4.2 The real plane

When we were building up the structure of the integers, we showed the traditional number line R².

with the negative integers to the left of 0 and the positive ones to the right. Really, though, this was just part of the real number line:

The line is one-dimensional in that we need only one value, or coordinate, to locate a point uniquely on the line. For a real number x, we represent the point on the line by (x). For example, the point (–2.6) is between the markings –3 and –2. We use or omit the parentheses when it is clear from the context whether we are referring to the point or the number that gives its relative position from 0.

I drop the decimal points on the labels now that it is clear we have real numbers.

4.2.1 Moving to two dimensions

Suppose the number line sits in two dimensions so that we extend upwards and downwards:

Since we...