NUMBER SYSTEMS AND ANALYSIS

Complex Numbers

Contents

Representation of Complex Numbers
Algebra of Complex Numbers
Complex Numbers are Not Ordered
Complex Conjugate
Powers and Roots
The Fundamental Theorem of Algebra
Sequences

Representation of Complex Numbers

A complex number z has the form
z = x + i.y,
where i2 = -1. The real number x is called the real part of z, and the real number y is called the imaginary part of z. We write:
x = Re(z), y = Im(z).
The complex number z may also be written in the form
z = r.cos theta + i.r.sin theta,
where r2 = x2 + y2, cos theta = x/r, and sin theta = y/r. The real number r is called the absolute value of z, and the real number theta is called the argument of z. We write:
r = | z |, theta = arg(z).
A complex number may be represented by a point in the complex plane (see Fig. 1).

Fig. 1.

Fig. 1. The complex number z in the complex plane.

Algebra of Complex Numbers

The algebra of complex numbers is the same as the algebra of real numbers, except that i2 is always replaced by -1. Let
z1 = x1 + i.y1 = r1(cos theta1 + i.sin theta1),
z2 = x2 + i.y2 = r2(cos theta2 + i.sin theta2).
Then we have the following laws:
Equality:
z1 = z2 if and only if x1 = x2 and y1 = y2.
Addition:
z1 + z2 = (x1 + x2) + i.(y1 + y2).
Subtraction:
z1 - z2 = (x1 - x2) + i.(y1 - y2).
Multiplication:
z1.z2 = (x1x2 - y1y2) + i.(x1y2 + x2y1)
= r1.r2.[cos(theta1 + theta2) + i.sin(theta1 + theta2)].
Division:
z1/z2 = [(x1x2 + y1y2) + i.(x2y1 - x1y2)] / (x22 + y22)
= (r1/r2).[cos(theta1 - theta2) + i.sin(theta1 - theta2)].
It follows from these formulas that the complex number system is a field. Let
z = x + i.y = r.(cos theta + i.sin theta).
Then we have in the field:
the zero complex number:
0 = 0 + i.0,
the negative of z:
-z = -x - i.y,
the unit complex number:
1 = 1 + i.0,
the reciprocal of z:
1/z = (x - i.y)/(x2 + y2) = (1/r)(cos theta - i.sin theta).

Complex Numbers are Not Ordered

Complex numbers cannot be ordered like the real numbers. If the complex numbers could be ordered, then the trichotomy law would give i < 0, or i = 0, or i > 0. None of these relations is acceptable:

Complex Conjugate

The complex conjugate z* of a complex number z is defined as follows:

Given

z = x + i.y = r.(cos theta + i.sin theta),
we define
z* = x - i.y = r.(cos theta - i.sin theta).
In other words:
Re(z*) = Re(z),   Im(z*) = -Im(z),   | z* | = | z |,   arg(z*) = -arg(z).
It follows that:
Re(z) = (½)(z + z*),   Im(z) = -i.(½)(z - z*).
The following formulas are useful:
(z*)* = z,   z.z* = | z |2,   1/z = z* / | z |2.

Powers and Roots

Let n be a natural number, and let
z = r.(cos theta + i.sin theta).
Then
zn = rn[cos(n.theta) + i.sin(n.theta)],
and the n-th roots of z are given by
z1/n = r1/n[cos phi + i.sin phi],
where phi = (theta + 2.pi.k)/n, with k = 0, ..., n - 1.

Note that z1/n has n different values given by the n values of k.

The Fundamental Theorem of Algebra

Although there is no real number x such that x2 = -1, there exists a complex number z satisfying the equation z2 = -1, namely z = i. More generally, there is always a complex number z which satisfies a given polynomial equation:
zn + a1zn-1 + ... + an-1z + an = 0,
where the coefficients ak are complex numbers. This important fact is called the fundamental theorem of algebra.

Let z1 be a solution of the above polynomial equation. Then the polynomial equation can be written in the form:

(z - z1).(zn-1 + b1zn-2 + ... + bn-2z + bn-1) = 0,
where the bk are new complex coefficients.

Now, applying the fundamental theorem of algebra again, we find that there exists a complex number z2 which satisfies the polynomial equation

zn-1 + b1zn-2 + ... + bn-2z + bn-1 = 0.
By repeating this process we find that the original polynomial equation can be written in the form:
(z - z1).(z - z2)...(z - zn) = 0.
The complex numbers z1, z2, ... zn are the roots of the original polynomial.

EXAMPLE

The polynomial z2 - 2.z + 2 has roots z1 = 1 + i, z2 = 1 - i. Therefore
z2 - 2.z + 2 = (z - 1 - i).(z - 1 + i).

Sequences

Let z1, z2, z3, ... be a sequence of complex numbers.

A complex number w is a limit of the sequence zk if and only if, for all epsilon > 0, there exists n such that k > n implies | zk - w | < epsilon.

Let Re(zk) = xk and Im(zk) = yk. Then the sequence of complex numbers z1, z2, ... has a limit if and only if the sequences of real numbers x1, x2, ... and y1, y2, ... have limits. Then we have:

lim zk = lim xk + i.lim yk.
The following properties of limits of complex sequences are the same as the properties of limits of real sequences:

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By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.