Ruminations on Infinity

Note: Haven't touched this page in ages. Don't know when I will again. For now, enjoy my discussions below and feel free to disagree. (One thing I ask -- if you do disagree, please e-mail me and let me know. I'm always open to other opinions and don't believe I'm always right!)

Infinity is a hard concept to grasp in the finite world we live in. Everything around us has a beginning and an end, and even things that seem to last "forever" are still quite finite. Infinity is more of a theoretical concept than anything else, but it is still useful to know about it. These are some short descriptions that try to explain infinity as well as describe some interesting facts about it.

Of course, many of these are mathematical in nature or use analogies from mathematics, because the nature of infinity can best be described this way. However, I hope this will not scare anyone away and is not too complicated for anyone not having an extensive background in mathematics.

Getting a Grasp on Infinity: The Number Line

Numbers are the easiest way to describe such an abstract idea as infinity. And the easiest numbers to start with are the numbers which are most familiar to everyone--the integers.

Integers are whole numbers like 0, 1, 2, 3 ... and -1, -2, -3 ... (not like fractions 1/2 or decimals 1.235). And everyone agrees that these numbers go on forever in both the positive and negative direction.

For example--Think of the BIGGEST number you can think of. And add 1. And add 1 again. And again. ... You get the idea. There is no end. But that isn't the half of it.

Now think about FRACTIONS. Just between 0 and 1 are INFINITELY MANY fractions! If you don't believe me, take ANY number you can think of. Then make a fraction out of it by putting a 1 over it:

If you chose 12, the fraction is 1/12. If you chose one million, the fraction would be one-millionth. (1/one million). And so on.

And of course, there are numbers BETWEEN these fractions. For example, between 1/2 and 1/3 are numbers like .4, .41, .401, .4001, etc. And there are infinitely many of these, too!

The key here is this:
No matter how many times you divide up infinity, each piece still has infinitely many parts!!. That is an awfully hard concept to grasp, but it gives you some idea of the extent of things I'm talking about.

Properties of Infinity

As we have just seen, if infinity is divided up into smaller parts, each part is still infinity. So infinity divided by anything is still infinity.

What if we take things AWAY from infinity? If we have an infinite number of things and we take away a million, you STILL have infinity. The same for a billion or ANY number you can think of. So infinity take away anything is still infinity.

And if you ADD anything to it or MULTIPLY anything times infinity, of course you still have infinity. It can't get any bigger. Even infinity times infinity is still just as big -- infinity.

So basically, no matter WHAT you do to infinity (as far as normal numbers go), you still have infinity. But are there any SPECIAL cases where infinity acts differently?

Here are a few I've come up with:

What is Infinity Times Zero??

After a discussion (quite a few years ago, while I was still in college) I came up with this theory on Infinity times zero, stating that I believed infinity times zero is 1. Recently I received an e-mail from an internet colleague of mine, explaining that my argument is a bit faulty ... I'll include both here, for your reading enjoyment:

I believe Infinity times zero is 1, and there are two possible ways to come up with this solution. Both involve the same idea, just one is more algebraic and one involves calculus.

The Algebraic Solution

Let's start with something that looks TOTALLY unrelated to the idea and see how it's relevant:
x * 1/x = 1.
This is an algebraic identity. For example, 2 * 1/2 = 1, 3 * 1/3 =1, 495 * 1/495 = 1. And so on and so on and so on. This holds true for every number, even a number as large as 10^100 (a 1 with 100 zeroes, called a google by some people). 10^100 * 1/(10^100) = 1.

Note that as x gets larger, 1/x gets smaller. 10^100 is "close" to infinity, and 1/(10^100) is close to ZERO. And as the number gets closer and closer to infinity, one divided by that number gets closer and closer to zero. But still, the identity holds: x * 1/x = 1.

So what about AT infinity? 1/infinity is ZERO. If the identity still holds for EVERY OTHER NUMBER (even those that are "almost infinity"), shouldn't it stay true AT infinity?

So infinity * 1/infinity = 1. So infinity * zero = 1.

The Calculus Solution

In calculus, there is a concept called the "limit". It tells you what happens in equations where you can't just plug in a number. For example, what happens to 1/x as x gets larger and larger? You think 1/x becomes closer and closer to zero. The limit function agrees. It says (in more "formal" lingo):

The limit of 1/x as x approaches infinity is zero.

And similarly, The limit of x as x approaches infinity is infinity.

These are limits taught in first-year calculus classes, if not before.

However, what if we take the limit of x * 1/x as x approaches infinity? You get infinity * zero.

But with limits, you can combine this into x/x, and the limit of x/x as x approaches infinity is 1.

But the limit is ALSO infinity * zero. So by definition, these two MUST be equal, and

infinity * zero = 1.

Again, here's the rebuttal to this argument.

Infinity and the Universe

Some people believe the universe is infinite, but that it had a beginning point, is expanding, and may possibly collapse. How can this be?

The analogy to this is a balloon, and we are on the inside surface of the balloon. We cannot escape into the "empty space" of it, we can only travel around the surface of the balloon. So we could travel straight ahead and never run into the edge of the universe. Of course, we could eventually end up where we started. But you can never go past the edge of the universe. We can continue to travel forever. So this is the sense in which the universe is "infinite". It's curved in on itself, creating a complete circle. And just like a balloon, it can be "inflated" (expanded) or "deflated" (contracted). The points along the surface get further apart or closer together, respectively, but there is still an infinite circuit within this realm.

There are some more properties that I have contemplated and discussed with friends of mine. These theories will soon be added along with an Acknowledgements section giving credit where it is due.

For now, enjoy the rest of my theories and the other pages I've created:

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