# Golden Ratio

I just had a question involving both psi and phi. I know that:

Ψ= (1-√5)/2 = -0.618033989...
Φ= (1+√5)/2 = 1.618033989...

And out of boredom, I decided to put into my calculator:
(Φ^Ψ) = 0.7427429446...

But my question rose from there: What happens if you do (Ψ^Φ)? I plugged it in and got ERR:Nonreal Ans and couldn't distinguish why that had happened. So, I went to my math teacher and he had no idea. I then went to my statistics teacher and he had said the only way he could see an error was some sort of correlation involving squaring a negative number, which may have some similarity with taking the square root of a negative number as well. To see if he was right, I tested:

(-√(5))^(√(5))

To get the same error. My question is, why does this happen and what exactly is this error?

Powers of negative numbers are not well-defined in the real number system.

Basically, we can make sense of $(-1)^2, ~(-1)^3$ and others, but that is only for integer exponents!

Once we come to non-integer exponents, then things stop being defined. Things like $(-1)^{1/2}$ or $(-1)^\pi$ are not defined anymore. This is sharp contrast with powers of positive numbers!

Of course, it is possible to extend the real number system to define expressions such as the above. This extension is called the complex number system. Things like $(-1)^{1/2}$, $(-1)^\pi$ or $(-\sqrt{5})^{\sqrt{5}}$ are defined there. They are complex numbers, but not imaginary.

If you want to play around with complex numbers and powers of negative numbers, you can always check wolfram alpha.

Powers of negative numbers are not well-defined in the real number system.

Basically, we can make sense of $(-1)^2, ~(-1)^3$ and others, but that is only for integer exponents!

Once we come to non-integer exponents, then things stop being defined. Things like $(-1)^{1/2}$ or $(-1)^\pi$ are not defined anymore. This is sharp contrast with powers of positive numbers!

Of course, it is possible to extend the real number system to define expressions such as the above. This extension is called the complex number system. Things like $(-1)^{1/2}$, $(-1)^\pi$ or $(-\sqrt{5})^{\sqrt{5}}$ are defined there. They are complex numbers, but not imaginary.

If you want to play around with complex numbers and powers of negative numbers, you can always check wolfram alpha.
Definitely makes more sense with an explanation rather than a simple inference, thank you so much!