I'm in the school library and my fingers are freezing so I thought I'd give everyone a headache, before starting my homework. I dub this edition a brief look at disconcerting math anomalies. There's a few hundred of these but I'll do as many as it takes for my fingers to warm up.
First, irrational numbers. A simple definition of irrational numbers is numbers which cannot be written as fractions. Pi is such a number. Pi is approximately 22/7 but not exactly. The truth is pi's exact value cannot be expressed. Such it is with all irrational numbers. If that isn't paradoxical enough, consider this:
The square root of 2 is another such number. The Almighty Google tells us that sqrt(2) is 1.41421356, but Google is only telling us the number up to 9 significant numbers. There are many, many more numbers that follow those 9. Square roots have a very empirical application with distances of right triangles. Suppose we have a triangle with an unknown hypotenuse with sides 1 unit long each, like so:
The Law of Cosines in its more specific form of the Pythagorean Theorem tells us the length of the hypotenuse of this triangle can be found by solving 1^2 + 1^2 = c^2 for c. What we get is c = sqrt(2). So then we might ask, how big is this length? Well 1.4142135 is close, but it's too small. We're leaving out .0000006 of the length. Ok, so it's 1.41421356 units long. No, still, we've said the length is shorter than it really is. This process can go on ad infinitum until the building I'm in falls down. I will get closer and closer to the length of the hypotenuse, but I will never express it exactly. The peculiarity comes in here. How can my approximations of the sqrt(2) produce numbers which increase infinitely when my eyes tell me that the hypotenuse of our triangle begins and ends?
Ok I only got one before my hands warmed up. I'm done boring everyone. Enjoy your pondering.
