Pythagoras' Theorem (Part 2)

Hello reader,

As you may have noticed in my last post, I didn't actually cover that much about Pythagoras' Theorem at all...yeh, sorry about that :p . That said, the theme of rigorous mathematical proof is something I'll return to in the near future, so it wasn't a complete waste of time.

So...Pythagoras' Theorem.

For starters here is a quick, non-rigorous, geometrical proof of Pythagoras' Theorem.

This is not a complete proof on its own however, as an accompanying set of algebra is required.You should be careful when seeing geometrical 'proofs' without an accompanying algebraic solution as these can be faulty (our eyes are not very good tools for mathematics).



(Source of animation: Wikipedia)



Something you may not have thought about, but which is crucially important to Pythagoras' Theorem is that it's converse is also true. This means that given 3 integers a < b < c iff (if and only if) a² + b² = c² then there exists a triangle with side lengths a, b and c with a right angle between sides a and b.

It is therefore natural for people to be interested in finding sets of integers which satisfy the equation a² + b² = c² . Such sets are called 'Pythagorean Triples'. For example, {3, 4, 5} is a Pythagorean triple because 3² + 4² = 5².

A Pythagorean triple generating formula was discovered by Euclid (a post-Socratic, Greek mathematician) and written in his book 'Elements' which is historically one of the most influential books ever written in science. The formula is as follows:

For any two integers m and n:

f (m , n) = {2mn , m² + n² , m² – n² }

is a pythagorean triple (try it out!)

I feel this would be a good time to let you in on a somewhat humbling fact. Almost everything the standard GCSE student knows about mathematics was known over 2000 years ago. We're playing catch up, and we've got a very long way to go.

For those of you who have further interest in Pythagoras' theorem or triples, I would recommend investigating its generalisations into higher dimensions and non-Euclidean geometry. But that will do for now on Pythagoras on this blog.

Thanks for reading :)