Wednesday, October 31, 2007
Wednesday Math, Vol. 3: The Pythagorean Theorem
Today, I'm giving a talk about graph theory, which a mathematical way to deal with connecting dots with lines. At least, that's the simplest way to look at it. In graph theory, you can always draw a picture to represent the situation, but the representation is just a metaphor. In geometry, where we also deal with dots, better known here as points, and the lines and line segments between them, the picture is more than metaphor. The picture by itself can be a proof.
The Pythagorean Theorem is often remembered by students as "a squared plus b squared equals c squared", and when prompted, the students might recall that a, b, and c have something to do with a right triangle. This nugget of knowledge has been independently discovered by civilizations from around the globe. There is good evidence the Egyptians had some idea of it at the time of the building of the pyramid. There are proofs from around the world, including China, India and the Middle East. The Western convention of naming it for the great Greek mathematician Pythagoras isn't incorrect, since he definitely studied right triangles, but he is by no means the first in history to do so.
The big square in the picture is (a + b)^2. I use the up arrow to represent exponents when I don't have a word processor that lets me easily write superscripts. When multiplying a binomial like a + b, we have to remember the middle terms produced by the FOIL method so what we get is a^2 + 2ab + b^2.
The little square in the middle that is rotated slightly is c^2. It is a square and not just a rhombus, because the two small angles in a right triangle must add up to 90 degrees, so if we get three angles that add up to 180 degrees (a straight line) and two of them add up to 90 degrees, the third angle itself must be 90 degrees. Since a triangle's area is (base * height)/2, the four copies of the right triangle add up to 4*ab/2 or 2ab to write it simply. by subtracting away the 2ab term from both ways of representing the big square's area, we are left with the famous equation for the theorem, a^2 + b^2 = c^2.
Using the precise rigor of proof as it is practiced today, the picture isn't a proof of the Pythagorean Theorem, but just a proof that the Pythagorean Theorem works for this one particular right triangle I drew. To make the proof complete, I would have to prove that I could choose any right triangle, make four copies of it and arrange those copies to make a picture with the same properties as the one above. This part of modern proofs is often given the prefix without loss of generality or w.l.o.g., and in this case the fact that the straight line means the "missing angle" will be 90 degrees, as 'splained in the paragraph above, is the fact that lets us construct a big square/little square/four triangle picture with any given right triangle.
The picture proof above is just one of many, many proofs of The Pythagorean Theorem, some of which have been known for many millenia. The picture of the trapezoid here to the left is just the picture of the square above cut in half along an oblique line, but if you know the formula for the area of a trapezoid, this will give us (a^2 + b^2)/2 = (c^2)/2, so it also counts as a picture proof of the Pythagorean Theorem. Credit for this proof goes not to the lasagna loving cat Garfield, but to James Garfield, the American president best known for getting shot by a crazy person who wanted to be ambassador to France. (See? You had to be crazy to want that job even back in the 19th Century!)
Whether this should count as a separate proof from the first proof is a matter of debate, kind of like whether My Sweet Lord is really different from He's So Fine, or if Cole Porter's tune Be A Clown is really different from Make 'Em Laugh, the tune Donald O'Connor sang in Singin' In The Rain.
Now playing: Tom Lehrer - That's Mathematics