Arithmetic with Straightedge and Compass

One way to look at arithmetic in terms of geometry is to let the lengths of line segements represent numbers. If we have two different line segments (or in our simplification, two different sets of points), we have two lengths, and it is easy to see how to add or subtract the lengths, given only a straightedge and compass (see the arithmetic exercises in Section 5.2).

Addition and subtraction is easy, but a problem occurs when we talk about multiplication--a problem having to do with units. If you have two segments, for example that are one meter long, if you multiply them, you'll get something that is a square meter. How can you compare the area of a square with the length of a line? In fact, you cannot--if you think the answer is 1, since $1\times 1 = 1$, why isn't the answer 10000, since the lines are 100 centimeters long, and $100\times 100 = 10000$? In fact, an equally good case can be made for any such result. Geometrically, you can only compare lengths with lengths, areas with areas, volumes with volumes, et cetera.

A similar problem arises with division, square roots, cube roots, et cetera. There are a couple of ways to get around the problems. The easiest is when something like multiplication or square roots occurs is to give an additional segment whose length is defined to be 1, and to work in terms of that. Another is so make sure that the units are right, so it is possible to construct a segment of length $AB/C$ or $\sqrt{AB}$, if $A$, $B$, and $C$ are lengths of given segments.