Binary is just counting when the largest digit you can use is 1.

So:

Regular Number   Binary Number
         0                     0
         1                     1
         2                    10
         3                    11
         4                   100
         5                   101
         6                   110
         etc...

why is a^p = a in Z sub p when p is prime?

That's a direct consequence of Fermat's Little Theorem. Have you learned that yet?

Also, if you divide f(x) by (x-a), why is the remainder always f(a)?

Hmm. I'm not sure about that one. You're assuming that f(x) is a polynomial? You can probably do something with the division algorithm, but I'm not sure exactly what.

Okay, but I don't understand that chart. How does 6 in normal numbers = 110 in binary?

Some computer calculator programs can convert to and from binary. I forget if the one that comes with OS X does that, but the Win XP does.

I had a tag recently that said "There are 10 types of people in the world, those who understand binary and those who don't." Gud's table explains the joke.

Also, if you divide f(x) by (x-a), why is the remainder always f(a)?

*This* is an understandable question, but "what up with binary code?" ISN'T? You mathy types are confusing.

I had a tag recently that said "There are 10 types of people in the world, those who understand binary and those who don't." Gud's table explains the joke.

Um. Okay.

Steph, binary counting is based on saying that every number can be expressed as the sum of powers of 2: 1, 2, 4, 8, 16, 32, 64, etc. The first digit (first on the right) is how many ones there are -- so every odd number will have a 1 there, and every even number will have a 0 there. The next digit is how many 2s there are. So 2 is 10, 3 is 11, since it's 2+1. And so on.

So, 9 = 8+1, so in binary it's 101. 10 = 8+2, so in binary it's 110. 19 = 16+2+1, so it's 1011. Etc.

Let's use eggs instead.

Now Pe R. Son uses a 10-egg egg crate, but the town idiot Comp U. Ter uses a 2-egg egg crate instead. So if you have 3 eggs then P. Son doesn't have enough eggs to fill a crate so he says he has 3 eggs. C. Ter says he has one crate and 1 egg.

So for Pe R. Son - No crates and 3 eggs or "3" for short.

For Comp U. Ter - 1 crate and 1 egg or "11" for short.

Binary translator: [link]

Pretty sure it was tommyrot who posted that originally. But I don't if it's helpful to you in referencing binary in text.