how it works, but I have an idea. Let’s back up several decades and think of it this way. If you have an object, for the sake of illustration let us say you have a rectangular eraser, than you can split it in half. You can split it in half again, and again and again. Theoretically, you can split it in half eternally. There is always that one piece of matter, and there is always one half of it to split. Once you get to the atomic level things might get more complicated, I don’t know. Leave it at that.

Now, what if you use a number? This might be slightly more difficult to fathom, because a number is not a piece of matter. It is more like an idea. Let’s use the number 12. Split is once and you have 6, split it again and you have 3, split it again and you have 1.5, split it again and you have 0.75, split it again and you have 0.375 and so on. We need not say that, theoretically, you can split a number infinitely, because numbers as we are dealing with them are already theoretical, and you can certainly split them infinitely. You can split the number 12 until your head splits, but I don’t aim to go that far. The point, that I have taken a vastly long time to make, is that a number can be split infinitely. Let’s move on.

This is a very simple number line that I have created with the help of Microsoft Paint, (a certain software that I have a love/hate relationship with.)

Now, stare at that beautiful piece of art and think about this very carefully for a moment.

IF YOU CAN SPLIT A NUMBER INFINITELY, THAN INFINITY EXISTS WITHIN THIS NUMBER LINE.

Did it click? If it didn’t than I’ll spend one short paragraph trying to explain it a little better.

There are an infinite number of points on that number line, because the points can always get smaller and smaller and smaller, (since you can cut them in half eternally.) So even though the number line has a finite length, infinity exists within it – on a very small scale.

If you don’t agree that there is infinity within a line, than you may as well stop reading right here. You can write me a comment, asking me to do a better job of explaining, or telling me I am dead wrong, and I’ll try to help.

If you do agree, than we’ll go on to the final statement of Part One.

Try to relate ‘infinity within a line’ with ‘an infinite number that is not a whole number or a fraction and never reaches the square root of 2.’ If you think very hard for about three minutes, you will probably be able to fathom how an infinite number can exist within a line. How the length of a line can actually be an infinite number. It all hangs on relating the two concepts I just mentioned. If infinity exists within a line, than it might just possibly be able to imagine how a number can be infinite within a line and never quite reach a whole number or a fraction that, when multiplied by itself, equal 2. In a way, the square root of 2 does not exist. Almost, but not quite. There is just no point on that number line that is the square root of 2. To me, at least, that begins to make a shadow of sense.

When I first came to this conclusion, with the help of the Physicist, my first objection, – I like making objections – which may possibly be what is going on in your mind right now, was that the square root of 2 goes on and on forever, but the line doesn’t. There may be infinity within the line, but the line itself isn’t infinite. I have no proof either way, but if you think about it, all of the numbers that come after 1.4 get very small very fast and it is only reasonable that they correspond with the points on the line accordingly. If that didn’t make any sense, and doesn’t help you, than just leave it alone and go on.

This is all conjecture – none of it is proof – but I have not yet even conjectured, much less proven, that irrational numbers are at all justified. All I have done so far is given an argument for the existence of an infinite number within a finite line. Which fact most mathematicians accepted several thousand years ago. It seems quite simple, actually, once you have thought of it, but it was very hard for me to fathom at first. Well, actually, I have done a little more. I have given an argument for the existence of an infinite number within a finite line never reaches anything that is exactly the square root of two. In other words, it never reaches anything that, when squared, that is, multiplied by itself, will equal 2.

By the way, before we go on, let me clear something up. If my repeated reference to a number that ‘never reaches,’ bothers you, let me put that straight. Most numbers don’t reach anywhere, but if you have an infinite number, it is rather difficult to speak of it without using the word ‘reaches.’ Since it is infinite, and goes on and on and on, it is actually, in a way, ‘going’ somewhere. Thus the word ‘reaches.’

But anyway, the conclusion we came to above is not necessarily an irrational number, I don’t think. An irrational number is not only infinite, it also, supposedly, never repeats itself.

Infinity within a line posed a problem to me for a while, but once you get over it, it becomes minuscule in the presence of this colossal problem. Because now that we have got over the argument that it cannot be, we are faced with the much more significant argument that it should not be. The math books try to tell us that that number that never ends and never repeats itself is random - absolutely and eternally random, and for me, that is far more difficult to fathom than infinity within a line. The universe is NOT random!

But this post is already four pages long, and even if there is anyone who has read this far, they will probably not be inclined to read any farther just now, so the rest of this will have to wait for a future post.

One more thing. If you read this post thinking, yes, of course, or I knew that all along, or just what I was going to say, and were sorely disappointed at being unable to participate in a controversy simply because there was nothing to argue about, than here are a few other points that might trigger a cyber debate.

Einstein’s General Theory of Relativity – is there an absolute by which to measure the motion of objects, or can we never discover whether the ball that fell from the train was traveling straight or curved or both or neither?

The Nature of the Universe – Is the Universe finite – or not? Is a finite Universe fathomable? What exactly do we mean when we say ‘universe?’

The Andromeda Galaxy – Can we read history in the light of a galaxy 2.5 million light years away? If that figure is accurate, than what are we really seeing? Can we look back in time?

I don’t necessarily have opinions on any of them, but I would love to have it out with someone and create an opinion on any or all of them.