# Pythagoras

Pythagoras lived in the 500s BC, and was one of the first Greek mathematical thinkers. He spent most of his life in the Greek colonies in Sicily and southern Italy. He had a group of followers (like the later disciples of Jesus) who followed him around and taught other people what he had taught them. The Pythagoreans were known for their pure lives (they didn't eat beans, for example, because they thought beans were not pure enough). They wore their hair long, and wore only simple clothing, and went barefoot. Both men and women were Pythagoreans.

Pythagoreans were interested in philosophy, but especially in music and mathematics, two ways of making order out of chaos. Music is noise that makes sense, and mathematics is rules for how the world works.

Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras is said to have proved that it would always be true. We don't really know whether it was Pythagoras that proved it, because there's no evidence for it until the time of Euclid, but that's the tradition. Some people think that the proof must have been written around the time of Euclid, instead.

Here is the proof:

The Pythagorean Theorem says that in a right triangle,
the sum of the squares of the two right-angle sides will always be the
same as the square of the hypotenuse (the long side). A^{2}
+ B^{2} = C^{2}. Try it yourself: if Side A is 4 inches
long, and Side B is 3 inches long, then 4x4=16, and 3x3=9, and 9+16=25,
and so Side C will be 5 inches long. Try it with other size triangles
and see if this is still true (you can use a calculator, or your computer,
to figure out the square roots).

But how can you know that this is always true, every single time, no matter what size the triangle is?

Take a straight line and divide it into two pieces, and call one piece a and the other piece b, like this:

Now make a square with this line on each side, like this:

and draw in the lines where A meets B on each side to make four smaller shapes. So now you have one square with area AxA (the big yellow one) and one square with area BxB (the little green one) and two rectangles with area AxB (the light blue ones). So the area of the whole square is (A+B) x (A+B) or the area is (AxA) + 2(AxB) + (BxB).

Or you might say that

(A+B)^{2} = A^{2} + 2AB + B^{2}

Now draw diagonal lines across the blue rectangles, making four smaller blue triangles. Call those lines C. Do you see that you have made four blue right triangles, whose sides are A, B, and C?

Now imagine that you take these triangles and rearrange them (or if you print it out you can cut them up with scissors and really rearrange them) around the edges of the square like this:

The little triangles take up part of the square. The area of all four triangles together is the same as the two blue rectangles you made them from, so that is 2AB.

The area of the pink square in the middle is CxC or C^{2}.

And the area of the whole big square is, as we have already seen,

A^{2} + 2AB + B^{2}

So A^{2} + 2AB + B^{2} = 2AB + C^{2}

We can subtract 2AB from both sides, so

that gives (ta da!)

A^{2} + B^{2} = C^{2}

Here's an animated short video showing another way to prove the Pythagorean Theorem.

Learn by Doing - Proving the Pythagorean Theorem

## Other mathematicians:

Aristotle

Anaxagoras

Euclid

Thales

and Aristarchus

## To find out more about Pythagoras, check out these books from Amazon.com or from your library:

What's Your Angle, Pythagoras? A Math Adventure, by Julie Ellis and Phyllis Hornung (2004). For middle schoolers.

Pythagoras and the Pythagoreans: A Brief History, by Charles Kahn (2001).

History of Greek Mathematics: From Aristarchus to Diophantus, by Thomas L. Heath (1921, reprinted 1981). A lot of Euclid, but also describes who the other major Greek mathematicians were and what they did.

Episodes from the Early History of Mathematics, by Asger Aaboe (1997).