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.