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LECTURE 12: COMPLEX NUMBERS
So far we have concentrated on analysis over the field of real numbers. In our
initial discussion, the construction of the real numbers was given in order to ensure
certain polynomial equations, such as x2 − 2, have solutions. Although we have
developed a rich theory, the real numbers are lacking in one (algebraic) aspect:
there are still certain polynomial equations, a key example being x2 + 1, which do
not have any solutions over R. Here we develop a system of numbers (i.e. a field)
which exends R and in which every polynomial equation has a solution. It turns
out that to do this, one simply has to add a solution, which is denoted i, of the
polynomial x2 + 1 and then extend the algebra of R in a natural way to incorperate
this new number. This leads to the field C of complex numbers.
We will not discuss the construction of C in detail, but describe it rather formally.
Definition. We define C to be the set of all formal expressions a+ib where a, b ∈ R.
We define addition and muliplication on C by
(a + ib) + (c + id) = a + c + i(b + d);
(a + ib) · (c + id) = a · c − b · d + i(a · d + b · c).
The motivation for this definition comes from our wish that i should satisfy
i2 = −1. Indeed, if we assume we can manipulate the expressions a + ib like real
numbers, then the distributive and commutative properties would force
(a + ib) · (c + id) = a(c + id) + ib(c + id) = ac + iad + ibc + i2 bd = ac − bd + i(ad − bc).
Of course, we’re working “the other way round” here, by defining the multiplication
above in such a way so that the field properties have a hope of holding.
Theorem. C satisfies the field axioms P1 - P9 with:
• 0 := 0 + i0;
• 1 := 1 + i0;
• −(a + ib) = −a + i(−b);
• (a + ib)−1 = a/(a2 + b2 ) + i(−b/(a2 + b2 )).
The field R naturally embeds1 into C by identifying a + i0 with a for each a ∈ R.
In general, if z = a + bi ∈ C then a = Rz is the real part of z and b = I is the
imaginary part. We will often write just a or ib rather than a + i0 or 0 + ib and
a − ib rather than a + i(−b), and so on.
The complex numbers can be naturally identified with points on the plane R2 .
In this case, each z ∈ C is identified with (Rz, Iz) ∈ R2 . With this picture, the
embedded real line corresponds to the x-axis.
As suggested by the picture of the complex numbers as a plane, C does not share
the order structure of the reals described by the axioms P10-P12. However, we have
a useful substitute given by the absolute value or modulus function.
Definition. If z = a + ib ∈ C, then the conjugate z̄ ∈ C is defined as z̄ := a − ib.
Note that z z̄ = a2 + b2 can be idenified with a non-negative real number; the
absolute value or modulus of z is then given by
p
√
|z| = z z̄ = a2 + b2 .
1We will not give a precise definition of what an embedding means - but we’ll rely on the
simple intution that we can inject R into C as a set in such a way that the algebraic structure is
preserved.
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LECTURE 12: COMPLEX NUMBERS
Viewing z ∈ C as a point in R2 , the modulus |z| corresponds to the distance of
z from the origin.
Theorem (Properties of conjugates and the modulus). Let z, w ∈ C. Then
(1) z̄¯ = z,
(2) z̄ = z if and only if z is real,
(3) z + w = z̄ + w̄,
(4) z · w = z̄ · w̄,
(5) |z · w| = |z||w|,
(6) (Triangle inequality) |z + w| ≤ |z| + |w|,
Let z ∈ C \ {0} so that z = |z|z 0 where z 0 := z/|z|. Writing z in this way, we
see that |z 0 | = 1 so that z 0 = cos θ + i sin θ for some choice of θ ∈ R. Thus, every
non-zero z ∈ C can be written in the form
z = r(cos θ + i sin θ)
where r = |z| > 0 and θ ∈ R. Of course, there does not exist a unique choice of
θ: if z = r(cos θ0 + i sin θ0 ), then z = r(cos(θ0 + 2πk) + i sin(θ0 + 2πk)) for any
k ∈ Z. Any choice of θ is called an argument of z. If we talk about the argument
of z, denoted arg z, then we refer to the unique value of θ which lies in the interval
[0, 2π).
Writing complex numbers in this way makes the multiplication operation more
transparent: if z, w ∈ C \ {0} are given by
z = r(cos θ + i sin θ)
w = s(cos φ + i sin φ),
then
z · w := rs (cos θ cos φ − sin θ sin φ) + i(sin θ cos φ + cos θ sin φ)
= rs(cos(θ + φ) + i sin(θ + φ)
by the double angle formula. Thus, multiplication by z correpsonds to a combination of scaling by |z| and rotating by an angle determined by arg z.
Theorem (De Moivre’s theorem). Let z ∈ C \ {0}. Then
z n = |z|n cos(nθ) + i sin(nθ)
for any choice of argument θ.
Corollary. Every w ∈ C \ {0} has precisely n complex nth roots.
Proof. Write w = s(cos φ + i sin φ). Then, by De Moivre’s theorem, z = r(cos θ +
i sin θ) is an nth root of w if and only if rn = s and
cos(nθ) + i sin(nθ) = cos(φ) + i sin(φ).
Since nth roots of positive real numbers are uniquely defined, there is only one
possible choice of r. On the other hand, θ can be any number of the form
φ 2πk
for some k ∈ Z.
θk := +
n
n
However, if k 0 = k + mn, for some m ∈ Z then it follows that cos θk + i sin θk =
cos θk0 + i sin θk0 . Consequently, θ0 , . . . , θn−1 yield n distinct choices of nth root,
and these are all possible roots.
Example. Let w = 125(cos π/6 + i sin π/6). Then the third roots of w are given
by
5 cos π/18 + 2πk/3 + i sin(π/18 + 2πk/3)
for k = 0, 1, 2.
LECTURE 12: COMPLEX NUMBERS
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Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]