Unit 1 · Lesson 1.2

The number system & the number line

When you solve 2x = 7 later on, the answer is 3.5. That might look like an odd answer, but 3.5 is a real, ordinary number. To see where it fits, start with one picture that holds every number you'll meet: the number line.

Draw a straight line and mark 0 in the middle. The counting numbers (1, 2, 3, and so on, the ones you'd use to count apples) step off to the right. Their negatives, the mirror image on the other side of 0, step off to the left. Every number you'll meet lives somewhere on this one line 1.2.f1.

The line is not only the evenly spaced ticks. Look between two ticks, say between 0 and 1, and there are more numbers sitting in the gap: 3/4 lands there, and so does 0.5. The in-between points are numbers too, and every gap is packed with them, no matter how far you zoom in.

Now let's name the points on the line, one group at a time.

Start with the counting numbers: 1, 2, 3, and so on.

Add 0 to that group and you have the whole numbers.

Now add the negatives too. That gives you the integers: …, −2, −1, 0, 1, 2, …, none of them split into parts.

Then there are the in-between points, the ones you can write as one whole number over another, like 3/4. These are the rational numbers. The integers belong here as well, since a whole number like 5 is just 5/1.

Each group grows out of the one before it, all on the same line.

Most points on that line are rational, but not all of them.

Some numbers have decimals that run on forever. That alone doesn't make them special: 0.333… runs on forever, yet it just repeats 3 over and over, and it's really the fraction 1/3. A few numbers, though, run on forever and never fall into a repeating pattern, so no fraction ever lands exactly on them.

The most famous is π (say "pie"), written π = 3.14159…, the number that turns up whenever you measure circles; its decimal never ends and never repeats. Another is written √2 = 1.41421…. That √ is the square-root sign, and √2 just means the number that, times itself, gives 2; you won't compute it here, so read √2 as the name of one point on the line.

Numbers like these are the irrational numbers.

Put the rationals and the irrationals together and you have the real numbers. That's every single point on the line, with no gaps left over. So the "whole line" the picture promised finally has a name, and it's the ground algebra is built on.

You've now met every kind of number on the line, from the counting numbers all the way out to the reals. Here's one last point, and it's a simple one once you see it. A fraction bar or a decimal point is just how a number is spelled, not what it is. The number 5 can be written 5, or 5/1, or 5.0: three spellings that all land on one point on the line. So you can't tell a number's type from how it's written. Work out its value first, then classify it.