This week's theme: the Square Root of Two

I mentioned before I have a Bachelor of Mathematics degree from the University of Minnesota, Institute of Technology.

It's amazing how a single viewing in ninth grade of "Donald Duck in Mathmagic Land" can alter your life forever :-)

In my junior year of college, I read a magazine article that said there were some 8,000 jobs for Mathematicians nation-wide and 80,000 jobs for Systems Analysts. I figured, wow,  that's like 8 times as many jobs in the computer industry (I actually never was very good at "figuring" - which might make my choice of Mathematics seem somewhat misguided :-)

Anyway, I recognized that my interest in a steady meal ticket was starting to exceed my interest in Lebesgue integration, so I started taking as many Computer Science courses as I could still fit in. I didn't actually "minor" in "CompSci" - Math majors didn't do "minors" back then - but I did take a fair amount. There were some crossover courses that counted as Math and others that counted as "technical electives".

Mathematics is sometimes called the "Queen of the Sciences" but actually gets short shrift from all the others. Something I attribute to pure jealousy :-)

For example there is no Nobel Prize for Mathematics - well, okay the story is Nobel's wife ran off with a Mathematician and he never got over it, but that's probably not really true. We are much maligned :-)

So, I ask, "who's your favorite Mathematician?" - besides me of course :-)

I guess I'll have to add - "over all time" - since I doubt anyone can name a famous living Mathematician.

Some might say Pythagoras with his "theorem" or Euclid with his "geometry".Others might say Sir Issac Newton and "The Calculus", but I bet it thins out pretty fast after that. Few would say Gauss, Pascal, Descartes, Leibniz, Euler, Fermet, Fibonacci, Napier, Reimann, Hilbert, Cantor, Boole, Venn, Russell, Polya, Goedel or von Neumann - or even Gosta Mittag-Leffler - the guy that ran off with Nobel's wife.

I have a favorite Mathematician - Hippasus of Metapontum (5th century BC).

I'm sure you've never heard of him because I've never heard of him and I'm the guy who's heard of him (???)

So anyway here's his story.

The ancient Greek Mathematician Pythagoras started a sort of religious cult based on Math - "Everything is number". Numbers are the only true reality - oh - and you should be a vegetarian, too.

Now the ancient Greeks didn't know about zero or negative numbers - they just had the so called "counting numbers" 1,2,3,4, ... and they knew about fractions, which they called "ratios", like 1/2, 2/3, 5/8, ... the "rational" numbers (i.e. "ratio"-nal numbers).

They thought that was it. That explained everything. Math was complete. Wrap it up and ship it. It's all done.

So along comes Hippasus. He proved that there is NO rational number which when squared gives you the number 2. If rational numbers are all there are - what's the deal?

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Here's his proof - it's a proof by contradiction - that is he assumes there IS such a number and then shows that can't happen.

(I'll cheat a little in two places and use the same cop out all my Math books always used - "this is left to the student to verify" - but trust me this is all okay. It's not a big cheat :-)

So - assume M/N is the square root of two - where M and N are both "counting numbers".

We can assume M/N is in it's lowest form since if not we could divide out the common factor and reduce it. Like if M/N were say 4/8 we can divide both by two to get 2/4 and again by 2 to get 1/2. This reducing doesn't change the actual value of (M/N) so it's okay to do this.

This means M and N cannot BOTH be even numbers - since if they were we could divide out 2 in both and reduce it.

So we have ((M/N)^2) = 2.

Which is the same as ((M^2) / (N^2)) = 2.

Now move (N^2) to the other side.

(M^2) = (2 * (N^2)).

Since (M^2) is 2 times something it must be an even number.

Here's my little cheat - if (M^2) is even - M must be even.

If M is even we can replace it with (2 * P) - since that's what even means. 
(P is just half of M).

Substititing that in our equation.

(2 * P)^2 = (2 * (N^2)).

Which is the same as (4 * (P^2)) = (2 * (N^2)).

Divide both sides by 2.

(2 * (P^2)) = (N^2).

Since (N^2) is 2 times something, it must be an even number.

Here's my little cheat again - if (N^2) is even - N must be even.

So M and N must BOTH be even - which is impossible. (Q.E.D.)

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On the number line, if you pick fractions just below where the square root of two should be and square them you get numbers just below two. If you pick fractions just above where the square root of two should be and square them you get numbers just above two. All are "rational".

But NO fraction exists AT the square root of two's position.

Why not? - Who knows?

This led to the invention of the idea of "irrational" numbers ("not"-"ratio"-nal numbers).

But do they actually exist? You can't write them down - they're non-terminating - non-repeating fractions - you can only approximate them. But are they real?

Hippasus only proved there's a hole where the square root of two should be. He didn't really prove there's an "irrational" number there.

We know now there are an infinite number of these holes in the number line. But are irrational numbers really numbers? Do they really exist? Or are they just holes?

Again - Who really knows?

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So how did the Pythagorean Math Cult respond to Hippasus' proof?

Exactly as you would expect true men of science and philosophy, in constant pursuit of knowledge and truth would respond - they pushed him overboard off a boat and drowned him.

That'll teach the little S.O.B. to wreck Mathematics :-)


Hippasus
From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Hippasus

If n^2 is Even, n is Even
http://mathforum.org/library/drmath/view/56084.html