Saturday, 6 March 2010

Some Proofs are Prettier Than Others

Just read a post about Nichomachus's Theorem at as a guest blog on Loren Shure's Matlab blog. It pointed out that the theorem was cleverly proven by the simple picture below.
The idea of a proof without words is that you can "see" what it shows.

The theorem is usually presented in pre-calculus classes when they cover sequence and series. In simple words it says the square of the nth triangular number is equal to the sum of the cubes of the first n integers. Cool math language to replace all that verbiage is .

Nicomachus lived in what would now be part of Jordan around 100 AD, and was a follower of the Pythagorean Cult. He wrote about arithmetic and his works were translated into Latin by Boetheus. His book on music and its relation to math is the earliest source of the story that Pythagoras came up with the idea of harmonic tones when he walked past a blacksmith pounding on an anvil. His "Art of Arithmetic" contains the equality above.

I like the "proof without words", but I think my favorite was by Charles Wheatstone. I first heard of Wheatstone as a young electronic trainee in the Air Force. He didn't invent, but did improve a device for measuring the resistance of ..well, almost anything, which is now called the Wheatstone Bridge. He also invented the Playfair cipher (math names are confusing, but Lord Playfair was a heavy promoter of the cipher), the concertina and the stereoscope (if you are older, you looked at cards at your grandmothers house through one of these that gave a 3-D view of the Taj Mahal or other beautiful scenery. He also came up with the following really clever proof of Nicomachus's theorem.

1 + 8 + 27 + 64 + 125 + ...
= (1) + (3 + 5) + (7 + 9 + 11) + (13 + 15 + 17 + 19) + (21 + 23 + 25 + 27 + 29) + ...
= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 ...

and since the sum of the first n odd integers is equal to n^2, we only have to see that by the method he has broken up the cubes there will be 1+2+3+...+n odd integers for the first n cubes... which seems really nice to me.
Post a Comment