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.

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Now playing: Tom Lehrer - That's Mathematics
via FoxyTunes

9 comments:

dguzman said...

Love that Pythagoras. I kinda dig the trig/geometry thing.

Matty Boy said...

I wrote a website all about Pascal's Triangle and the binomial coefficients, but even I would say that the single most common and useful theorem in all of mathematics is The Pythagorean Theorem.

Karla said...

Tino took his Maths at Berkeley with a heavily accented professor from Russia.

He didn't know until much later that the "Peter-GOR-us" his instructor constantly mentioned was actually our triangular friend!

Matty Boy said...

Heavily accented professor story: I took accounting from a Nigerian. He was discussing "deposition", which I faithfully recorded in my notes, not exactly knowing where it was going but assuming he knew best.

When he finally wrote it on the board while speaking it, he was talking about "depreciation", which makes MUCH more sense in an accounting class. Every other student I talked to also assumed it was "deposition" until he wrote it and spoke it at the same time.

FranIAm said...

Oh Matty just the site of this post makes me break out in hives.

Math phobic plus!!

Karla said...

Hey Franiam - I was lucky enough to grow up in a mathy house with mathy folks like our blogfather, so bad teachers couldn't ruin my interest in it.

But they've ruined so many others, and especially womens interest, that it's a damn shame.

I think we're going to have to ask our friend Matty Boy to consider writing a regular folks maths book that would help everyone get into the wacky fun of math.

I'm not kidding - it's justsofun if you learn yer maths from the cool people.

Distributorcap said...

but the SCARECROW, who has a brain (like Bush made of straw) says in the movie:

"The sum of the square roots of any two sides of an isoceles triangle, is equal to the square root of the remaining side."

but leave it too the scarecrow to be a communist since he cannot mention the word "right" so uses isoceles instead.

ps -- i loved geometry, but those poor secants, co-secants and co-tangents, never used, always the evil stephildren

Matty Boy said...

DCap: secants are useful once you start doing the trig functions in calculus, since the derivative of the non-stepchild tangent is secant^2.

My take on the Wizard of Oz scene is that the writer of the film wanted to say that "smart" people are just confident bullshitters. A populist sentiment, not unlike villagers with pitchforks and torches are a populist sentiment.

FranIAm said...

Karla- I would love to see a book like that. I think that even with just my electronic and blogic (o a made up word!) knowledge of Matty, that he would be the one.

It is sad- I grew up in that whole "new math"era. Plus at that time, "girls" were discouraged from such things! I did excel at readin' and writin' (not that you'd know that now) and that is where I put my efforts.

My 11 year old stepdaughter loves her maths and I am embarrassed to admit that I find myself surprised by that from time to time!

This is a cool maths guy...Mathman.

But there is ...
Only
One
Matty

Write the book man! I will sign up for a class when it is available.