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.

::braces for inevitable crosspost::

Captain! The shields are buckling! She can't take any more crossposting!

No let's a box to the egg scenario. Pe R. Son uses a box that holds 10 crates. Comp U. Ter uses a box that only holds two egg crates. So when presented with six eggs Pe R. Son still can't fill a crate so he says he has 6 eggs. Now Comp U. Ter fills up three 2-egg egg crates. Two of those egg-crates go into his 2-crate box, leaving one 1 unboxed crate. Therefore, the Comp U. Ter says he has 1 box, 1 crate, and 0 single eggs.

So for Pe R. Son - 6 single eggs or "6" for short.

For Comp U. Ter - 1 Box, 1 Crate, and no single eggs or "110" for short.

I"m not going to try and explain it because I think the mathy people here have it down. I'll just say that for me, it was understanding base 12 that got me to understand binary and all bases below 10. My math prof used t and e for the new numbers that came after 9 and then it clicked.

(Not sure if that post made sense)

For me, the only part of binary and base eight that every stuck was the basic definition. I always hated those parts of math classes where they'd make us do calculations in base eight.

What is the purpose of base eight?