You can do fractional exponents by transforming to a log, then multiplying by the exponent & exponentializing it but I don't think a number system exists that combines both fractional components & negative base numbers. The imaginary number system sort of satisfies this demand but I am not sure it can be generalized to include all possible fractional powers. Figuring out what to do with the imaginary numbers returned would also be a challenge.
Some power functions in some languages can sort of do this but what they are really doing is invoking different arithmetic sub-functions depending on whether or not the exponent is integer or not & also depending on whether or not the base is negative. A fractional component is usually handled by either a power expansion approximation using integer powers or by using the log but there is usually a check to return a error code if both fractional exponents & negative base numbers are given to the routine at the same time.
Even without a log function, you can calculate the log using a power expansion like a Taylor expansion. I believe the recommended expansion is given in most books of math functions, tables, intregals, etc. The inverse function is also given. I bet some of the math libraries on the web also have formulae for this.
Edited by Jack Lothian (2004-01-27 05:01 AM)
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Jack
