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.