Representation of Complex Numbers
Algebra of Complex Numbers
Complex Numbers are Not Ordered
Powers and Roots
The Fundamental Theorem of Algebra
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. The complex number z in the complex plane.
z1 = x1 + i.y1 = r1(cos theta1 + i.sin theta1),Then we have the following laws:
z2 = x2 + i.y2 = r2(cos theta2 + i.sin theta2).
z = x + i.y = r.(cos theta + i.sin theta).Then we have in the field:
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.
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.
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.
z2 - 2.z + 2 = (z - 1 - i).(z - 1 + i).
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.