Nick,

Base 0 does not work, of course. 0ⁿ is undefined for n=0 and 0 for n>0.

a negative base,

Negative bases work really well - in some ways, better than positive bases, because no minus sign is needed and you can count through all the integers starting from 0 (though determining order is harder.) I'll write [153:7] to mean 1x7² + 5x7 + 3 (etc) where the number after the colon is always base ten.

[0:-2] = 0
[1:-2] = 1
[10:-2] = -2
[11:-2] = -2+1 = -1
[100:-2] = 4
[101:-2] = 5
[110:-2] = 2
[111:-2] = 3
[1000:-2] = -8 (and so on)

or a fractional base,

A base less than 1 is problematic, as 1+b+b²+... is a geometric series with a finite limit, and if N is the maximum numeral we can express nothing greater than N/(1-b). However, using a radix point, base 1/10 is just base 10 written backwards! So 23.5 would be [5.32:1/10].

Fractional or irrational bases greater than 1 work. For instance, one can use the Golden Ratio as a base! However, all or most integers will have to be written using a "point" and negative powers. Whether the expressions are terminating or nonterminating may be an interesting question.

never mind any other base.

People have looked into place value systems using complex-number bases. They do work, and the theory involves has a fascinating tie-in with "dragon curve" fractals. The idea seems to have been invented by the famous computer expert Donald Knuth as a science fair project when he was about 17.

http://en.wikipedia.org/wiki/Complex_base_systems

Good Hunting!
RD

Base -2 would be possible no?
With digits 0 and 1,the base 10 numbers
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
would become
1, 110, 111, 100, 101, 11010, 11011, 11000, 1001, 11110;
while with digits 0 and - (for -1) the sequence would be
--, -0, --0-,--00, ----, ---0, -00-, -000, -0--, -0-0.

With positive integer bases, we are used to the convention that
there will be symbols for the digits 0, 1, ... up to the base minus one.
For negative integer bases, we can put in symbols for 0, 1, ... up
to the absolute value of the base minus one, or digits
0, -1, ... down to the base plus one.

How about using digits outside these values? We already do it
when we use a microwave: The seconds entry is supposed to be
one of the base 60 ``digits'' from 00 to 59, but we can use 90
instead of 1:30 to represent a minute and a half.

There are nice mathematics to be done if we allow extra digits
to be used: Suppose that we count in base 2, but we can use
the digits 0, 1 or 2. Then some numbers can be represented in many
ways:
0 can be only represented as 0,
1 can be only represented as 1,
2 can be written 10 or 2,
3 can only be written as 11,
4 can be written 100, 20 or 12,
5 can be written 101 or 21,
6 can be written 110, 102 or 22,
7 can only be written as 111,
8 can be written 1000, 200, 120, 112,
9 can be written 1001, 201, 121,
10 can be written 1010, 1002, 210, 202, 122,
...
If we look at the sequence of how many ways we can represent the
numbers 0, 1, 2, 3, ..., we get
1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, ...
The sequence of ratios of consecutive terms in this sequence is
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, ...
Every positive rational appears exactly once in this sequence
(see for instance http://www.cut-the-knot.org/blue/Fusc.shtml)

Claude