Don't Hate Me Because I'm Irrational

An very old math post redone and reposted.

Some rational people turn tail when they hear the term irrational numbers. They figure if they can’t understand rational numbers, how are they going to understand crazy numbers that fly off the handle with the slightest provocation! But there’s no reason to be fearful in the presence of an irrational number, because irrational when applied to numbers has nothing to do with unreasonableness. Rather, it tells us that these are numbers that can’t be represented as a ratio of two integers.

We’ve known about rational numbers almost as long as we’ve known that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse, which you probably recognize as the Pythagorean theorem. The story of the discovery of rational numbers goes something like this: Mr. Pythagoras of the Pythagorean theorem loved numbers. He loved them because he thought they were beautiful, perfect, and pure, so beautiful, perfect and pure that every single number could be represented as a ratio of two whole numbers; and so beautiful, perfect and pure that the sum of the squares of the sides of a right triangle was exactly equal to the square of the hypotenuse.  

Unfortunately, Pythagoras had a smarty-pants student who began fooling around with the pythagorean theorum, starting, rationally enough, with a right triangle with sides of one unit, just like the one in the picture above. After a bit of calculation, he discovered—ta da!—that the hypotenuse was exactly √2 units in length. As any good student of Pythagoras would do, he decided to put √2 in the form of a ratio of two whole numbers.

He worked things out something like this:

• √2 = m/n, in which m/n is a fraction in lowest terms.

This much was assumed by definition, since the Pythagoreans assumed that all numbers could be expressed by a ratio of two whole numbers. Next, he squared both sides, and came up with this:

• 2 = m²/n²

Then he multiplied both sides by n² —and reversed sides— and got: 

• m² = 2n² 

This told him that m must be a multiple of 2, because we know that the square of an even number is even, and since 2n² is even, m must be even. This means we can substitue 2q in the equation for m, since m is a multiple of 2:

• 4q² = 2n²

Then he divided both sides by 2:

• 2q² = n²

Which showed him that n must be a multiple of 2 as well. (See the reasoning regarding squares of even numbers above.)

And therein lies the rub. Remember, the student started out with the assumption that that m/n was a fraction in lowest terms. If both m and n are multiples of two, then m/n can’t be a fraction in lowest terms, and there’s no way to represent √2 as a fraction in lowest terms. Meaning, of course, that √2 is not a rational number, and if it’s not rational, it’s got to be irrational, right? 

But let’s leave the math behind and get back to the story. Mr. Pythagorus’ student did what any student would do. He said, “Hey Teach! Look at this!” Unfortunately for the student, his proof undermined the entire belief system of his famous teacher. As the story goes, Pythagoras did what any perfectly rational person would do: He had the student executed rather than take a chance that the secret of irrational numbers would get out to the world. He also insisted that the rest of the class take an oath of secrecy with the threat of death for any tattletale.

If I were you, I’d be a little sceptical of this tale, since there are about a million different versions of it floating around, proving that mathematicians can embellish with the best of them. What we know for sure is this: The Pythagoreans discovered irrational numbers1, and they used the proof given above to prove that √2 was irrational. Evidently, their confidential info eventually leaked out, proving that mathematicians can’t keep a secret either.

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1Although the Egyptians may have beaten them to the punch.

Sources

• Asimov on Numbers by Isaac Asimov
• Irrational Numbers at Wolfram Math World
• Irrational Numbers at Math Is Fun
• Irrational Numbers from Jim Loy.