A Brief Review of Complex Number Notation
The complex numbers C = {x + iy : x , y ∈ R} are the real vector space R
2
spanned by a basis {1, i}
where i
2
= −1 is a ’number’ satisfying i
2
= −1. To multiply complex numbers, simply expand out
and replace i
2
with −1: for example
(2 + 3i)(1 −i) = 2 − i + 3i − 3i
2
= 2 + 2i + 3 = 5 + 2i
Complex conjugate: Given a complex number z = x + iy ∈ C, its conju-
gate z = x − iy is the reflection of z in the real axis.
Modulus (length): r =
|
z
|
=
√
zz =
p
x
2
+ y
2
is the distance of z from
the origin.
Argument (angle): If z = 0, then θ = arg z is the angle measured
counter-clockwise from the positive real axis to the ray
−→
0z.
0
0 1 2
z = 1 +
√
3i = 2e
iπ
3
|z| = 2
arg z =
π
3
i
2i
Polar form: z = re
i θ
= r cos θ + ir sin θ. The complex exponential obeys the usual exponential laws
and is 2πi-periodic: for instance
• e
i θ
e
i ψ
= e
i (θ+ψ )
• e
i θ
= 1 ⇐⇒ θ = 2πk for some integer k
The modulus and argument are the usual polar co-ordinates of a point in R
2
. The exponential
laws show that the polar form behaves nicely with respect to complex multiplication:
|
zw
|
=
|
z
||
w
|
and arg(zw) ≡ arg z + arg w (mod 2π)
Euler’s Formula & the Unit Circle: When r = 1 we have Euler’s formula:
e
i θ
= cos θ + i sin θ
the source of the famous identity e
i π
= −1. These complex num-
bers comprise the unit circle
S
1
=
z ∈ C :
|
z
|
= 1
=
e
i θ
: θ ∈ [0, 2π)
−1 1
S
1
e
iθ
1
θ
i
−i
Rotations: Let θ be a fixed real number. The complex function rot
θ
(z) = e
i θ
z has the effect of rotating
z about the origin by θ radians. To see why, write z = re
i ψ
in polar form and observe that
rot
θ
(z) = e
i θ
re
i ψ
= re
i (θ+ψ )
has the same modulus r, but has had θ radians added to its modulus. For this reason, the unit circle
S
1
can be thought of as the set of rotations around the origin.
