Where arrows are rotated

Arrows correspond to numbers; positive numbers point right, negative point left, and they rotate each other when you multiply them. When we multiply two numbers, we add their angles and scale their magnitudes (arrow lengths).

How do we get other arrows, with angles other than 0° and 180°? First, nothing prevents us from constructing one directly; here's one: 4(45°). The picture of it is below:



This is an arrow of length 4. If we multiply this arrow by itself, we will get 16(90°). If we multiply that by itself again, we get 256(180°), which is -256.

So we found a number, an arrow, which, when multiplied by itself four times, gives -256. That's pretty amazing, because normally, when we multiply something by itself, we get a positive number, but here we multiplied something by itself twice, and we still got a negative number. If we multiply -256 by itself, we'll get 65536, and by then the interesting part is over -- all the numbers will be positive from then on.

A side on terminology: just like repeated addition has a word (multiplication), repeated multiplication is called exponentiation. Also, the way to say "X multiplied by itself four times" is "X to the power of 4".

What are all the arrows that, to the power of 4, give 256(180°)? One is 4(45°), that much we already know.

But there are others, in fact a total of four. That's because angles wrap around at 360. If we start, say, with an arrow whose angle is 135 degrees, after the first multiplication the angle becomes 270 = 135 + 135, then 405 = 135 + 135 + 135, then 540 = 135 + 135 + 135 + 135. But 540 is the same as 180, so after four rotations we arrive at the same result: 256(180°), but from a different starting point. Here is how you can find all four starting angles:

180 / 4 = 45
(180 + 360) / 4 = 135
(180 + 360 + 360) / 4 = 225
(180 + 360 + 360 + 360) / 4 = 315

Visually, they look like this:



So we see that arrows are more than "numbers"; they are more interesting, in fact. When you compute with arrows, you'll end up with a regular-looking number every once in a while, but only if the angle is 0 or 180. Arrows are to regular numbers are what a color monitor is to a black-and-white one: numbers only have two directions (black and white, left and right, negative and positive), but arrows have all the shades of gray in between.