Pythagoras' Theorem (Part 1)

Hello reader,

Every student who has ever studied mathematics has come across Pythagoras' theorem. The sum of the squares of two sides equal blah blah blah...

And by using the theorem to solve pages upon pages of monotonous questions, most students learn the formula as just that...a formula to solve questions on maths exams. But it's so much MORE than that!

Pythagoras came up with his famous theorem in some time around 500BC. Does it not strike you as odd then that it has not been disproved yet? For example, around the same time, biologists thought the human body was made of clay and earth, and physicists thought the earth was the centre of the universe (and would do for another 2000 years). Yet, what mathematicians were discovering then is considered as true today as in 500BC and will be considered true forever more. So what is it about mathematical theorems which gives them such longevity?

The answer is 'proof'. In every other field of study, a problem is considered solved when a theory has enough evidence to back it up. For example, people wondered how human beings came into existence. Darwin studied organisms, saw evidence of genetic lineage and natural selection (eg. Galapagos Islands) and so came up with the theory of Evolution. Problem Solved. 

However, this approach doesn't work for mathematics. The Riemann Hypothesis (a very, very important and famous problem in mathematics) has been checked for billions of numbers but this is not even close to what constitutes proof in mathematics. This is because there is an infinite number of numbers. So even if we check Graham's number (This is the largest non-trivial number currently known. It is so big, there is not enough atoms in the universe to write it out as a tower of powers, let alone in full!) of numbers, there's still infinite numbers to go!

So how can you prove something is true in maths? You have to generalize, and all students have done this countless times. Every time you use x in some equation, you are proving something true for all numbers, because if it works for x, it'll work for any number. What's more, once you prove it, your theorem (like Pythagoras') can never be disproved because there are no opinions, new evidence or anything else, only logical proof which can never change. Today mathematicians use methods of proof which are considered 'rigorous' (something which I will devote an article to soon):

Pythagoras' Theorem is a lovely reminder that mathematical theorems will last forever. So, for those of you who crave immortality, prove something new in mathematics (much, much easier said than done :p )