The Real Line

All the numbers we are used to (and some numbers that we just don't care about too much) are really quite interesting. They have heaps of interesting properties that mathematicians like to talk about. And when they want to talk about all the numbers, they usually start by saying Consider the real line... . The real line is a representation of all those numbers, all placed on a straight line. (The reason why they're described as real is because there's another bunch of numbers called imaginary numbers, and they have even more interesting properties. But I'll talk about them another time.)

So here's a picture of the real line.

I added a 3 just because it's cute, and also because 3 is a number, and numbers are on the real line. There's a little vertical line below the 3 to show us which point on the line is for the 3. Nice.

The is there to remind us it's the real line. Though lord knows why. It's pretty bloody obvious, right? However, I'm not one to rock the establishment, so there it is, and all. (if we wanted to talk about imaginary numbers, we'd draw the line vertically and put an on it. Not so strange, eh?)

It's a representation of the real thing, of course, as the real (or actual) real line goes off to infinity to the right. And to the left, it goes past 0 down to negative infinity. Like this.

Ok, so I just added the symbols for 0 and plus and minus infinity. But they remind us that the real line goes on forever, in both directions.

What's interesting is that I've left the 3 there to show that all the number are represented on the real line. Here the number 3 is just a point, but what's also there is the number 3.5 (which is to the right of 3), the number 2.5, the number 3.14, etc.

In fact, if you think about it a little bit more, there's an infinite number of numbers just around 3. There's 2.5, 2.6, 2.7, 2.75, 2.9, 2.99, 2.9999, 2.99999999, etc. They're all there.

So, what if we're talking about all the numbers less than 3? Well, I'm going to colour them in with red, so that the line would look like.

That's it, there are all the points less than 3. In the bit that's red. Notice how I haven't included the little vertical line for the 3? It's not red because 3 is not one of the numbers less than 3. So I didn't colour it red.

Now what if the mathematician doesn't have a red pen? Well, they'd mark it out using little curvy lines. Like this.

Since we don't include 3, we use a curvy line, kinda like a cap, that includes everything to the left. Hence all the numbers less than 3.

If we wanted to include the number 3 (if we were considering, say, all the number less than or equal to the number 3), we'd use some square brackets, like this.


Now we've got a handle on the square and curvy brackets on the real line, let's do something neat with them. Let's try and stake out all the numbers that are less than 3, but also greater than or equal to 1? How would you draw this? Yup, that's right, like this.


Finally, let's draw on the ideas I discussed in the last post on sets. How could we describe all the numbers less than 3 using sets? Easy. We'd say this.



Now A is represented by the red bit above four pictures ago. If we wanted to talk about all of the numbers greater than or equal to 1, but less than 3, as in the last picture, we'd simply refer to them like this.



Notice how I gave it a new label B?

Clever.

Sets

A set is a collection of things. That's right, it's that simple. What those things are, depends on what you're talking about. For example, you can have a set of fractions, or a set of numbers, or a set of integrals, or a set of whatever (Sometimes mathematicians try to be cool and talk about a set of unicorns, or pink elephants, but that's just stupid.)

Sets are denoted by a set of curly brackets, like this: { }. And you put the things in your set inside the brackets, so that a set of two fractions would look like this.



Sometimes you might want to give it a name, (but not an imaginative one, because we're mathematicians) like A, so that you would write



That's all fine and dandy if you can write down the list of things you want to put in you set. But what if you want to talk about a whole bunch of things, where's there's a lot of them. Say an infinite number of them. Like, all the fractions between 1 and 3. Obviously there 3/2, and 4/5, and 8/3. But there's a whole lot of them. How can we describe such a set?

Well, we write down inside the curly brackets the rules for what's in the set and what's not. (This is excitingly close to the important concept of the Axiom of Choice (what a great name; "The Axiom Of Choice"), but I'm not going to talk about it, because I'm not an expert, and it deserves it's own blog, as it was the sort of idea and revelation that shook the very foundations of mathematics. (If we really want to know, it's about whether you can even talk about an infinite number of objects or not. Or something like that))

So, getting back to it, we write down a rule for what's in the set, and hope that someone else knows what we're talking about it. For example, we might want to talk about all of the numbers between 1 and 3 (There should be an infinite number of them, so we'd better make a rule!). We would write this like



One would read this out loud (if one was a mathematician) as

A is the set of all things which are bigger than 1 and less than 3.

The colon is useful there when you want to be more specific about what sort of objects you're applying the rule to. For example, if you wanted to talk about all the fractions between 1 and 3, where is the set of all fractions, you'd write



where, as you can probably guess, the symbol means `is one of', or in maths speak, `is in'. In other words

A is the set of all numbers that are fractions () and are bigger than 1 and less than 3 ( ).

More to come.

I never saw `Puppetry of the Penis', and heard some good things. Here's the next wave. Not for the faint of heart. ;)

There's going to be a maths blog later this afternoon.

some maths terminology

So, as I head off to sleep for the first time in this new year, I leave you with these thoughts.

"There are things. Things do stuff, and stuff can be done to things.

If I become more specific about these `things' and `stuff', do I sacrifice generality or introduce structure?"

Here endth the stuff.