Back from Maine with Squares and Triangles

6 October 2009

Ah, how I’ve missed my little blog.  Sorry about the hiatus, but now I’m back from the number theory conference with a head full of new ideas.  Unfortunately, most of the topics of the conference have far too many prerequisites to fit in this blog.  Let’s just say I saw many beautiful things and was reminded (in case I had forgotten) why I am a number  theorist.

There was one historical talk, in which David Cox lectured on Galois theory according to Galois.  If you aren’t a math major, don’t worry, you’ve probably never heard of Evariste Galois.  So inspired was I by this talk, and by the beauty of  the ideas at play in what Galois brought to light, that I want to share the heart of Galois theory with all of you. This will take quite a few posts to realize, working our way there one vignette, one thought experiment at a time. Fasten your seatbelts, ladies and gentlemen . . . the next few weeks will be interesting.

I did learn one extremely clever thing which is suitable for this audience “right out of the box”.  The inimitable Steve Miller showed me the following purely graphical proof that $\sqrt{2}$ is irrational.

What would it mean for $\sqrt{2}$ to be rational? It would mean that $\sqrt{2}=m/n$ for some integers m and n, which we can choose to be in lowest terms.  In other words, there is a square of integer side length (m) whose area is the same as two squares of another integer side length (n), and furthermore we couldn’t find smaller integer squares with this relationship.  Place the two smaller squares in opposite corners of the larger square as sshown in the picture. By our setup, the two light purple squares together have the same area as the large square.  This means that the uncovered area (the two white squares) must account for the same area as the doubly-covered area (the darker purple square).  If the original squares have whole-number sides, then so do these.  And the new squares are obviously smaller than the new ones, since they’re physically inside the new ones.  But we had supposedly chosen the smallest possible integer squares with this property.   Contradiction.

Neither is this trick is limited to $\sqrt{2}$.  The following picture can be seen as a demonstration of the irrationality of the square root of 3, if you look at it right.  I leave that to you. If you want a further challenge, try to find proofs in the same spirit that $\sqrt{6}$ and $\sqrt{10}$ are irrational.

The Pythagorean Theorem and the Square Root of 2

20 July 2009

Most of you probably remember the Pythagorean Theorem, and the rest of you probably remember that there was a time when you remembered it. The Pythagorean Theorem relates the lengths of the sides of a right triangle. If $a$ and $b$ are the lengths of the shorter sides (the legs) of a right triangle and $c$ is the length of the longest side (the hypotenuse), then $a^2+b^2= c^2$.

In other words, if you draw three squares based on the side lengths of the triangle, the total area of the smaller two is equal to the area of the largest.

The Pythagorean Theorem is named for Pythagoras, a Greek mathematician. To be fair, he was not the only person to do so. This fact, so fundamental to geometry and measurement, was known independently to many ancient cultures, and in some places there is evidence that the fact was known long before Pythagoras.  I like to think that the theorem bears Pythagoras’ name because he is the most colorful choice for a namesake — mathematician, numerologist, leader of a secret society (yes, really!).

What you might not expect is that it is actually quite simple to see why the Pythagorean Theorem is true. You don’t have to be an ancient mathematician shrouded in mystery to discover it.  Actually all it takes is some paper squares cut into colored pieces. (How this illustrates the Pythagorean Theorem is explained after the jump . . . but try to see it for yourself before reading on.)