Pythagoras, Venn, Euclid and Complex Numbers

[This is a back-issue of one of this site’s newsletters]

Last week, I showed you how a puzzle about rectangles gives you a simple formula for pythagorean triplets. This week, I want to show you a bit more about that formula.

But first, have you seen this? It’s proof that even people in a marketing career need to know a bit of mathematics:

As someone said: “Once you’ve seen this as a Venn Diagram, it’s hard to un-see”. Poor Reuters!

Ok, back to pythagorean triangles. The formula went like this:

  • Start with a number, T
  • Work out 1/T
  • Then, work out T + 1/T and T – 1/T.
  • Divide these both by 2, and put them over a common denominator.
  • Then, if (T+1/T)/2 = R/Q and (T-1/T)/2 = P/Q, then P2 + Q2 = R2.

If you make T equal to M/N, then this process gives P=M2-N2, Q=2MN and R=M2+N2. That’s a much more explicit formula, and is known to us from Euclid’s “Elements”, a famous collection of ancient Greek mathematics.

There’s a much simpler way to come up with this formula, which uses complex numbers. Yes, complex numbers make some things simple!

  • Start with a complex number M + iN.
  • Square it: (M+iN)2 = (M2-N2) + (2MN)i.
  • Take the absolute value of both sides: |M+iN|2 = |(M2-N2) + (2MN)i|.
  • Expand out the squared absolute value: M2+N2 = |(M2-N2) + (2MN)i|.
  • Let P=M2-N2, Q=2MN and R=M2+N2: R = |P+iQ|
  • Square both sides of this: R2 = |P+iQ|2
  • Expand out the last absolute value: R2 = P2+Q2.

So there you have it – Pythagorean triples from complex numbers. Who would have thought!