Golden Ratio

by eric.l
Tags: golden, ratio
 Mentor P: 18,240 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.
 Quote by micromass 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.