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# Complex Numbers and Their Integer Roots

Module by: Stephen Kent Stephenson. E-mail the author

Summary: Derivation of integer roots of complex numbers.

Complex number z=a+bi=r( cosθ+isinθ ), where r= a 2 + b 2 :=| z | and i= 1 . Substitute Eüler's Formula,  e iθ =cosθ+isinθ, then z=r e iθ . Complex number z=a+bi=r( cosθ+isinθ ), where r= a 2 + b 2 :=| z | and i= 1 . Substitute Eüler's Formula,  e iθ =cosθ+isinθ, then z=r e iθ .

The terminal rays of θ and θ+2πkk integer, are the same, so z=r e iθ =r e i( θ+2πk )  repeats values for k0. The terminal rays of θ and θ+2πkk integer, are the same, so z=r e iθ =r e i( θ+2πk )  repeats values for k0.

With n integer, the complex roots are  z n = z 1 n = r 1 n e i( θ n +2π k n ) . If k starts at 0 it can increase only to n-1 before the roots start to repeat; i.e., there are n unique roots coresponding to k={ 0,1,2,...,(n1) }. With n integer, the complex roots are  z n = z 1 n = r 1 n e i( θ n +2π k n ) . If k starts at 0 it can increase only to n-1 before the roots start to repeat; i.e., there are n unique roots coresponding to k={ 0,1,2,...,(n1) }.

Those n complex roots lie on a circle of radius r, centered at the origin of the Argand Plane, starting at an angle of  θ n  and spaced  2π n  apart. Those n complex roots lie on a circle of radius r, centered at the origin of the Argand Plane, starting at an angle of  θ n  and spaced  2π n  apart.

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