Intro

It took me a while to come up with an intuitive explanation for infinity. It is a difficult concept to grasp fully, yet it is crucial when learning calculus. Integrals have infinite elements; each element is infinitesimal. What does that mean?

My Intuition

About a week ago, I thought of a way to look at infinity. Infinity is a number that is outside our range of detection. Measuring instruments must have some upper bound, and infinity is the number that exceeds that upper bound. In essence, any number that exceeds the range of our measuring instruments is infinity for all practical purposes. No matter what number we can come up with on our instrument, infinity is larger. To make this all clear, let's take an example:

Suppose we have an odometer that can go up to 999,999 miles. Suppose that the odometer can't overrun, and stays at 999,999 (all we can do is reset it). If we start at zero and drive 1,000,000 miles, the reading is 999,999 - the largest number possible, but it still doesn't capture the distance of a million miles. In this case, 1,000,000 miles is essentially "infinity". We can't measure it with our instruments.

Similarly, I have an intuition for an infinitesimal number. It is any number small enough (in magnitude or absolute value) that we cannot distinguish it from zero with our instruments. However, it cannot be zero. Using our odometer analogy, suppose we drive half a mile. Our odometer still shows 0, because we have not yet crossed the 1 mile mark. However, we have driven a distance - an infinitesimal distance :). [Note: suppose the odometer doesn't turn as you move along. At the one-mile mark it clicks to 1, at 2 miles it clicks to 2, etc.]

Another interpretation is that of a digital scale. Some weights are so small that they do not register as any weight. Some weights are so large that they max our the scale, perhaps filling up all digits. Either way, both weights cannot be represented by the scale, so they are effectively "infinitesimal" and "infinite", respectively.

Is this useful?

Is this interpretation useful? I think it is. If a number is so large that we have no way of measuring its magntiude, it might as well be infinity. If a number is non-zero, yet we cannot distinguish it from zero, it might as well be infinitesimal. How would we detect that a supposedly infinitesimal number is non-zero? We multiply it by an infinite number :).

In our example, we can choose "infinity" to be 2 million, and "infinitesimal" to be 1/4. Our odometer cannot detect either number. However, if we drive 1/4 of a mile 2 million times, we end up with 500,000 miles, a number which we can detect. In a similar manner, an "infinite" amount of "infinitesimal" things can be a finite, detectable number. In reality, the numbers aren't truly infinitesimal or infinite, just beyond our range of detection.

Note: What if we multiplied 1/4 by 20, a finite, detectable number? We would get 5, yet we didn't have to multiply an infinitesimal by a infinite number. We could then deduce that the original infinitesimal was (5/20). The number is still undetectable, but it loses some of its mystery. Read on.

Other ideas

I'm going to take this concept a little further. Those who have studied infinity formally know there are ways to represent the different "sizes" of infinity. For example, there are an infinite number of natural numbers (0, 1, 2, 3...). However, it seems that there is an even greater number of reals. We need some way to show that one infinite number is larger than another. In the world of math, this is done with omegas and alephs but we won't get into it.

Using our intuition, we see that it is simply that one unmeasurable number is greater than the other. To us, both are "infinite", but we may be able to detect some relation between them (such as an inequality).

In the previous section, 1/4 was simply one of the infinitesimals, and a rather large on at that. In the odometer world, the concept of a fraction probably doesn't exist -- in that case, 5/20 is like a representation of infinity (like the infinity symbol is to us, or omega, or aleph-naught). 5/20 wouldn't be a meaningful number to anyone in the odometer world. However, it is meaningful to *us*. Want to learn more?

The heart of the matter

It's nice to consider infinity in a world where we count in integer amounts using limited odometers. However, we count in real numbers, without a limit -- right?

Well, there really *is* a limit, even conceptually, to how large our numbers can get. Suppose that we think of the most compact, efficient way to represent large numbers -- scientific notation will do. If we write 10^10^10^10^10^10... on a piece of paper, we get a gigantic number. Still, it isn't the biggest. We could have continued to a second piece of paper, or a third, or a millionth, etc. Eventually we use up all the resources (ink, space and time) in the univerise. However large the number we get, there is still a larger one out there. Remember the old "Think of the largest number you can. Add 1" trick? Well, eventualy we reach a limit. We can no longer write down the number (run out of matter in the universe to write it on), or it may take an "infinite" number of years to write.

My point is that infinity is a number so large that we cannot even conceptualize it. Our brains are the odometers, and infinity is out of its range. The same goes for an infinitesimal number: we cannot distinguish it from zero. If we could, it wouldn't be infinitesimal, it would be a number we know (it would simply be 0 + the difference). This is the hard pill to swallow, but I think that is the essence of infinity. If we could ever explicitly represent an "infinite" number, it would no longer be infinite. Thus, we cannot work with infinite numbers except to combine them in some way with infinitesimals.

In the odometer world, 5/20 can be infinitesimal because the representation is meaningless to them. That doesn't work for us, but the idea is the same. A unrepresentably small number can be infinitesimal (and similarly for infinty).

Applications

The most obvious application, and perhaps most useful, is a good understanding of calculus. I am reinforcing my own intuition behind dx, derivatives, and integrals which are combinations of infinity and infinitesimals. In fact, one modern way to solve calculus proofs using limits (delta-epsilon) involves stating that a certain things will always be out of our detection, no matter how small we make our changes. I will post articles about these topics specificially. I hope that this interpretation, while not a rigorous proof, satisfies you. I never was into omegas or alephs anyway.

Summary

The big picture: infinity is a number we cannot represent or work with. An infinitesimal is a number so small that we cannot distinguish it from zero. However, combining infinite numbers with infinitesimals can yield real, detectable numbers.

Links

This stuff is very deep. Any helpful links I find will be put up here.