Lies, Damn Lies

Maths part 2- a guide to numbers

It is valentines day today, so I thought I'd talk about numbers.

See, there are various types of numbers. The first set we are introduced to, in our infancy are the natural numbers, or the counting numbers. The ones that go 1,2,3,4,..... Basic stuff, you might think, although if you go to the wikipedia article about the natural numbers you will notice sometimes quite heated debates about whether 0 should be a member of the natural numbers. It shouldn't be, and anyone who feels differently is unfortunately a fool. They are denoted with a blackboard bold N, if you care about notation.

Of course, these aren't enough numbers, and the integers are pretty important to- these contain the natural numbers and 0, and all the negative whole numbers. A pretty simple extension, and not a terribly interesting one either. They're denoted Z.

From here we go to the rational numbers, which are basically all the fractions- 1/2,6789/956676 etc, etc. Notice that the integers are contained in this set. Essentially, any number which can be expressed as an integer above a natural number is considered to be a rational number. And, actually, it seems like this should probably be enough- these are the basic numbers you can concieve just thinking about it on a common sense of level. Indeed, many greeks thought that these were the only numbers.

Sadly, it's a bit more complicated than that. Now, as I'm sure you all know, pythagoras has a neat law describing the lengths of right angled triangles- a^2=b^2+c^2, where a is the length of the longest side. Well suppose the length of b and c were both 1 (1 of what is a reasonable question, although it does not really matter- say centimetres if it makes you happier). Then the length of a is the square root of 2.

Well, fair enough, so by our reasoning from before, as it is certainly true that the square root of 2 exists, so it must be expressable as a fraction as above, eh? Well.... no. Euclid proved that the root of 2 was irrational. In fact, you can see the proof here. So we now have numbers that can be written down as decimal sequences that will never terminate or repeat periodically, and certainly cannot be expressed as a fraction.

This is pretty odd, but if we accept the basic assumptions made by mathematicians, all this follows. Still, at least in the case of these weird irrational numbers they will be the solution of some kind of equation- in this case x^2=2 (incidentally, the solution of an equation is a value for the letter which will make is true- so in this case x equal to root 2 is a solution. As is -root2. Incidentally). Except.... some of them don't.

See, if a number satisfies those nice equations with x to the power of something, we call it algebraic. But there are actually some numbers that don't even do that. Pi is the most famous number- an example of the transcendental numbers.

One might ask how one generates pi. Excellent question. Well if you can find the area of a circle accurately, and divide by it's radius squared, you will get pi. Or if you sum 4-4/3+4/5-4/7+4/9-......... you will also get it. Of course, you can't actually express pi, but it certainly has physical reality, in that it turns up in circles all the time. Actually pi turns up literally everywhere in mathematics, as I may or may not explain to you.

So there we have it. The natural numbers, the integers, the rational numbers, the algebraic numbers and the transcendental numbers. Put them all together and you get the real numbers, a nice smooth number line. Phew. We're surely done now?

Well.... no..... Unfortunately someone got curious about the root of -1.

posted by Mr K @ 2:55 pm