Interesting Maths Stuff

Pythagoras theorem is probably most famous amongst non-mathematicians. You will in all probability know it. It goes, take a right angled triangle (a triangle in which one of the internal angles is 90 degrees). Measure the length of the shortest side, and multiply it by itself ("squaring" the number). Measure the next longest side and square it as well. (You will notice that the two shortest sides are the ones that meet at the 90 degree angle. This always happens.) Add the two squared numbers together. Now, you have a much bigger number.

You now need to work out what number you need to multiply by itself to get that big number. This is known as the square root of the number. You can either use the \(\surd\) button on your calculator, or you can work it out gradually by guessing ANY number, dividing the big one by it, and seeing if the result is the same as your guess. If it is then a) you already knew the answer, b) you have experienced a massive coincidence, or c) your subconscious knows more about maths than you do. If, more likely, the result is NOT the same as your guess, guess another number between your original guess and the result of the division. Divide the big number by that. Again if the result is not the same as the new guess then guess again between the result and the new guess. Keep doing this - guess new ones between the last guess and the last result of the division into the large number, and the true root will make itself known.

I have made a spreadsheet in Openoffice which demonstrates this, and you can download it here. Without doing anything apart from guessing and dividing, you can force the square root of the number to come out of the shadows. You pop the number you want to find the root of in the blue box, and your first guess in the orange box. The spreadsheet will then run through the operation described in the previous paragraph 25 times, showing you the results each time. Despite the fact this is completely random, it forces the root out. I have also put a cube root finder in as well, which works in exactly the same way. The only difference is that it divides the first number by the guess twice, rather than once.

If that wasn't cool enough already, now measure the longest side of the triangle. Guess what its length is - exactly the number which we just guessed! That is a round about way of saying that the square of the hypotenuse is the sum of the squares on the other two sides. The classic triangle to refer to is one with the longest side length 5, and the other two sides length 3 and 4. (Don't worry about which units we are using, it doesn't matter). 5 fives are twenty five, which is nine (3 threes) plus sixteen (4 fours). That is Pythagoras theorem, and it is justifiably famous. Now, you can do a lot with this equation. You can invent engineering and build the modern world for one thing, but at its core it just predicts lengths of triangles. That isn't very interesting to the Universe.

A possible contender for most interesting formula, and one of great interest the universe is, of course, \(E=mc^2\). This probably vies with Pythagoras for most well known. It says very simply that [E]nergy and [m]ass are exchangeable with an exchange rate of the speed of light (c) multiplied by itself. This one is a bit more profound than Pythagoras, but at a basic level, it lets our engineers build nuclear power stations.

For what I would call a profoundly interesting equation you want this:

\(e^{\pi i}+1=0\)

If there is a formula which demonstrates the profound connections which underlie the structure of our mathematical universe, it is that. It is called Euler's Identity, and it isn't an identity. I'll try and explain that in due course. To try and establish why I think it is interesting, I am going to break it down into its constituent parts, stare in fascination at the fact they all combine, and then work out why it is that they combine.