



 
Nick, Base 0 does not work, of course. 0^{n} 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 1x7^{2} + 5x7 + 3 (etc) where the number after the colon is always base ten.
or a fractional base, A base less than 1 is problematic, as 1+b+b^{2}+... is a geometric series with a finite limit, and if N is the maximum numeral we can express nothing greater than N/(1b). 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 complexnumber bases. They do work, and the theory involves has a fascinating tiein 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!
Base 2 would be possible no? With positive integer bases, we are used to the convention that How about using digits outside these values? We already do it There are nice mathematics to be done if we allow extra digits Claude  


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