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April 15

New problem (negative number raised to a fractional exponent)

Microsoft Excel defines a negative number to any fractional power as undefined. I find it natural to define a negative number to a fractional power defined if the fraction is the ratio of 2 integers p/q where q is odd. Which is more natural to you?? Georgia guy (talk) 21:55, 15 April 2015 (UTC)[reply]

Yes, -8^1/3 = -2. However, a computer might have a hard time figuring that out, since it would resolve 1/3 to something like 0.3333333333333333331243, which isn't quite the cube root. They just would have to write the software more carefully to avoid this type of precision error. In this case, that would mean processing the exponent numerator and denominator separately, and keeping them as integers. StuRat (talk) 22:13, 15 April 2015 (UTC)[reply]

This is probably because the complex logarithm has some weird behavior.--Jasper Deng (talk) 23:05, 15 April 2015 (UTC)[reply]

This is an old discussion that generates heated debate. The cleanest answer that I can find is that exponentiation cannot be defined consistently over all useful values without breaking the key identities, even considering only real numbers. The two useful but distinct versions on real numbers seem to be: any real base (with a caveat for zero) to an integer power, and positive real base to real exponent. So, essentially, one has to choose which of these two functions to use. —Quondum 23:16, 15 April 2015 (UTC)[reply]

(I added to the title to at least hint at the question.) StuRat (talk) 23:58, 15 April 2015 (UTC) [reply]

Interestingly, Microsoft Excel seems to determine what version of function to apply from the exponent value. With a negative base, if the exponent evaluates to an integer or to a value that is sufficiently close to the reciprocal of an odd integer (but not to another ratio), it proceeds quite happily assuming that it is exact. So, for example, it is happy with (-2.1)^(1/9999), but not with (-2.1)^(3/5). It is thus effectively choosing between three distinct definitions of exponentiation based on the value of the the exponent: real to integer, positive real to real, and nth root of real. —Quondum 00:31, 16 April 2015 (UTC)[reply]

The J (programming language) generates a complex third root of minus eight, and also Quondums examples above

   _8^%3
1j1.73205
   _2.1^%9999
1.00007j0.000314214
   _2.1^3%5
_0.482296j1.48436

That is how I like it. Bo Jacoby (talk) 06:33, 16 April 2015 (UTC).[reply]

and precisely the opposite of how I like it (rant deleted). —Quondum 14:12, 16 April 2015 (UTC)[reply]

The important expression e^2πix can be computed as (-1)^2x , without reference to transcendental numbers e and π, and showing that for rational x the expression is an algebraic number. Bo Jacoby (talk) 23:03, 16 April 2015 (UTC).[reply]