XX. The Pythagorean Theorem

Most people think computers area a pretty good idea. The Pythagorean Theorem was a bigger idea!! It's the basis for most of what we do with mathematics. It allows us to calculate the distance between two points.

WARNING - I talked about Pythagoras at an teaching job interview once. At the end of the interview the principal pulled me aside and told me they were looking for a math teacher, not a historian. The rest of this small section is going to be about the Pythagorean's and not the Theory - if this is offensive, skip it and move on to the next paragraph. (bitter - I'm not bitter)

We don't know too much about Pythagoras, in fact - we don't even know if he existed. We do not there was a secret society named the Pythagoreans who discovered the theory. And then kept it a secret for almost a century. They were upset about the hypotenuses of an isosceles triangle having the square root of two in it. One disciple pointed out that there was no way around it. They killed him.

They were vegetarians who would not eat beans. They were anti-beans because they believed that every time you passed gas - you lost a bit of your soul. (I know what you're thinking - no, I didn't include this part in the interview, it is possible that the preceding paragraph about them drowning the disciple was a bit off topic for an interview. But hey, I think it adds color.

And on to Pythagorean Theorem

Right triangle

The above illustration shows a right triangle (meaning that two of the sides join at 90 degrees. The sides are labeled a, b, and c.

The Pythagorean's found that the square of side a plus the square of side b equals the square of side c. The BBC have a great demonstration of this theory at BBC2. It's only a minute and a half - really worth clicking on the link. No advertisement!

If you plot any two points on a graph, one at (v,w) and the other at (x,y) then a would be (v-x) and b would be (w-y), the distance would be the square root of the square of a plus the square of b using the Pythagorean Theorem.

Why were the Pythagorean's upset with this?

These were religious fanatics. They believed mathematics was the language of God (actually, so do I). If you have a right triangle (90 degree triangle), with 2 equal sides (isosceles triangle), then you get a bothersome answer for the hypotenuse (the side opposite the right angle).

Let's say the two equal sides have a length of 1. So, 1 squared plus 1 squared equals the third side (the hypotenuse) squared. 1 squared plus 1 squared equals 2, so 2 equals the hypotenuse squared. To find the hypotenuse I take the square root of both sides and get the square root of 2. The square root of 2 is 1.41421356237.... as far as we can tell, this goes on forever without repeating the sequence. We've checked it out to 10 trillion digits (as of 2016). So, the Pythagorean Theorem says that on an isosceles triangle (a triangle with two equal sides) the hypotenuse will be the length of one of those equal sides times the square root of 2. And, we're pretty sure that the square root of two, like pi, is irrational (it can't be expressed as a fraction).

The Pythagorean's thought it should be a rational number (one that could be accurately represented by a fraction). So - they became upset. Of course we have the same problem with circles where the circumference divided by the diameter is pi - another irrational number. In addition to 2, the square root of 3, 5, 6, 11, 13, 17, 19, and many more primes are irrational numbers.