The Pythagorean Theorem
Perhaps the most famous theorem of all time, the Pythagorean theorem says that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It s usually written A2 + B2 = C2.
Origins of the Pythagorean Theorem
At least 5000 years ago, someone noticed that if you take cords of lengths 3, 4, and 5, and stretch them to make a triangle, one angle (the one between the cords of length 3 and 4) will be 90 degrees, what is known as a right angle. But for thousands of years, no one went further. It was not until the Greeks came along that someone, probably not Pythagoras, extended this to a theorem – saying that for ALL right triangles, of ANY size, A2 + B2 = C2. There are many many proofs, but I am not giving any of them. It’s easy to find them on the internet.
Sets of numbers that satisfy A2 + B2 = C2 are known as Pythagorean triples. By far the most famous set is 3, 4, and 5. You can tell, easily enough, that this is correct: 3*3 = 9; 4*4 = 16, and 5*5 = 25, and 9 + 16 = 25.
When presented with something like this, the natural urge is to play. At least, that’s the natural urge if you have any math feeling. It’s true for 3, 4, and 5. Perhaps it’s true for any set of consecutive numbers? No. It doesn’t work with 4, 5 and 6. 16 + 25 is not 36, But if addition doesn’t work, maybe multiplication? Let’s try doubling: 6, 8 and 10 work fine! 6, 8 and 10 are a Pythagorean triple because 36 + 64 =100. Perhaps we are on to something?
Try tripling, halving, play around. It always works. You can play with bigger numbers using Excel or a calculator. In Excel, in column A, put 3, 4, and 5. In column B put whatever number you like, 3 times. In column C, put = a1*b1, and drag that down three rows. Then in column D put C1^2 and drag that down. The sum of the first two rows of column D should be the third row of D.
Well, you can play all you want, and you won’t find an exception. But that’s not proof.
Here’s what we are saying, algebraically:
3q^2 + 4q^2 = 5q^2.
If we divide both sides by q, we get back to 3, 4, and 5 (just don’t divide by 0).
More bizarre Pythagorean triples
It’s easy to get numbers that are too big for hand calculation. But now that you have proof, you know it’s true. And if you are asked if 3333, 4444 and 5555 are a Pythagorean triple, you can answer yes, without any calculation.
Although Pythagorean triples are usually considered to be positive integers (whole numbers), it works with fractions and negative numbers too. -3, -4 and -5 are a Pythagorean triple; so are 4.5, 6 and 7.5. It even works with imaginary numbers! -3i, -4i and -5i are a Pythagorean triple, because -9 -16 = -25.
The above is a very simple mathematical proof. If you know what squaring means, you can follow this. But it yields something profound: We know that something is true of an infinite set of numbers.
More Pythagorean triples
Are these all the Pythagorean triples? No way!
Here are more:
5, 12, 13
7, 24, 25
9, 40, 41
These all follow another pattern – the square of the first number is divided in half, and rounded each way to give the second and third numbers. Thus, 5^2 = 25. Half 25 is 12.5. Round down to get 12, up to get 13.
Can you prove it?
What other triples can you find?