Last December, we took the time to really analyze infinity, and we ran into a little problem. We didn’t bring it up too much, but all of the infinite series were the same size! To be more specific, they were all equal to alef naught, we called it. Even the number of fractions couldn’t break this barrier. But we haven’t exhausted our toolbox yet!

Let’s look at the set of real numbers between zero and one. Say you had a list and said that it contained all of the real numbers on it. I’ll just put a few down here:

1) .6283185...

2) .2718281...

3) .1618034...

4) .3141592...

5) .1428571...

6) .4000000

7) .3333333...

I claim that there is a number that is not on your list. Of course, you can notice many off the top of your head, like .666666, .1, .2, ,3, and so on, but there is a systematic way to generate a new number. Take that first number, .6283185, the number tau over ten. Let’s change the first digit of the first number to something different. I’ll just keep it simple by adding one.

8. = .7...

Now, look at the second number on the list, .2718281. This happens to be the number e over ten, but that’s a complete coincidence. Let’s change the second digit of the second number by adding one.

8. = .78...

Let’s continue this process. Take the golden ratio over ten, .1618034. Add one to the one and get two. Keep going until you have gone through the full list.

8. = .7822614...

And we have a brand new number. Could it have been on the list? No. Why not? Well, let’s prove it by contradiction (actually, it is kind of a proof by logic if that actually exists, but whatever). Say it were the thousandth number on the list. Well, the thousandth digits of these two numbers must be different since that is how the process works, so they cannot be equal. Therefore, you cannot pair up the natural numbers (1, 2, 3, 4...) and the real numbers between 0 and 1.

In our other infinity post, or our alef naught post, I could say, we saw that you must be able to pair each number with a natural number in order for the size of the set to be equal to alef naught. You can’t do that with this list. Therefore, it is bigger than alef naught, and is its own number. We denote this number with the letter C, for continuum

What about the numbers between zero and two? That ought to be double of zero and one. Well, prove it. Look at the graph of y = 2x.

<graph>

Now, draw a laser beam from the origin up to the two line. This beam hits a point on the one line and then a point on the two line. Pair those two real numbers up. Now, draw another beam. Pair these two numbers up. Keep doing this, and you will find that no matter how many beams you do, there is always a number on the one line to match up with the two line.

With alef naught, we tried to pair everything up with the natural numbers. With C, we try to pair everything up with a real number between zero and one. Here, we have paired up the real numbers between zero and two with these numbers, so this set is size C.

What about all of the real numbers? That should do the trick! Well, same thing, graph any function that has a domain (x-coordinates) between zero and one and a range (y-coordinates) that goes from positive infinity to negative infinity, then you’ve paired up the real numbers between zero and one with all of the real numbers.

If that is equal to C, then what is more than C? Well, how about the points in the unit square; the square that has points (0,0), (0,1), (1,1), and (1,0). That must be more than C. But we actually can pair it up. Let’s say your points was the following:

x = 0.acegikmo...

y = 0.bdfhjlnpr...

Pretend that those letters are digits. Say they were the following:

x = 0.123456789

y = 0.987654321

We can pair this with a real number between zero and one, namely:

0.abcdefghijklmnop...

So, the example we had would pair up with this number:

0.192837465564738291

Basically, these two quantities are equal! We could go on forever, or to infinity I should say, with different quantities and test them to be equal to alef naught or C. There are some quantities that actually are bigger than C, but I won’t go into that now.

Infinity is such a cool concept to wrap your mind around, since you can’t define it or explain it, but you can actually start to grasp it after looking at some of these things. I think it even gives you a better handle on numbers, and understanding how many there really are. Considering that you learn about infinity (without actually saying you are) in first grade, it must be pretty cool!