Representation of Complex Numbers

Algebra of Complex Numbers

Complex Numbers are Not Ordered

Complex Conjugate

Powers and Roots

The Fundamental Theorem of Algebra

Sequences

z = x + i.y,where

x = Re(z), y = Im(z).The complex number

z = r.coswhere rtheta+ i.r.sintheta,

r = | z |,A complex number may be represented by a point in thetheta= arg(z).

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

zThen we have the following laws:_{1}= x_{1}+ i.y_{1}= r_{1}(costheta_{1}+ i.sintheta_{1}),

z_{2}= x_{2}+ i.y_{2}= r_{2}(costheta_{2}+ i.sintheta_{2}).

- Equality:
- z
_{1}= z_{2}if and only if x_{1}= x_{2}and y_{1}= y_{2}. - Addition:
- z
_{1}+ z_{2}= (x_{1}+ x_{2}) + i.(y_{1}+ y_{2}). - Subtraction:
- z
_{1}- z_{2}= (x_{1}- x_{2}) + i.(y_{1}- y_{2}). - Multiplication:
- z
_{1}.z_{2}= (x_{1}x_{2}- y_{1}y_{2}) + i.(x_{1}y_{2}+ x_{2}y_{1})

= r_{1}.r_{2}.[cos(*theta*_{1}+*theta*_{2}) + i.sin(*theta*_{1}+*theta*_{2})]. - Division:
- z
_{1}/z_{2}= [(x_{1}x_{2}+ y_{1}y_{2}) + i.(x_{2}y_{1}- x_{1}y_{2})] / (x_{2}^{2}+ y_{2}^{2})

= (r_{1}/r_{2}).[cos(*theta*_{1}-*theta*_{2}) + i.sin(*theta*_{1}-*theta*_{2})].

z = x + i.y = r.(cosThen we have in the field:theta+ i.sintheta).

- 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)/(x
^{2}+ y^{2}) = (1/r)(cos*theta*- i.sin*theta*).

- Suppose i < 0. Then -i > 0, and we have (-i).(-i) > 0, which gives -1 > 0. This is false.
- Suppose i = 0. Then the laws of order give i.i = 0. But this is false because i
^{2}= -1. - Suppose i > 0. Then we have i.i > 0, which again gives -1 > 0. This is false.

Given

z = x + i.y = r.(coswe definetheta+ i.sintheta),

z* = x - i.y = r.(cosIn other words:theta- i.sintheta).

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.(cosThentheta+ i.sintheta).

zand the^{n}= r^{n}[cos(n.theta) + i.sin(n.theta)],

zwhere^{1/n}= r^{1/n}[cosphi+ i.sinphi],

Note that *z*^{1/n} has *n* different values given by the *n* values of *k*.

zwhere the coefficients^{n}+ a_{1}z^{n-1}+ ... + a_{n-1}z + a_{n}= 0,

Let *z*_{1} be a solution of the above polynomial equation. Then the polynomial equation can be written in the form:

(z - zwhere the_{1}).(z^{n-1}+ b_{1}z^{n-2}+ ... + b_{n-2}z + b_{n-1}) = 0,

Now, applying the fundamental theorem of algebra again, we find that there exists a complex number *z*_{2} which satisfies the polynomial equation

zBy repeating this process we find that the original polynomial equation can be written in the form:^{n-1}+ b_{1}z^{n-2}+ ... + b_{n-2}z + b_{n-1}= 0.

(z - zThe complex numbers_{1}).(z - z_{2})...(z - z_{n}) = 0.

z^{2}- 2.z + 2 = (z - 1 - i).(z - 1 + i).

A complex number *w* is a limit of the sequence *z _{k}* if and only if, for all

Let Re(*z _{k}*) =

lim zThe following properties of limits of complex sequences are the same as the properties of limits of real sequences:_{k}= lim x_{k}+ i.lim y_{k}.

- The limit of a complex sequence (when it exists) is unique.
- If lim w
_{k}= w and lim z_{k}= z, then lim(w_{k}+ z_{k}) = w + z. - If lim w
_{k}= w and lim z_{k}= z, then lim(w_{k}.z_{k}) = w.z. - A complex sequence has a limit if and only if it is a Cauchy sequence. Therefore
**the complex number system is complete**.

Home Page Natural Numbers Real Numbers Real Variables Complex Variables

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