Algebra with compass and straightedge

Thought: could you use a compass and a straightedge to do basic math?

Let's see. First we need to define a couple of things. Like zero. Let's use a zero-length point as the zero. Draw a long straight line, put a mark on it somewhere in the middle. There's your zero and line of numbers.

How to represent numbers? Line segments, right? Take the ruler and draw a few short lines. You can move the short lines to your line of numbers using the compass. Measure line with the compass, move the compass to the line, make a mark with the other end, there you go. Our numbers are lines that are on the number line.

With that, we can add positive numbers. Suppose you have three lines: A, B and C. To add them together, you'd first copy A to the number line, then B starting at one end of A and C at one end of B. The resulting long line is the three lines added together.

To get a proper group going, we're going to need a way to negate lines. So, let's add a direction to the lines. Put a short line A on the number line with one end at zero. Jot a small arrowhead at the other end. Now you can negate A by flipping it around so that the arrow points in the opposite direction. A + -A would go from zero to A, then all the way back to zero. To draw A and -A, first move A to start at zero, then rotate the compass 180 degrees and make a mark on the other side of zero. That's -A.

To add these arrows together, we need to add a rule. You need to attach the non-arrow start of a line to the arrowhead of the previous line. And the result of addition can be represented as a single arrow, going from the start of the first line to be added with the arrowhead at the arrowhead of the last line.

Ok, we have numbers and their negations. We can add lines and subtract lines. And they seem to obey associativity A + (B + C) = (A + B) + C and commutativity A + B = B + A. The addition operator requires us to define A as any arrow pointing to the same direction as A and that has the same length as A. Other group axioms, A + 0 = A seems to be ok, ditto for A + -A = 0.

Right, I think we have an additive group. How about multiplication?

You can multiply two of these vectors together by doing the addition but then rotating the second arrow by 90 degrees. Then add the first vector after the second, rotate it, and repeat once more for the second vector. Now you have a rectangle. The area of the rectangle is the result of the multiplication.

Now wait a minute, how do you convert that into a vector on the number line? It's an area, a completely new dimension, totally different unit from the number line segments. Right. Well, maybe this is not the kind of multiplication we really want. Let's try to find something else.

Ok, first, define one. It's some vector. Probably pretty short. Or maybe long. It's arbitrary. So we have our one.

Let's draw a multiplication line. It's just a long straight line. Jot a zero mark on there, copy the one-vector on the line. Now draw another one-vector at the end of the previous one, but rotated 90 degrees. Draw a dotted line from the zero mark to the end of the second one. There we go, 1 multiplied by 1. Is 1.

Now take some other lines, say 4 and 5. The 4-line is 4 times longer than the one-line. And the 5-line 5 times. Multiply 4 by 5. Draw the 4-line in the place of the second one-line. So that it's standing up, one unit away from zero. Then draw a line from zero through the end of the 4-line. Right, that's the line that multiplies numbers by 4. When it's one distant from zero, it's 4 units up. When the distance is two, it's 8 units up. And so on.

Now copy the 5-line on the multiplication line. And draw a normal to the multiplication line through the end of the 5-line. This is the line that multiplies numbers by 5. Where the 4-multiplier line and the 5-multiplier line cross, there's the result. The normal vector from the multiplication line to the result point is the number 4 times 5. To convert it into a number line number, rotate it 90 degrees so that it's flat on the number line.

Division is, well, there's a trick to it. You want to create the number 1/A. So let's do the multiplication line thing again. This time, draw an upright one-vector A units away from zero. Draw a line from zero through the end of that vector. Then draw a normal line one unit away from zero. The crossing point of those two lines is 1/A.