SOLUTION: Find the 3 cubic roots of unity (1 + 0i) in complex standard form. the 0 is a zero, not a theta

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Question 1133322: Find the 3 cubic roots of unity (1 + 0i) in complex standard form.

the 0 is a zero, not a theta
Found 2 solutions by ikleyn, rothauserc:
Answer by ikleyn(53751)   (Show Source): You can put this solution on YOUR website!
.

In the plane of complex numbers, 1 has the modulus  and the argument values ,, ,.... 


Taking the cube root of the complex number 1, you have the modulus  (positive value) and 


three argument values ,  and . 


It gives three values of the cube root of the complex number 1: 


1,     and   . 


Figure shows complex (real) number 1 in red and two other roots in blue.


Note that these tree points are vertices of the equilateral triangle.


    


    Figure. Cube roots of the complex number 1


------------------

There is a bunch of my lessons on complex numbers
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
    - How to take a root of a complex number (*)
    - Solution of the quadratic equation with real coefficients on complex domain
    - How to take a square root of a complex number
    - Solution of the quadratic equation with complex coefficients on complex domain

    - Solved problems on taking roots of complex numbers
    - Solved problems on arithmetic operations on complex numbers
    - Solved problem on taking square root of complex number
    - Miscellaneous problems on complex numbers
    - Advanced problem on complex numbers
    - Solved problems on de'Moivre formula
    - Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)
    - A curious example of an equation in complex numbers which HAS NO a solution
in this site.

Of these lessons,  the most relevant to your problem is the lesson marked  (*)  in the list.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Complex numbers".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
Assume the cube root of 1 is z, that is,
:
z = 1^(1/3)
:
now cube both sides of the =
:
z^3 = 1
:
rewrite as
:
z^3 -1 = 0
:
this cubic can be factored
:
(z-1)(z^2 +z +1) = 0
:
use quadratic formula
:
z = -1 +square root(1^2 -4**1)/2*1 = -(1/2) +i * square root(3)/2
:
z = -1 -square root(1^2 -4**1)/2*1 = -(1/2) -i * square root(3)/2
:
z = 1
:
*******************************************************************
The 3 cubic roots of unity (1 + 0i) in complex standard form are
:
1+0i, (-1+i*square root(3))/2, (-1-i*square root(3))/2
*******************************************************************
:
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