Can you have a base pi, or a base -10, or a base 5i? For that matter, can you have an abstract base, such as base X? Finally, is either a base 1 or a base 0 meaningful?
Well, as a complete non-mathematician I’m going to take a stab based on the best logic I can possibly muster up (though I know the real mathematicians will be here shortly to bash my every word)
First off, it would seem to me that Yes, bases must be positive integers. Well, actually maybe not. I was just trying to picture how a non-integer base would look on an abacus, and it occurred to me that I guess, well I don’t know. OK, eg. base 1.5, well it seems to me you would need a .5’s column. Wait, no, I just had another thought. In base 1.5, 1.5 would be represented by 1, 1.4 would be represented by 1.4/1.5 = .9333repeating, no actually that’s not right either because 1.5 would not be represented by 1 in base 1.5. Well I guess you have to decide on the smallest whole units you’re using in this system. Is this helping?
OK, I think I got something here. In base 1.5, 1.5 is represented as 10, right, and in base 1.5 10-1 = .5, so I guess it could work. I have absolutely no idea what I’m talking about.
But, in response to your latter question, NO, I can’t see how base 1 or base 0 could possibly have any meaning.
Umm, mathematicians? a little help here?
Yes, you can even have imaginary bases, irrational bases, negative bases, etc.
1 is an acceptable base, but 0 is not, because no matter how many zeroes you add together, the sum is still zero.
Variables such as X cannot be used as bases to represent anything except variable numbers, but I have seen a number of math texts that use variable bases (when discussing properties of number systems, not surprisingly).
Refer to Donald Knuth’s magnum opus, “Art of Computer Programming, Vol. 1”, where he has an extensive discussion of number bases.
So, how can base 1 work? Is this a finite system? because how could you represent 1. It would seem that any number within the system would be <1
It basically works just like you expect; zero is written “0”, one is written “1”, two is written “11”, three is written “111”, etc. Not much fun for large numbers (large being any number over about, oh, 4, maybe).
What is possibly confusing you is that base-10 has 10 symbols, “0” thru “9”, base 3 has 3 symbols “0”, “1”, and “2”, so base-1 ought to have 1 symbol, just “0”.
Which immediately begs the question, why did I use “1” up above? I answer, base-1 is an exceptional case, a boundary condition, and has to have 2 symbols to be meaningful.
I forgot to mention, base-1 can represent integers only.
I get it. When I read your post my immediate response was, how does that differ from base 2. But 2 in base 2 is 10.
So in base 1, what does 10 =?
The base doesn’t even have to be a constant! I’ve worked with a factorial numbering system, where the first digit was base 1!, the second digit base 2!, the third, base 3!, etc.
For example, the number 1234 would be written as 141120 (16!+45!+14!+13!+22!+01! = 1234). Digits to the right of the “decimal” point were worth 1/(N!).
I think it was Euler who showed that every rational number has a unique representation with this system. The nifty thing is that every rational number is representable with a finite number of digits- there are no repeating digits. For example, 1/3 in decimal is 0.33333333…, but 0.02 in the factorial base (2/3!).
In general, there are lots of ways to assign values to each digit position so that there’s a unique representation for every number. Some are more useful than others, though 
Arjuna34
In base 1, 10=1. In base 1, position of the digit does not connote significance; you can represent 1 as 1, 10, 010, 1000, or any other variation as long as you only have one nonzero digit. Likewise, 2 could be represented as 11, 101, 1010, etc.
Way back in high school I participated in a math contest that had both individual and team “events.”
The team problem asked each team to explore the feasibility of a base -10 number system, and to try to come up with workable techniques for basic arithmetic, etc.
Each “place” in a number represents (-10)[sup]n[/sup], so every other digit had a different sign, effectively. This led to, among other problems, non-unique ways to represent numbers. For example, 1[sub]10[/sub] can be represented by either 1[sub]-10[/sub]or -19[sub]-10[/sub]:
1*(-10)[sup]0[/sup] = 1;
(-1) * {1*(-10)[sup]1[/sup] + 9*(-10)[sup]0[/sup]} = (-1)*(-10 + 9) = 1.
Basically, it can work. Counting, addition and subtraction are pretty straightforward (you just carry/borrow negative values). Multiplication is kinda tricky, but a workable algorithm can be come up with. Division proved to be utterly intractable for us.
We were discussing our conclusions with the professor who’d written the question afterwards, and he agreed with us on that point - coming up with a simple method for dividing one base -10 number by another is almost impossible.
My logic teacher’s sister wrote about negative bases and worked up algoriths for the operations. I do not know her name. She was a nun and I think her last name was Windolph
What about base vagina?
Oops, 2 windows open at once, I’m sure you can all relate.
I’ll agree with Sleazey here, I can’t think of any conceptual reason that a base has to be a positive integer, except for the fact that using a non-positive integer base would probably be a pain as far as writing individual numbers, and I can’t imagine where it would be useful (though of course some mathematics expert will probably come along and show me some examples of the advantages of using base pi.)
Sleazey: I’d never even thought of a base 1 before! Did you logically deduce this from other bases or have you seen it mentioned in a mathematics textbook?
You can have any system you want and define it any way you want but I am not sure I buy the base 1 deal.
Our decimal positional system has 10 symbols and their value depends on the position within the number. x^n as has been said.
Now, the binary system has two symbols and works exactly the same way.
A unitary (base 1) system is not possible. All you can do is write as many ones as the number you are trying to represent and this subverts the basis of the positional system and is more like Roman numerals.
Precisely, sailor. That’s why no one uses such an asinine scheme. The base-1 discussion is just an extension of base-n principles to a very silly case–mostly as an examination of boundary conditions. We actually discussed this case when we studied base-n math in 4th grade.
I’m not sure I buy the base-1 concept either. Of course you can represent numbers that way, but I’m I little picky. I think a valid “base” system should have a UNIQUE representation for each number, but maybe I’m just too finicky Only being able to represent integers also disqualifies it for me.
Arjuna34
To be a useful base, I agree that the base should have unique representation for every number, and that the base should support fractions. I also think that the base should be reasonably small, though–yet the Sumerians used base-60, and managed well enough to inflict all manner of weirdness on us.
I would define a base-n notation as one in which numbers can be expressed as multiples of powers of n arranged in sequence. I could be wrong. Does anyone have a text on hand with a more rigorous definition?
Arjuna, can you refer me to a cite for this system and its properties? I heard about it long ago, but haven’t had a lot of luck finding more info lately.
-Ben
Why does the umpire always say “out” after I pass one? Why would he want me to leave after I was so nice and didn’t step on the pretty white bag? Why would I want it to get dirty?
::basks in her own ignorance::