The Rationals Aren’t the Only Ones…

It was Pythagorus who said, all is number in the world,

That’s why in ancient Greece the math geek always wound up with the girl.

He meant the markings on a ruler could in principle assign,

A fraction of whole numbers to each point on the line.

But then Hippasus came along and proved it simply wasn’t true,

In fact you couldn’t even do it for the square root of two.

Well Pythagorus’ buddies tossed him into the sea,

I’m going to prove it for you now, but please don’t do that to me!

When I prove that…

The rationals aren’t the only ones

The rationals aren’t the only ones

You need, you need, the reals.

Take the diagonal of a square with a side length of one,

If you could write it asp over q you’d be done

Where p and q are standing in for integers who

Share no common factors but the one that’s less than two.

We will assume this and be forced to conclude something absurd,

That’s called proof by contradiction as I hope you’ve all heard.

That’s how I’m going to prove…

The rationals aren’t the only ones

The rationals aren’t the only ones

You need, you need, the reals.

I’ve got a question but don’t worry cause its rhetorical,

“Must p be even or be odd or is it flexible?”

By Pythagorus’ Theorem it is very plain to see,

That p squared must be twice what q squared happens to be.

Thus p squared must be even so p cannot be odd,

That makes it twice some number k if you’re still with me please nod.

You plug 2k in for p each place that symbol occurs,

But to find out what you get you’ll have to hear the next verse.

When we finally prove…

The rationals aren’t the only ones

The rationals aren’t the only ones

You need, you need, the reals.

A little algebra is all that’s necessary to see,

That q squared must be twice what k squared happens to be,

By now you must be feeling quite a sense of déjà vu,

So I’ll cut to the chase, q’s a multiple of two.

At this climactic juncture let’s review where we’re at,

P to q’s a ratio of even integer’s that,

We very carefully assumed was completely reduced,

Yet this common factor 2’s exactly what we’ve deduced.

There’s your contradiction and in my opinion a whopper!

Those of you who brought champaign, now’s the time to pull the stopper.

You can tell the theorem’s proved when angels burst into song,

And if you find my proof convincing won’t you please sing along

Now that we all know…

The rationals aren’t the only ones

The rationals aren’t the only ones

You need, you need, the reals.