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algebra complex number

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2 Answers

According to the fundamental theorem of algebra, the equation z7=1 has 7 roots, called the roots of unity. Only one of them is obvious: z=1. The other six require polar form. First write 1 in polar form:
1 = ePi*in, where n is any integer.
 
Then set
z7=ePi*in
and take the seventh root:
z=ePi*in/7
Now substitute in values for n. You only need the values n=0,1,2,3,4,5, and 6, because the answer will repeat itself for all other n (due to the periodicity of complex exponentials).
 
For n=0, you get the obvious solution,
z0=e0=1.
 
For n=1, you get
z1=ePi*i/7,
which you can write in standard form using Euler's identity:
z1=cos(Pi/7)+i sin(Pi/7)
 
In this way, you get the seven seventh roots of unity.
 
If you graph them in the complex plane, you will find they all lie equally spaced on the unit circle.