Definition 6.13.
The
n
th
roots of unity
are all of the complex numbers
z
, such
that
z
n
= 1, or equivalently,
z
n

1 = 0.
27
To make our discussion of roots of unity much simpler, we will introduce
some basics about the polar coordinate system. Recall that a polar coordinate
has two dimensions: (
r, θ
), where
r
is the radius (the distance from the origin)
and 0
≤
θ <
2
π
is the angle from the horizontal axis going in the counter
clockwise direction. One can imagine a polar coordinate system superimposed
onto the complex coordinate system. Using this idea, we can convert complex
coordinates into polar coordinates.
Notice that the radius of a polar coordinate is equal to the distance between
the origin and the corresponding complex coordinate pair.
Thus, for some
z
= (
x, y
),

z

=

re
iθ

=

r

e
iθ

. We will need a better understanding of
e
iθ
to
compute

r

e
iθ

.
To find
θ
for some complex coordinate pair, we can set up a right triangle
between Re(
z
), Im(
z
) and the modulus of
z
. Using trigonometry, when

z

= 1 it
follows that, the Im(
z
) = sin
θ
, and Re(
z
) = cos
θ
. Therefore
z
= cos
θ
+
i
sin
θ
.
By Euler’s famous formula (see below), we have cos
θ
+
i
sin
θ
=
e
iθ
.
This
powerful formula allows us to represent complex numbers easily in polar form.
Here is the statement of Euler’s formula:
Theorem 6.14. Euler
0
s Formula
.
Let
z
=
x
+
yi
be a complex number and
let
0
≤
θ <
2
π
. Then
z
=
re
iθ
where
cos
θ
=
x
,
sin
θ
=
y
and
r
=

z

.
Now we may finish our computation of

e
iθ

:

z

=

re
iθ

=

r

e
iθ

=

r
 ·
1 since

e
iθ

=
p
cos
2
θ
+ sin
2
θ
= 1
.
Thus,

z

=

r

. Note that
r
is always positive.
For
n
th
roots of unity, by Proposition 6.12 (1), we have

z
n

=

z

n
= 1.
Notice that

z

is a real number. Therefore,

z

= 1 for all roots of unity.
Example 6.15.
Consider the complex number
z
=

1
2
+
i
√
3
2
. Let us compute
28
the modulus:

z

=
v
u
u
t

1
2
2
+
√
3
2
!
2
=
r
1
4
+
3
4
= 1
As is,
z
looks somewhat intimidating. However, we convert
z
into polar form
by using cos
θ
= Re(
z
) =

1
2
.
Thus
θ
= 2
π/
3, and so the polar form is
re
iθ
=
e
2
πi/
3
.
Theorem 6.16. Fundamental Theorem of Algebra
. Every nonconstant
singlevariable polynomial with complex coefficients has at least one complex
root. Consequently, every nonzero singlevariable polynomial with complex co
efficients has exactly as many complex roots as its degree, as long as all multi
plicities of roots are counted.
Note that monic polynomials factor over the complex numbers as
(
z

r
1
)(
z

r
2
)
· · ·
(
z

r
n
)
where each
r
i
is a root of the monic polynomial.
Theorem 6.16 allows us to
easily factor polynomials whose roots are the
n
th
roots of unity. The theorem
also allows us to conclude that
X
n

1 = 0 has precisely
n
roots over the
complex numbers.
Remember that
z
is an
n
th
root of unity if
z
n
= 1.
Converting
z
into
polar form yields
z
n
= (
e
iθ
)
n
=
e
iθn
= 1. We must find values of
θ
for which
e
iθn
= 1. Using Euler’s formula again, we must find
θ
values for which
e
iθn
=
cos
nθ
+
i
sin
nθ
= 1. Since there is no imaginary component to the number 1,
we know sin
nθ
= 0. Therefore
n
must sastify cos
nθ
= 1. These two previous
equations are satisfied only when
nθ
is any multiple of 2
π
. However, due to the
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 Business, Number Theory, Interest, Prime number, congruences