Saturday, December 17, 2011

To infinity and beyond!! With numbers I mean...

We have done a lot with infinite series already, we used it to supposedly prove the Communitive Property wrong, to practice systems of equations in the problem of the week, and even prove that the prime numbers fall into this category. However, we haven't really looked at infinity. Is it a number? Can we define it? What is greater than infinity?

Let's look at an infinite series. Take the natural numbers; 1, 2, 3, 4, 5, 6, 7, ... This sequence takes us out to infinity. How about 10, 20, 30, 40, 50, 60, 70, ... This takes us to infinity also. However, which sequence has more terms in it?

One side of you is saying, for every term in the multiple of ten series, you need ten terms from the natural number series to get that high. Therefore, the first series must be ten times bigger. The other side of you is saying, for both series, we are going out forever. This means they must be equal. When I first saw this, I was leaning greatly towards the first side. However, this isn't quite right.

For each number in the first series, we can pair it with a number in the second. For example, we can pair 1 with 10. Then 2 with 20. Then 3 with 30, 4 with 40, 5 with 50, and so on. If the first side is greater, then you must run out of terms on the second side. However, the second side is infinite also. If the second side were greater (which I cannot make an argument for that), then we would run out of terms on the first side. This means that the sides must be equal.

In fact, any series is the same size as the set of natural numbers if you can write it out with no infinite gaps in between. For instance, the integers:

... -3, -2, -1, 0, 1, 2, 3 ...

You can rewrite this to get:

0, 1, -1, 2, -2, 3, -3, 4, -4...

Since there are no infinite gaps here, that must mean that it is equal to the set of natural numbers.

What about the fractions (you could say the rational numbers, but rational numbers are simplified and fractions can or cannot be)? It's a big statement, but we can try it.


This is a table of the fractions. Is there a way to write this table without any infinite gaps? Turns out there is. If we draw lines diagonally, and put it all together, it will have no infinite gaps. It would look like this:

1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, ...

This means the amount of fractions is the same! This quantity is denoted with the hebrew letter alef, also called alef zero, or alef naught. However, do you think there is another type of infinity, or is every set equal? I will come back to this in a couple months, and we will find the answer and why.

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