$(-1)^{0.2}=0.8090 + 0.5878i$ how can this be?

Question

I'm working on a numerical analysis project (working with matlab a lot) and I noticed that when I ask for matlab to compute the exponent of a negative number, it gives wrong output when the exponent is not an integer.

For example like the title, when I input $(-1)^{0.2}$ in my calculator, it says $1$, which is the correct answer, if I know anything about how exponents work.

This is because $(-1)^{0.2}=((-1)^2)^{0.1}=1^{0.1}=1$

However, when I put it in matlab, it gives the output $0.8090 + 0.5878i$ how can this be?

On the other hand, $(-1)^{0.2}=((-1)^{0.1})^2$ and $(-1)^{0.1}$ is indeed a complex number. So on the one hand it has an imaginary part, and on the other hand it doesn't...I'm very confused.

Answer 1

This is a very common misconception about powers of negative numbers. The rule $x^{ab} = (x^a)^b$ only works for positive $x$. The reason we restrict to positive $x$ is that powers become a bit mysterious in some sense if $x<0$. Solving $x = (-1)^{.2}$ can be thought of as solving for those $x$ so that $x^5 = -1$ instead. Is $1$ a solution to this equation?

Answer 2

Using complex number $(-1)^{0.2} = (-1)^{\frac{1}{5}}$, and view it as taking the $5th $ root of $-1$. DeMoivre's theorem?

Answer 3

$$\Large\sqrt[n]x\quad\it\text{ is a SET of n numbers, and NOT a single solution !!!}$$

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\big($ Just sayin' $\ldots\big)$

The reason mathematical software systems return complex valued quantities for odd-order radicals of negative numbers is because they always seek out the one with the smallest argument $($i.e., the one which creates the smallest angle with the real positive semiaxis$)$. Obviously, the angle made by the negative and positive semiaxes is $180^\circ$, which is greater, in this case, than $\dfrac{360^\circ}5=72^\circ$, and the same goes for all $n>2$.

Answer 4

Here is a demonstration of the problem with this convention for roots of negative numbers:

$$ (-1)^{2/5} = 1 $$ $$ (-1)^{19/47} = -1 $$ $$ (-1)^{2 \sqrt{2} / 7} = \text{undefined} $$

which would mean

$$ (-1)^{0.4} = 1 $$ $$ (-1)^{0.404\ldots} = -1 $$ $$ (-1)^{0.404\ldots} = \text{undefined} $$

That's a pretty drastic difference for numbers that are so close together, isn't it?

Really, if you want to stick to floating point real numbers, then negative numbers raised to floating point numbers is nonsense, since negligible errors in the exponent make a dramatic difference in the actual value... or if the value is even defined at all!