Thursday, January 05, 2006

Transcendental Numbers

Maths is a philosophical concern for us, because it tells us of the many mysteries in the universe. One of these mysteries is transcendental numbers. They're the ones that come up on the calculator and you can't write down because they go on forever. They're also known as irrational numbers, so called because they seem to defy reason.

The discovery of irrational numbers has a curious story about to it. Apparently, it was discovered by someone who belonged to the school of Pythagoras (the well known ancient Greek mathematician). When the person who discovered it told the other Pythagoreans, they did not want anyone else to know about it, so they drowned him out at sea. I have heard many different reasons as to why they might have done this, but I think that it might have been because it went against everything they believed in. The Pythagoreans studied and worshipped Mathematics like a religion. To them, (and to the few who appreciate it nowadays), it was perfect and beautiful and complete. The idea that there were numbers that you couldn't express as fractions would have horrified them.

Here is the proof of the irrationality of the sqrt(square root) of 2. It is a reductio ad absurdum (latin for "reduction to absurdity") , which is an argument that shows that something cannot be true because it leads to a contradiction. In this case, we show that the root of 2 cannot be expressed as a fraction, by first assuming that it can:

Assume that the sqrt2 can be expressed as a fraction, say a/b (where a, b are whole numbers)
Let a/b be irreducible so that a and be have no common divisors (like 1/3 but not 2/6)

Now, we calculate:

sqrt2 = a/b

2 = (a squared) / (b squared) ________ (squaring both sides)

2 x (b squared) = (a squared) _________ (multiplying both sides by "b squared")

Since the left side is divisible by 2, the right side must be divisible by 2. The square root of a positive number is also a positive number, so we can now express "a" as "2c" for some integer c:

2 x (b squared) = (2c) squared ________ (by substitution)

2 x (b squared) = 4 x (c squared) __________ (expanding RHS)

(b squared) = 2 x (c squared) ___________ (dividing throughout by 2)

Since the right side is divisible by 2, the left side must be divisible by 2. The square root of a positive number is also a positive number, so b is divisible by 2.

But this contradicts the fact that a and b have no common divisors! It means that sqrt2 cannot be expressed as a fraction, which in turn means that it has an infinite number of decimal places.
The beauty of the classical proof is that we can know that such numbers go on forever, without actually checking that they do (which is impossible anyway).

For your interest - sqrt2 looks like this:

sqrt2 = 1.4142135623730950488......

On a line that is two units long, sqrt 2 is impossible to find. You can find an estimate, but you can't pin-point it because it exists way too deeply within the line. Now you can see why they're called transcendental numbers: they seem to transcend the number line. These sorts of numbers exist between every mentionable number. Other well known irrationals include pi, the exponential e, and other square roots like sqrt 3.


Doctor Marco said...

I see mathematics as the link between the Universe and the mind. Irrational numbers have the beauty of teaching us that is impossible to describe them completely but at the same same time we can use them if we accept them as they are. Just as the Universe.

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