Re: Calculation of powers (TI-86)
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Re: Calculation of powers (TI-86)
Kevin DeGraaf writes:
> How does the 86
> calculate e to non-integer/negative powers? I'm thinking there is an
> easy way to do this with higher math than I know (Functions, Stats,
> and Trig). Also, how are natural logs (and common ones too, for that
> matter) calculated?
>
This sort of calculation is based on the idea of infinite series expansions
worked out originally by Wallis, Newton, et al about 400 years ago and put
on a sound basis by Euler and others about 200 years later. This is part of
what is nowadays called "calculus". But there has been a lot of progress
in thinking about numeric methods (particularly in the last few decades)
which has not yet shown up in the standard Calculus curriculum. A lot
of thinking and discussion is currently going, in fact, into the topic of just
what sort of content should go into post-secondary instruction in
mathematics, what is the "soul" of calculus, etc. etc.
A book that might be of interest, which contains both historical background
on the development of the idea of logarithms and, as well, some of the
most modern of thinking on the nature of number itself, is John Horton
Conway and Richard K. Guy's "The Book of Numbers" (Springer Verlag, 1996).
This book should found accessible by anyone who isn't totally mathophobic.
> Finally, what about negative bases (can't take the ln() of those, if I'm not
> mistaken)?
An excellent question. You then are led back to consider again what a
fractional exponent means, and what it could possibly mean to write
something like 2^x where x is an _irrational_ number (regardless of
whether we have a fancy formula to calculate a value for such an
expression). And whether this same sort of thinking can be applied to
develop a meaning for the expression (-2)^x for an arbitrary number x.
It turns out that while it is clear that one can find integral powers of
negative numbers, as soon as you start trying to consider fractional powers
you run into considerations of complex numbers, multiple roots, etc. and
the whole situation gets much more messy (and interesting, of course).
In the context of complex numbers you _can_ take the log of a negative
number. But this is water that rapidly shoals into another deep
mathematical area...
I think this topic is a good illustration how a seemingly simple question,
relentlessly pursued, can lead one into all sorts of strange mathematical
country. All around us are rocks that haven't been turned over. Who
knows what we might find underneath them? All it takes is the courage
to ask questions!
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<