0

I need to know the results of those expressions (I need them for my Scheme interpreter).

$$ (-9)^{1/2} $$ $$ (-9)^{2/3} $$ $$ (-9)^{-1/2} $$ $$ (-9)^{-2/3} $$

The value can be any negative value, and the exponent can be any fraction.

I came up with this equation (not on my own) that seems to work the same as my reference scheme implementation, but I'm not sure if this is correct:

$$ a^b=|a|^{1/2}i,\text{ when } a < 0\ and\ b = 1/2 $$

$$ a^b=-\frac{1}{|a|^{1/2}i},\text{ when } a < 0\text{ and } b = -1/2 $$

and

$$ a^b=\ |a|^{b}\cos(b\pi)+|a|^{b}\sin(b\pi)i,\text{ when } a < 0\ \text{ and} b\ \not = \pm 1/2 $$

The first two are to get an exact number without any floats and rounding errors.

Are those equations correct?

Antony Theo.
  • 4,716
jcubic
  • 235
  • If you are asking "Are those equations correct?", the answer is yes. But if you want one to prove them correct, etc, you need to rephrase your question. Please edit:) – ShadowKnight700 Jan 19 '25 at 15:45
  • 2
    Beware with your equations: you get $((-1)^3)^{1/2}\neq((-1)^{1/2})^3$. – Christophe Boilley Jan 19 '25 at 15:54
  • 1
    a^(1/k) is a kth root of a, and there are k of them in the field of complex numbers, as you can multiply by any of the k roots of unity and still have a kth root of a. In other word, you need to be mindful that there are k kth roots in $\mathbf{C}$. – Simon Goater Jan 19 '25 at 16:58
  • 2
    I.e., fractional powers of negative numbers aren’t uniquely defined – J. W. Tanner Jan 19 '25 at 17:16
  • @ShadowKnight700 no I only want to know if they are correct. I've created them based on ChatGPT Code that seems to work as expected. But wanted confirmation. – jcubic Jan 19 '25 at 18:03
  • If you have access to a book on complex analysis, (e.g. "An Introduction To Complex Function Theory" by Bruce Palka), the subject that you raised may be completely resolved in the book as early as chapter 1. – user2661923 Jan 19 '25 at 18:08

0 Answers0