# Dimensional representation of Roots

• B
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?

Last edited by a moderator:

vanhees71
Gold Member
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".

Leo Authersh
fresh_42
Mentor
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.

Leo Authersh
Mark44
Mentor
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}##.

Leo Authersh