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For a personal project, I had to figure out what happens when $-1$ (negative one) is raised to fractional powers lying between $0$ and $1$. I thought that if I get a power $x = 0.a_1a_2a_3...a_n$ (where $a_1,...,a_n$ are digits), then I could express it as:

$$ x = 0.a_1a_2a_3...a_n = \frac{(a_1a_2a_3...a_n)}{10^n}. $$

Then, by examining whether $a_n$ is odd or even, I could tell if $(-1)^x = 1$ or a complex number. If $a_n$ is even, then $(-1)^x = 1$, else it's some complex number.

Basically, if I have a number like $(-1)^{0.222}$, I could write it as

$$ (-1)^{0.222} = (-1)^{222/1000} = ((-1)^{222})^{1/1000} = 1^{1/1000} = 1 . $$

Using Math.pow(-1, 0.222) on Java also gives the result $1$. However, putting in the same expression (i.e. $(-1)^{0.222}$) on Google's online calculator gives the result

$$ (-1)^{0.222} = 0.766493007 + 0.642252653 \ i. $$

What is going on here? Why is a complex number given as output?

Nilabro Saha
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    Might be relevant: http://math.stackexchange.com/questions/1612298/what-is-the-right-way-to-calculate-a-power – Brian Tung Jan 14 '16 at 18:11
  • http://math.stackexchange.com/questions/438/why-sqrt-1-times-1-neq-sqrt-12 could also be relevant for playing with powers here as don't forget that $a^.5=\sqrt(a)$ – JB King Jan 14 '16 at 18:13
  • Thanks, Brian Tung! It is very helpful! It wouldn't've been possible for me to dig out that relevant post, since that post has a very peculiar name. – Nilabro Saha Jan 14 '16 at 18:14

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