Dimensional representation of Roots

[Leo Authersh]

If the square root as two coordinate axes in the complex plane, does the cubic root has 3 coordinate axes and so on for nth root?

@vanhees71 Can you please explain this?

 

[vanhees71]

A complex number has real and imaginary part, which you can intepret as cartesian coordinates of vectors in a plane. I don't understand what you mean by "the square root has 2 coordinates".

 

[fresh_42]

If the square root as two coordinate axes in the complex plane, does the cubic root has 3 coordinate axes and so on for nth root?

@vanhees71 Can you please explain this?

No. One complex number is simply one complex number. The fact that complex numbers can be written as $x+iy$ with real $x,y$ is due to the fact, that $\mathbb{C}$ is a two-dimensional real vector space and $\{1,i\}$ a basis.
All complex numbers can be written in this way, no matter how often or which root you take.

 

[Mark44]

If the square root as two coordinate axes in the complex plane, does the cubic root has 3 coordinate axes and so on for nth root?

You are extrapolating with a sample size that is too small. A given complex number has two square roots, three cube roots, four fourth roots, and so on. Each of these roots can be expressed in the form x + iy in Cartesian form (also called rectangular form).

For example, the complex number -1 + 0i has these cube roots: $\frac{\sqrt 3} 2 + \frac 1 2 i, -1 + 0i$, and $\frac{\sqrt 3}2 - i\frac 1 2$. In polar form, these are $e^{i\pi/3}, e^{i\pi}$, and $e^{i 5\pi/3}$.