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.

Teppy, in regular numbers, when you get to 9, you're run out of numbers and have to add a place to say that you've hit 10. each time you hit 10 you increase that digit (in the tens place) by 1, until you run out of digits, and then you add the Hundreds place.

Binary does that same thing, only you run out much faster, as you only have 2 options for digits to put in a place, 0 and 1. So, rather than 10 (ten), meaning 1 ten and 0 units, you have 1 two and 0 singles. 100 would be 1 four, 0 twos, and 0 singles

How does 6 in normal numbers = 110 in binary?

Say you have 143. 143 in regular #s = 1*10^2 + 4*10^1 + 3*10^0
or 1 x 10^2 + 4*10 + 3

Now say you have 110 in binary. That = 1*2^2 + 1*2^1 + 0*1^0
or 2^2 + 2^1 + 0*1

Steph, 100 is four, 10 is two, so 110 is 6. That link I gave looks pretty good -- you might want to give it a try. I was confused by your original question because there's the binary numbering system -- which is just like counting, only instead of having a units place, tens place, hundreds, etc. you have a units place, twos place, fours, etc. -- and then there's using binary numbers for coding because then you can write everything in terms of on or off.

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

THANK YOU! I knew it was something simple, but I couldn't find what it was! So x^p+a is always factorable in Zsubp because (x^p+a) = (x+a)^p, is what I'm putting, because the intermediate coefficients are always multiples of p. The other thing should totally use the division algorithm, only I can't figure out why either.

(I knew once we figured out what was being asked, it would be a veritable orgy of binary-explaining.)

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

They're just different notations for the same value. (110 in binary isn't "one hundred and ten" it's "one-one-zero.")

Think of them as different languages. "Cat" and "chat" are different notations for the same animal.

::braces for inevitable crosspost::

Steph, in base ten, you have ten digits to use to count, and their value depends on their position in the number.

So if the number is 1, the 1 in the rightmost position means ... one. If the number is 11, the 1 in the rightmost position still means one, but the 1 next over means ten. And eleven is one plus ten.

Similarly, 23 means two tens plus three ones--twenty three.

With binary, you only have the two digits (0 and 1).

So a 1 in the rightmost position still means one. But in 11, the first 1 means two (because you're in base two, just like it meant ten in base ten). So 11 in binary is two plus one--three.

Why is this important to computers? Because for storage of information, we have only the two choices. I can represent something as either on or off. So I can't store things in base ten--I only have two digits (which, for sake of a lark, we shall call 0 (off) and 1 (on)).

So it's all stored as zeroes and ones in the deep dark intestines of the computer, and translated to whatever us klunky humans need to see. And by all I mean data and programming -- it's all translated to binary with various representations -- ASCII is one code, for instance, that says when storing the alphabet, 114 (or 1110010, if I get my binary right) really means "r." There are other codes, but ASCII is the text representation one that will come up most often.

*This* is an understandable question

Really only to a couple people, to be fair.

So far the egg explanation is the only one I vaguely understand.