The "miraculous" in math

[slimshady2357]

Aw, too slow

 

[Art Vandelay]

Yeah, you (plural ) got it. I was wondering if anyone would try to prove it. This is known as the Simpson paradox.

 

[Felice]

I feel like a Pavlovian mathematician. Someone says prove and I can't help myself

 

[Matabiri]

Actually the mandelbrot set itself is finite in area but the perimeter is infinitely long . Some very clever person has proved this I believe .

While I can't actually remember how to do it (i.e. step 1 below, mathematically), I think it's fairly simple:

1) Prove the perimeter is infinitely long (easy to do intuitively - every time you zoom in you see more convolutions)

2) Prove the area is finite (easy - draw a circle around your Mandelbrot set. The area of the set must be finite and smaller than that of the circle)

3) Combine 1 and 2.

Many areas of mathematics such as imaginary numbers, infinities, and clock arithmetic were invented to solve specific mathematical problems and have no relation to the real world. I think you're going to end up with a very long list indeed...

Imaginary numbers are used all the time in engineering, particularly electronic engineering, because you can use the

exp(i theta) = cos(theta) + i sin(theta)

relation to do all kinds of funky simultaneous and useful calculations.

Edited to add: My favourite bit of maths is the "Hairy Ball" theorem, as in

You can't comb a hairy ball.

http://www.math.hmc.edu/funfacts/ffiles/20005.7.shtml