Does infinity have any use?

Have you experienced infinity? Even in life, I believe you have. But, let's get to that after we experience the concept of infinity and make use of it in the elegant and abstract world of math, once again. Later, we will extrapolate to life. If you missed it, you can also read our first two encounters with infinity here (there are obviously infinitely many numbers, right?) and here (maybe not so obviously, there are infinitely many prime numbers!).

Maybe you are familiar with calculus and already know that calculus is built on top of what we call "infinite series." If not, read on. They are simpler and more interesting than you might think. 

Here is the challenge [Thanks go to Strogatz author of a very fine book on calculus: Infinite Powers]. Someone delivers a pie which has a square shape! You have three hungry kids and you need to divide the pie into three equal parts. How would you do it? 

1. Let's start with cutting it into two equal halves with a vertical cut.

2. Now, we can give it a horizontal cut and get four equal slices. 

3. Each slice is 1/4 of the original pie. We got four slices. Give one to each child. 

But, we don't want to waste the remaining slice. What to do?

Well, to use a wonderful concept from computer science: just recurse! Meaning, repeat the steps 1, 2 and 3 on the slice left over. We get four smaller slices. Give one to each child. Now, let's see what does a child have:

1/4 + 1/4/4. What fraction of the pie did each child get? I don't know about you, but, I dislike doing long division. So, instead, let's repeat the process:

1/4 + 1/4/4 + 1/4/4/4 + ...

By the way, mathematicians have a short cut for such long sums:

Sigma(n from 1 to N) = 1/ 4^n

This is an infinite sum with infinite powers... You might imagine that it would take an infinitely long time to the division in each term and then some them all up. That's reasonable. But, the powers of math, a big piece of the power of the mind, will surprise us and we'll find a short cut, right here, right now. Right below!

Forget about this infinite series for a moment. Instead, reconsider what we are doing: each of the three children are getting many slices, but, please notice that:

1. The successive slices are much smaller as we repeat the cutting and slicing.

2. Every child gets the same number of slices (infinitely many!).

3. Every slice that each child gets is identical with the slices other two get.

I hope they are all clear. Please check to make sure you find them to be true. Please convince yourselves. I can't really do that for you. I've already said too much :-)

Now, we can take them to the next obvious step: each child gets one third of the pie!

But, you might object, that would take a long, very long wait, in fact as long as forever! Yes and no. Remember the first point: the size of the slices we get shrinks very quickly. 1 in four. 1 in sixteen. 1 in sixty four. 1 in two hundred sixty four... Even when we divide into two many times, say 10 times, we get to less than 1 in a thousand (1/1024 to be exact.) 

So, let's go back and recap. We found out that by dividing our pie into four infinitely many times, we actually divided it into three! The sum of the infinite series is 1/3. 

I myself find this to be a beautiful find. There is beauty and truth in numbers and how they relate. Peaceful. Eternal. Orderly. Kind. 

Hope you enjoyed it. But, if you think this is too impersonal, give the personal version a read here: Experience infinity for yourself!