Shevek (shevek) wrote,

Where do numbers come from?

Most numbers don't really exist. We can't express them. We can only say how to get to them, we don't actually have a way of saying the number itself. For example, "The square root of 2" isn't a number, it's a property of a number. It means "a number which, when multiplied by itself, gives 2". Well, it does exist, if you have enough numbers, and this is how we invent them.

What this doesn't automatically tell us is that
(a) such a number exists
(b) uniquely
(c) we can represent it
(d) we can compute it.

So I'm going to ramble on a crash course about number systems.

The overall pattern is this:
1) Find some numbers.
2) Make an operation on that set of numbers.
3) (optional) Find the inverse of that operation.
4) Apply the operation (or its inverse) to numbers it doesn't "work" on.
5) Define these new numbers to exist, submit paper to best journals.

Actually, step 5 doesn't happen (or at least, not like this) because we've invented all the numbers now. So here are the first few.

The Natural Numbers: I'll start with simple numbers: 1, 2, 3, 4, 5, ... which we get by counting fingers. Let's skip the history, and include 0. These are the natural numbers, written N (or \mathbb{N} in LaTeX). We have a representation for them (a pile of rocks), and we can confidently say that this representation works for arbitrarily large numbers. We get addition, and from addition, we derive subtraction.

The Integers: But subtraction lets us get at numbers we couldn't get at by piling up rocks. We can write "3 - 5" but the number doesn't exist in our current number system. So we do what mathematicians always do: We define a new number system which contains all of these new numbers, and we invent a representation. So we have -2, -3, -4.... and we invent the integers, \mathbb{Z}. The generated structure is an example of a monoid (which will turn up later in Galois theory if you're revising in a hurry and using the abbreviated definitions).

The Rational Numbers: When we add multiple times, we get multiplication, and the inverse of multiplication is division. We can divide 8 by 2 in our number system, but we can't divide 7 by 2. So, again we solve problem (a) by defining a new load of numbers, and we have fractions. Note that decimals cannot represent all rational numbers, since not all rational numbers have a (finite) decimal representation. The generated structure is an example of a field. There are an infinite number of these things, but we can count them using Cantor's diagonal argument.

The Real Numbers: Now we get to numbers which really don't exist, so mathematicians get their revenge by calling them "real". We can write down a rational number which is close to the number we mean. For example, 1 is close to sqrt(2): 1*1 = 1. But 3/2 is closer: 3/2 * 3/2 = 2.25. But 142/100 is even closer: (142/100)*(142/100) = 2.0164. And so on. A Cauchy Sequence is an infinite sequence of numbers like this which gets closer to a value which we can't (necessarily) write down. So as long as we can find a Cauchy sequence of rational numbers which gets forever closer to a number, we say that number exists, and is real. Aside: We now have an example of a continuous space which has the rationals as a dense subset and is therefore a separable space.

A quick aside: Where do we find these sequences? Newton's Method gives a fast approximating sequence for the square root. There are other similar methods.

The Complex Numbers: We have now invented a new operator, square root, which only works with some of our numbers. Just as we did with addition to generate negative numbers, and division to generate the field, we apply square root to things we "shouldn't", let's start with -1. Now we truly have a number which doesn't exist, so we call it i, or j if you're an engineer and have to use i for something more mundane. When the real line goes along, i goes sideways... and I think I'll write more of this another time.

I'll add links to this post on request, but any of the terms in bold should hit some explanation via google.
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