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 and are the lengths of the shorter sides (the legs) of a right triangle and is the length of the longest side (the hypotenuse), then .
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.)
Explanation: The gray triangles are all the same size and shape, and could be any right triangle. The areas of the colored squares are the squares of the lengths of the sides of the triangle. The two smaller squares and four triangles, together, have the same area as the largest square and four triangles.
[Red Area] + [Blue Area] = [Purple Area]
And that’s precisely what the Pythagorean Theorem says.
In order to appreciate the significance of this to the Pythagoreans, you have to understand that for Greek mathematicians at this time, algebra and arithmetic were viewed as inseparable from geometry. Numbers were lengths, or ratios of lengths. Multiplication of two numbers is the computation of the area of the rectangle formed by two sides. Multiplication of three numbers is a volume. (This explains why we talk about “squaring” or “cubing” a number and also why we hardly ever see “fourth powers” in mathematical writings of the time.)
A fundamental belief at the time about the nature of lines (or what is the same thing, about the nature of numbers) is that all lengths are commensurable. The idea is that given any two lines, we should be able to choose a unit of measurement so that both lines have whole number lengths. In other words. So if we think of a number as a ratio of two line lengths, then the mathematicians of the time believed that all numbers were fractions, or rational numbers. (If you’ve never noticed the connection between the words rational and ratio, notice it now.)
Look at the following simple figure.
If the sides the square has length and the diagonal has length , then the Pythagorean Theorem says , or in other words the ratio .
So an ancient Greek who believed that the side and the diagonal were commensurable would now conclude that there must be a fraction which is the square root of 2. So how to find it?
It so happens that is at least near . Could it be that ? Well, that would be the same as saying that . Notice that is in lowest terms, so that the factors of the denominator (7) don’t divide the numerator. After squaring, this is still true. The denominator of has a factor of 7; the numerator doesn’t. So though is near 2 on the number line, it certainly doesn’t reduce to 2.
And there is nothing special about . Let be any fraction in lowest terms whatsoever which is not an integer (in other words, .) So there is a number which evenly divides , but which does not divide . What happens when we square? . But divides and it does not divide . So after we reduce the fraction as much as possible, the denominator will still be divisible by . In particular, there must still be a denominator; is not an integer.
We have proven the following.
Theorem. The square of any non-integer fraction is another non-integer fraction.
At this point it might not be obvious why I am mentioning this. But notice that what I just said is equivalent to the following.
Theorem. The square root of an integer is never a non-integer fraction.
Or, better yet:
Theorem. If is an integer than either is an integer or is irrational.
(Notice how powerful a tool it can be to restate a question in slightly different words. If you are staring at thousands of digits of on a computer screen and I ask you, “is that number a fraction?”, it’s not at all clear how to decide. But if I ask you “if you square a fraction, do you ever get the number 2?”, it’s much easier to see that the answer is known. All that’s changed is the wording. Actually, this fact is even more general than we’ve said; it is not specific to square roots. If the cube root, or the fifth root, or the nineteenth root, of an integer is not an integer, then it is not any kind of fraction.)
Uh-oh. the square root of 2 is obviously not an integer (1 is too small to work, but 2 is too big). So it’s not a fraction at all. Which means, if you’re an ancient Greek at heart, that the square root of 2 is not a number (it’s like “the second Tuesday of next week”). This is huge blow to your worldview. You have to abandon your deeply-held notion that all lengths are commensurable, or else you have to deny the existence of squares.
This story has certain parallels to the much better-known stroy of Galileo Galilei, who (rightly) concluded that the Earth revolves around the sun (and not vice versa) at a time when the official position of the Church was that the universe was Earth-centered. It is one thing (indeed, I believe it can be a very good thing) to have faith about a matter which is impossible to prove or disprove. But what happens when what was an article of faith is now known to be false. Galileo was excommunicated.
It is known that the existence of irrational numbers was a big deal in the time of Pythagoras, especially for those in his secret society. Stories are told that at least one person lost his life over the controversy, but it is impossible to be sure about that.
In some sense, this situation is more dramatic than that of Galileo. Telescopes were exceedingly rare in those days, so it was easy to dismiss someone like Galileo if you had a vested interest in his being wrong. But anyone, anywhere can draw the two diagrams in this post, and be forced to the same conclusion.