Ruminations on Infinity

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??

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.

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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