A math dilemma

How do you calculate the value for (-1)^pi?


Or rather, how do you calculate (-1)^a, where "a" is irrational.

Well, according to my calculator, it is -0.9026853619-0.430301216i, so yeah. Not exactly sure how to calculate it, sorry =/.

I don't know the exact method, but I know it involves using logs to change the base of -1 to make it one number to the power of an integer.

Hmm just searched wolfram-alpha.

you just use these identities:

e^(i*pi) = -1
e^(i*x) = cos(x) + i*sin(x)

so (-1)^pi = e^(i*pi)^pi = e^(i*pi² = cos(pi² + i*sin(pi²

so it's the value posted by Corn, yeah.

[COLOR="Red"]Wolfram states it's equal to -1.

Edit: beat me to it.
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Imagine and numbers as vectors on a complex plane (x-axis is real and y-axis is imaginary).



Usually when you multiply a real number with (-1), the direction simply flips once (a 180 degree turn or pi radian) and flips twice when you multiply (-1) twice (a 360 degree turn or 2 pi radian). Since you're spinning it pi times, you simply rotate the vector 180*pi degrees or pi² radian. You then use trig to find out where the vector ends which yields

cos(pi² + i*sin(pi²

It's probably easier to visualize the problem and tackle it geometrically than to think about it arithmetically.