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+2πk, k integer, are the same, so
z=r
e
iθ
=r
e
i(
θ+2πk
)
repeats values for k≠0.
The terminal rays of θ and θ+2πk, k integer, are the same, so
z=r
e
iθ
=r
e
i(
θ+2πk
)
repeats values for k≠0.
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,...,(n−1)
}.
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,...,(n−1)
}.
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.
