Tuesday, May 13, 2014

Roots

Most people know that there are two square roots for every number, but for some reason we insert a disconnect and start lying to students that there is only 1 root for roots with an odd index and 2 for roots with an even index. Why is that? Isn't the truth better? Doesn't it make sense that there are n nth roots for every number? Isn't that elegant? Isn't that beautiful?

Oh, it's harder and we like our students to be idiots. I keep forgetting that.

You're right, it's much better to have people learn mathematics as a series of abstract rules that make no sense because understanding was never valued in American mathematics classrooms. Let's approach mathematics the same way as we do the Ten Commandments (#HandedDownFromOnHigh).

So, every number can be visualized as a circle. Specifically, let's imagine the number 1 as the unit circle, the circle with radius 1 centered on the origin.  Then the second root (or square root) of 1 are the two numbers located on the circle, such that when we square them, we get back to one.  In this case, it turns out that -1 and 1 are the second roots of 1, and these clearly split the circle in half.

For third roots, these are the three locations on the circle that when cubed, get back to one.  We can easily visualize this by rotations on the circle.  If you travel 1/3 of a revolution from 1, you get to the first 3rd root of 1, another 1/3 gets to the second 3rd root of 1, and a final 3rd gets us back to 1, the third 3rd root of 1.

For fourth roots, we rotate 1/4 of a revolution, fifth roots, 1/5 of a revolution, and so on. This idea that we can visualize roots as rotations is beautiful in and of itself, but it gets even better.

In order to conceptualize this two-dimensional location as a single number, we need to append an entirely different dimension to our original numbers. We can do this by attaching an arbitrary direction variable, say i. If we let i be the variable that represents traveling 1 unit "up", then -i would be 1 unit down, 5i would be 5 units up, and we can immediately start talking about a two dimensional location with a single "number", say 1+3i, which would represent traveling 1 unit right, followed by 3 units up.

According to this then, the fourth roots of i would be i after one-quarter of a revolution, -1, after another one-quarter, -i, after a third one-quarter revolution, and finally back to 1 after four one-quarter revolutions.

If we wanted to find the values of the third roots of 1, we could turn to trigonometry to find values of the unit circle at any given angle. What? We're incorporating trigonometry into our geometrical discussion of the algebraic principle of roots? Oh my God, it's almost like mathematics is completely consistent and true, and yes, again, beautiful and elegant. Trig tells us that a 1/3 rotation would result in traveling 1/2 of a unit to the left, and \sqrt(3 / 2 units up (where "\sqrt(3)" means "the square root of 3").  Then the first third root of 1 could be expressed by the single number -1/2 + \sqrt(3) / 2 i.

You could plug this into any calculator that understands complex numbers (which is what these numbers I've sneakily described with vectors actually are), cube it, and the calculator would tell you that the result of your cubing is the number 1.