User:Aleksander Stos/ComplexNumberAdvanced

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Revision as of 08:36, 13 August 2007 by imported>Aleksander Stos (expo)
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This is an experimental draft. For a brief description of the project and motivations click here.

Complex numbers are defined as ordered pairs of reals:

Such pairs can be added and multiplied as follows

  • addition:
  • multiplication:

with the addition and multiplication is the field of complex numbers. From another of view, with complex additions and multiplication by real numbers is a 2-dimesional vector space.

To perform basic computations it is convenient to introduce the imaginary unit, i=(0,1).[1] It has the property Any complex number can be written as (this is often called the algebraic form) and vice-versa. The numbers a and b are called the real part and the imaginary part of z, respectively. We denote and Notice that i makes the multiplication quite natural:

The square root of number in the denominator in the above formula is called the modulus of z and denoted by ,

We have for any two complex numbers and

  • provided

For we define also , the conjugate, by Then we have

  • provided
Geometric interpretation

Complex numbers may be naturally represented on the complex plane, where corresponds to the point (x,y), see the fig. 1.

Fig. 1. Graphical representation of a complex number and its conjugate

Obviously, the conjugation is just the symmetry with respect to the x-axis.

Trigonometric and exponential form

As the graphical representation suggests, any complex number z=a+bi of modulus 1 (i.e. a point from the unit circle) can be written as for some So actually any (non-null) can be represented as

where r traditionally stands for |z|.

This is the trigonometric form of the complex number z. If we adopt convention that then such is unique and called the argument of z.[2] Graphically, the number is the (oriented) angle between the x-axis and the interval containing 0 and z. Closely related is the exponential notation. If we define complex exponential as

then it may be shown that

Consequently, any (non-zero) can be written as

with the same r and theta as above.

This is called the exponential form of the complex number z.


The trigonometric form is particularly well adapted to perform multiplication. If <math>z_1=r_1(\sin(

  1. in some applications it is denoted by j as well.
  2. In literature the convention is found as well.