Negative bases

[Mat]

Yes. I'd say there is a pretty good argument for *consistency*:

Base | Digits
  10  | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
   9  | 0, 1, 2, 3, 4, 5, 6, 7, 8
   8  | 0, 1, 2, 3, 4, 5, 6, 7
   7  | 0, 1, 2, 3, 4, 5, 6
   6  | 0, 1, 2, 3, 4, 5
   5  | 0, 1, 2, 3, 4
   4  | 0, 1, 2, 3
   3  | 0, 1, 2
   2  | 0, 1
   1  | 0

[BrewManNH]

There is no zero in a unary (base 1) system, actually there's technically no 1 in a base 1 system it usually uses something to represent the count, not numbers per se.

[Mat]

As already shown above, that something should be 0 if you look at how the bases above it work.

Your argument could be extended to say: Binary doesn't need numbers, it could use + and - instead to represent count and not-count:

-+--+----++--+-+-++-++---++-++---++-++++--+-++----+------+-+-+++-++-++++-+++--+--++-++---++--+----+----+

Perfectly true, but for consistency with the notation for other numeric bases we use 0 and 1. Look at the table posted above and you'll see why I say that base 1 should use zero.

[czardas]

Base 1 should definately use zero - there can be no argument about that. However using 1 as a numeric base is highly questionable in the first place. Does a simple tally system really have the credentials to be considered a numeric base?.

Edit

Hypothetically speaking, base zero would have to use infinity to represent all values.

Edited December 21, 2011 by czardas

[Mat]

Does a simple tally system really have the credentials to be considered a numeric base?

Since it can't represent all real numbers I am very tempted to say no.

[MvGulik]

Base 0 ... Erm ... something based on nothing ... Might work ... some day.

Base 1 with Zero(as value) is ... mmm, better left as hypothetical funny case ... to.

---

PS: Think its better to look around the internet for some info on these two case than thinking about it yourself. (unless you have a really good math brain of course.). ... As some (other) math brains probably already though and wrote something about these tricky cases.

[Edits: ... lots ... ]

Edited December 21, 2011 by MvGulik

[JohnQSmith]

Since it can't represent all real numbers I am very tempted to say no.

Why can't it? If you take a mark, be it 1 or 0 or whatever you want to use, and define it as a fractional amount then it would work for real numbers as well.

Using the classic fractional example of a pie cut into eight pieces, if you defined each mark to be 1 pie then IIIII would be five pies. However, if you defined each mark to be a slice of pie, then IIIII would be 5/8 of a pie.

[jvanegmond]

|||/// = 3.3

Lol.

( Better 3 + (1/3) or something )

[Mat]

Ok. Perhaps I shouldn't have said that base 0 was a tally.

Let's look at the theory. From other bases we can see that there should be only one character (by looking at the pattern of characters I have already said this should be zero, ofc that looks silly in practice and doesn't actually work). The positions are in increasing powers of 1 as you go to the left, and decreasing to the right. That's a rather useless thing though as 1^x == 1. Representing any decimal is therefore not possible as all powers on the right are still 1/1 = 1, so you can't represent something less than 1. 1 ^ infinity should be 1 (undefined by wolfram alpha, but log laws (inf*log(1) = inf*0) mean it could be anything, but would usually be defined as 1). From that I conclude no decimal less than 1 can be represented by base 1, so it's not a real number system.

If you define it any other way (so a tally = epsilon or whatever) then it's different. It's not a numeric bases system in the same way roman numerals aren't.

[czardas]

With things like this I generally try to visualize the process by associating it with something tangible. A mechanical clock, or simple difference engine, is more familiar to me. Each time the base value is reached, the next cog in the gear sequence rotates one click and the preceeding wheel is reset. Since there is only one value in the 'hypothetical' base 1, we end up with an error and the system breaks down. Of course there is no rigour in such an argument.

This could also be a question of semantics: all numeric bases can be used as tally systems. I don't see this expression as holding true when reversed.

Edited December 23, 2011 by czardas