I like graphing the sum of various sine functions in Excel. I first found out about it in Physics, when doing constructive and destructive interference of waves. If you combine about three of different frequencies and amplitudes, you can get some really nice patterns.

Last edited by mathsyperson (2005-08-10 01:08:32)

Oops! OK, fixed it. Thanks.

A simple proof that no integer of the form 4n-1 can be written as the sum of two squares.

If n is even, say n = 2r, (2r)² = 4r² = 0 (mod 4).  (Leaves remainder 0 when divided by 4.)
If n is odd, say n = 2r + 1, (2r + 1)² = 4(r² + r) + 1 = 1 (mod 4.)

So the sum of two squares, mod 4, is always equal to 0 = 0 + 0, 1 = 0 + 1 or 1 + 0, or 2 = 1 + 1; never 3.  (Of course 4n - 1 = 3 (mod 4.))

In fact, for the square of an odd integer, we can write (2r + 1)² = 4r(r + 1) + 1.
Since one of r and r + 1 must be even, 4r(r + 1) is divisible by 8. 
So the square of an odd integer leaves remainder 1 when divided by 8, which is sometimes useful.

sin x° is a wave that repeats for every 360 units.
sin x is also a wave, but it repeats every 2pi units.
Adding these gives a wavy wave (hard to describe, easy to replicate in Excel )
The only way I could think of to make it a JPG was to copy and paste it into paint and save it from there, so I'm unsure of how good the quality will be, but I've uploaded it into the image gallery.

The combined function would repeat itself every (lowest common multiple of 360 and 2pi) units, but as pi is transcendental that will never happen. So, it will be a wavy wave that looks almost exactly the same for whatever range of x values spanning 360 you choose, but it will actually always be slightly different.

Last edited by mathsyperson (2005-08-10 01:13:14)