REVIEW OF COMPLEX NUMBERS
1. Complex numbers
A complex number has the form
z = x + iy
where x and y are ordinary real numbers. Here the symbol i stands for
an “imaginary” number satisfying i2 = −1. The real numbers x and y
are the real and imaginary parts of z:
x = Re z,
y = Im z.
The set of all complex numbers is denoted by C. I will assume that
you have a basic familiarity with arithmetic of complex numbers.
1.1. Cartesian representation. Complex numbers can be identified
with points in the Cartesian plane by identifying z = x + iy with the
point (x, y), as illustrated in Figure 1.
The modulus or absolute value of a complex number is
p
√
|z| = |x + iy| = zz = x2 + y 2 .
Geometrically, this is the distance from z to the origin.
The complex conjugate of a complex number z = x + iy is given by
z = x + iy = x − iy.
Geometrically, conjugation is reflection across the real axis as shown in
Figure 2. You can easily check the following important properties of
Imaginary axis
z=x+iy
iy
Real axis
x
Figure 1. Cartesian representation
1
REVIEW OF COMPLEX NUMBERS
2
z=x+iy
|z|
z=x−iy
Figure 2. Conjugation and modulus
conjugation:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
z=z
z+w =z+w
zw = z w
zz = |z|2
Re z = 21 (z + z)
Im z = 2i1 (z − z)
|zw| = |z| |w|
(Triangle Inequality) |z + w| ≤ |z| + |w|
1.2. Division. Property 4 above can be used to calculate the quotient
of two complex numbers. Here is the idea. It is easy to divide by a
nonzero real number, and by multiplying the numerator and denominator both by the conjugate of the denominator, you can arrange, by
property 4 above, for the denominator to be a positive real number:
z
zw
zw
=
=
.
w
ww
|w|2
Exercise 1. Calculate
with a and b real.
1+2i
.
3+4i
Express your answer in the form a + ib
1.3. Polar representation. Representation of complex numbers in
polar form is facilitated by Euler’s Formula
eiθ = cos θ + i sin θ.
REVIEW OF COMPLEX NUMBERS
3
z=rei θ
r
θ
Figure 3. Polar representation
You can take this as the definition1 of eiθ . You should check, using sum
formulas for the sine and cosine that
ei(θ+φ) = eiθ eiφ .
If the complex number z has polar coordinates (r, θ), then its real
and imaginary parts are
x = r cos θ,
y = r sin θ
so
z = x + iy = r cos θ + ir sin θ = reiθ .
Here r = |z|. The polar angle θ is called the argument of z, and is
denoted by θ = arg z. Of course this comes with the usual ambiguity
associated with the polar angle: It’s only defined up to addition of a
multiple of 2π. The representation z = reiθ is the polar representation
of the complex number z. It is illustrated in Figure 3.
1It
may bother you to take this as the definition of eiθ , since the exponential
function already has a definition. However, the exponential function you studied
in calculus was only defined when the exponent was a real number. Here we’re
assuming that θ is real, so the exponent iθ is pure imaginary. The only time the
exponent is real is when θ = 0, and in this case Euler’s Formula gives a result of 1,
which is consistent with the “old” definition for real exponents. In more advanced
courses, the exponential function is defined by the power series
ez =
∞
X
zn
n!
n=0
for any complex number z. If you took this as the definition, then Euler’s Formula
would require proof! You are invited to supply the proof yourself. Hint: Use the
power series representations for the sine and cosine that you learned in calculus.
REVIEW OF COMPLEX NUMBERS
4
2. Algebraic completeness
A polynomial is a function on C of the form
p(z) = an z n + an−1 z n−1 + · · · + a0
with a0 , . . . , an ∈ C. If an 6= 0, then n is the degree of the polynomial
p.
A complex number λ is a root of p if p(λ) = 0.
Theorem 1 (Fundamental Theorem of Algebra). Every non-constant
polynomial has a root.
Corollary 2. If p is a polynomial of degree n > 0, there are complex
numnbers a, λ1 , . . . , λn such that
p(z) = a(z − λ1 ) · · · (z − λn ).