Precisely Irrational

The Solipsist is a closet math-geek. (Yes, yes, there's nothing "closeted" about our geekdom. But you may not have known about the "math" part.) As an English instructor, YNSHC finds something pleasing about the fact that math provides an opportunity to arrive at actual solutions. And every now and then, we discover a philosophically interesting tidbit amidst the mathematical minutiae.

Take the Pythagorean theorem (please!). Now, as any math teacher, engineer, or 7th grader could tell you, the Pythagorean theorem states that the sum of the squares of the two legs of a right triangle will equal the square of the hypotenuse: A-squared + B-squared = C-squared.

(Digression: You'll have to put up with the fact that the Solipsist has no idea how to do superscripts or other math-y symbols. EOD)

Now, in practice, what this means is that if you know the lengths of any two sides of a right-triangle (a triangle with a 90 degree angle), you can calculate the length of the third side fairly easily. If you know, for example, that the two legs of said triangle are 3" and 4", you can get the length of the hypotenuse:

3-squared + 4-squared = C-squared.

9 + 16 = C-squared

25 = C-squared

The square root of 25 = C

C = 5

Easy peasy eggs and cheesy.

Of course, not all right triangles provide such a nice pat answer. Sometimes, the sum of the squares of the legs will not equal a perfect square, i.e., the answer will be irrational. Take, for example, an equilateral right triangle, the legs of which are each 1":

1-squared + 1-squared = C-squared

1 + 1 = C-squared

2 = C-squared

C = Square root of 2

Now, on the test, you could usually get away with saying that root-2 is the answer. Some instructors might want you to whip out your handy calculator; the Solipsist's gives him an figure of 1.4142135623730950488016887242097.

Approximately.

Because it is approximate, of course. You could keep adding digits until your figure could encircle the globe. Twice. And you still would not get the exact answer for the square-root of 2.

But here's the thing: You could certainly DRAW a right triangle with legs of 1" and 1". You could then draw the line--the hypotenuse--connecting those two legs. And you know how long that line would be? EXACTLY root-2 inches long.

Think about that: The quantity is precise. It is exact. It is logically sound. And yet, it is completely impossible ever to assign to this quantity a precise numerical value. There are more things in mathematics, than are dreamt of--or at least nameable--in our philosophy.