Base Systems?

[ssha]

I'm being bugged by thoughts of non-base 10 numeric systems. Specifically, I'm attempting to remember how they work. For instance, in a base 3 system, would the count go: 1 2 3 10 11 12 13 20, etc., or would it be 1 2 10 11 12 20, etc.? I'm thinking the former, but am unsure. Someone who remembers middle school math to the rescue, please?

Comments

[chamois_shimi]

In a base x system, you don't get to the x. So for base 3 it's 0, 1, 2, 10, 11, 12, 20, 21, 22 ...

(Just like base 10, you count 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, then start over at 10, 11, 12, 13, 14, 15, 16, 17, 18, 19)

Of course, in base 3, 10 doesn't mean what 10 does in base 10, it means 3 (1 three and 0 ones), 11 means 4 (1 three and 1 one), etc.

Which is sort of counter-intuitive, we tend to think in terms of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, because 1 is the starting place for most things in real life. If you're counting how many apples you have, you don't start with zero, you start with one. But mathematically speaking, things start with 0.

I guess there's no real reason you couldn't count it starting with 1 ... but the Math Gurus seem to think it has to be 0.

In duodecimal bases, you use letters in place of the double-digit numbers, because the double digit numbers aren't really single entities.
base 12:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, etc.

[kieferskunk]

I guess there's no real reason you couldn't count it starting with 1 ... but the Math Gurus seem to think it has to be 0.

The technical definition of a number system uses 0 as the "origin", so it would be correct to say that in Base 3, 0 (or 00, if you prefer) means 0 3s and 0 1s. That is considered the starting point in the actual numbering system - if you started counting at 1, that just means that you're starting one unit up from the origin.

The only number set that does not include zero is the natural numbers (which starts at 1). But number sets really don't have anything to do with number systems - number systems (like octal, base-3, binary, decimal, hexadecimal, etc.) are all just ways of representing the same actual numbers differently.

[active_apathy]

Base 3 is 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, ...
Base 4 is 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, ...
Base 16 is 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, ...

[wrog]

you can also have mixed-radix systems, where each place/digit/bit/whatever can be a different base.

E.g., the way we tell time (11:02:37) is like this, i.e., the hour part is base-24 (*) while the minutes and seconds are base-60

(*) or base-12 if you're using AM/PM (which itself is a bit / base-2-digit if we read AM as 0 and PM as 1).

you can also have the digits run over some other range than 0 .. base-1,
e.g., there's balanced ternary which is just like for ternary (base 3)
except that in each of the places you use -1 .. +1 instead of 0 .. 2,
and then counting from 0 looks like

0, +, +-, +0, ++, +--, +-0, +-+, ...

(where, e.g., +-+ is 9-3+1 = 7 in decimal).
What's notable about this is you can do the negative numbers
without needing a separate sign bit

0, -, -+, -0, --, -++, -+0, -+-, ... (for 0,-1,-2,-3,-4,-5,-6,-7,-8...)

more weird possibilities here

[ssha]

You just broke my brain.

Poor brain...