most people don’t like math. it’s just one of those things. it seems like doing math requires a certain isolation that most people find unnatural. it’s not just isolation from other people, it’s isolation from the world of ideas. euclid laid down the five axioms of geometry more than 2000 years ago, and those five axioms have served as the laws of the universe in high school geometry ever since. it all feels very restrictive.

this restriction and isolation extends to how we teach the individual subject areas of math in school- geometry for a year, algebra for a couple of years, maybe some trigonometry and calculus. the different areas build on each other to some extent, but since you’ve generally forgotten most of what you learned the previous year, it’s hard to see the ties that bind the different subjects together. and this is the great tragedy of math education, because there’s a secret behind all of this seemingly different math stuff that most people never find out about: it’s all the same thing. it all ties together.

i’m sure that sounds strange, but it’s true, and it’s true in a way that we don’t really fully understand yet. we still only have these glimpses of the connections between the different areas. let me give you my favorite example of this, using three of the weirdest numbers that most people ever encounter.

first, let’s take pi, that number that is kinda sorta approximately 3.14159… most people come across pi when they are learning how to calculate the circumference or the area of a circle, but pi actually shows up everywhere in mathematics, often in really unexpected places. introducing people to pi as the way we calculate the area of a circle is like introducing someone to the adventures of huckleberry finn by showing them how effectively it can be used as a beverage coaster.

next, let’s take i, which is a number that is defined as the square root of negative one. it always feels to me like i is the red headed stepchild of algebra. i have this image of some mathematician going along a couple of hundred years ago, working out the quadratic equation or whatever, and then he comes across an equation like x^2 + 1 = 0, and has one of those oh shit moments. believe it or not, these moments are somewhat common in math. the greeks had this religious cult called the pythagoreans who actually executed this guy who figured out that the square root of 2 was an irrational number. i have to wonder if there was some other guy around that time who wondered about the square root of negative one, but thought of the poor square root of 2 guy and kept it to himself.

the last number in this triumvirate is e, which most people don’t come across until calculus. it’s a magical number, not unlike pi, that shows up all over the place in math. the various ways it’s defined can be found at mathworld, but the way in which it’s usually used is pretty familiar- we usually come across e as a base for an exponent. In the same way that 2 raised to the fifth power is 2 * 2 * 2 * 2 * 2 = 32, we usually end up raising e to some power in order to do…something. sometimes we do it to get a job at google. sometimes we do it to find something like this:

e raised to the (i*pi) = -1.

it’s perfectly understandable if your reaction is sort of like this, that was my reaction the first time I ever saw it.

three of the weirdest numbers in mathematics, each of which arises from a separate subject area (trigonometry, algebra, calculus) combined together in such a way that you end up with one of the simplest numbers around. it’s extraordinarily beautiful and extraordinary obscure, and that’s a shame. it seems like we teach math in such a way that students much suffer for years of seemingly pointless formula manipulation in order to get just a glimpse of the really good stuff behind it all. and we wonder why we have a hard time getting kids interested in math or finding people who want to teach it.

books related to this post: pi in the sky

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