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Topic: Transcendental exponents
Replies: 6   Last Post: Feb 3, 2010 2:54 PM

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Nathan

Posts: 188
Registered: 12/13/04
Re: Transcendental exponents
Posted: Feb 2, 2010 8:05 PM
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On Feb 2, 5:47 pm, Shepherd Moon <shepherdm...@yahoo.com> wrote:
> If n is an positive integer, then 2^n means repeated mulitplication of
> 2 (n times). So 2^4 = 2*2*2*2.
> If n is 0, then 2^n = 1.
> If n is a negative integer, then 2^n means to invert the base. So 2^-3
> - 1/(2^3) or 1/(2*2*2).
> If n is a fraction, then 2^n expresses a root because of the law of
> multiplying exponents**. So 2^(1/2) = sqr(2) because 2^(1/2)^2 = 2^
> ((1/2) * (2)) = 2.
> But if n = pi, then I'm not sure what 2^pi means. Does it really mean
> multiple 2 by a little more than 3 times - such as 2*2*2...(?)


Pi is irrational, but can be approximated as
closely as you want by fractions. So for
example, you could look at the sequence of
values 2^3, 2^3.1, 2^3.14, 2^3.141, etc.,
each of which has a definition as you gave.
This sequence approaches a limit, and
it turns out that that limit doesn't depend
on the exact approximations for pi you use.
So it makes sense to define 2^pi to be that limit.

In practical computation one uses exponential
and logarithmic functions to calculate these values.



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