The Question Why

By Sebastiaan

Another poem:


The Question Why

Linear time passes equally dull
as all other times -
straight forward,
immer gerade aus.

What's new becomes old
over time and
in symphonic obedience
of listening, telling, and

knowing what to do,
I startle, and question

why?

Pythagorean Theorem

By Sebastiaan

It's been long since I wrote about math, so here's a rather simple subject which almost everyone knows of, but taken to a bit higher level. I'm talking about the Pythagorean Theorem.

Let me first bring back the long ago acquired knowledge of the theorem. When you have a right-angled triangle, or simply a right triangle, the theorem states that the sum of the squares of the legs belonging to the right angle are equal to the square of the side opposite the right angle.

Right, that's a pretty large piece of information, but what I'm actually saying is, with reference to the picture above, that:

AB² + AC² = BC²

or

b² + c² = a²

The classic example is AB = 3 and AC = 4. The squares of these sum up to 25 (9 + 16 = 25), and the square root of 25 is 5. So BC = 5.

Now, a more interesting question comes up when one asks oneself, "why?". For most of us this is pure magic, but it is most definately proveable. Not only is it proveable, it has been proven in many, many different ways. I will actually try to take some of these proofs and show them to you, in understandable language.


This first 'proof', is actually just to make the matter a bit more comprehensible. It's good to get a bit more knowledge about the theorem.

With reference to the picture above:

If AC = 3, then the area of the square with side AC, is 3 x 3 = 9.
If AB = 4, then the area of the square with side AB, is 4 x 4 = 16.
If AC = 5, then the area of the square with side BC, is 5 x 5 = 25, which is indeed 9 + 16. If you work this out neatly, you'll find by measuring that this is indeed true.

But this doesn't actually prove the theorem. If you replace '3' with 'b' and '4' with 'c' and '5' with 'a', then you get b² + c² = a².

A nicer prove, in my opinion, is the following:

With reference to the above picture:

(1) The area of each of the triangles are given by 1/2 times a times b, so 1/2*(a + b), so all four triangles sum up to 4*(1/2*(a + b)) which is 2*a*b or simply 2ab.
(2) The area of the square with sides equal to c is given by c*c, or c².
(3) The area of the square with sides equal to a + b is (a + b)*(a + b) or (a + b)².
(4) The area of the square is also given by combining (1) and (2), which makes c² + 2ab.
Now we can combine (3) and (4), and you get, magically (or perhaps not so magically?), the Pythagorean Theorem:

(a + b)² = c² + 2ab
a² + b² + 2ab = c² + 2ab
a² + b² = c²

I will probably post more proofs, hopefuly more elegant ones, in the future.