A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that would be impossible with only real numbers.

Introduction and definition

Complex numbers have been introduced to allow for solutions of certain equations that have no real solution:

the equation x2+1=0

has no real solution x, since the square of x is 0 or positive, so x^2 + 1 cannot be zero. Complex numbers are a solution to this problem. The idea is to enhance the real numbers by adding a number i whose square is −1, so that x = i and x = -i are the two solutions to the preceding equation.
[edit]Definition
A complex number is an expression of the form

where a and b are real numbers and i is a mathematical symbol which is called the imaginary unit. For example, -3.5 + 2i is a complex number.

Elementary operations

Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.
Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:

(a+bi)+(c+di) = (a+c)+(b+d)i

Similarly, subtraction is defined by
(a+bi)-(c+di) = (a-c)+(b-d)i

Multiplication and division

The multiplication of two complex numbers is defined by the following formula:

(a+bi)(c+di) = (ac-bd)+(ad+bc)i

In particular, the square of the imaginary unit is −1:
i^2 = ii = -1

The division of two complex numbers is defined by the following formula:

(a+bi)/(c+di) = (ac+bd)/(c^2+d^2) + i*(bc-ad)/(c^2+d^2)

Complex Matrix

Definition: A matrix whose elements may contain complex numbers.

Let A =  4+i    -2+3i           B =   2-i       3-4i              C = 1+2i        i

            6+4i    -3i                   5+2i   -7+5i                     3-i         8

                                                                               4+2i     1-i

(a)      A+B =  (4+i)+(2-i)           (-2+3i)+(3-4i)       =     6             1-i
                 (6+4i)+(5+2i)          (-3i)+(-7+5i)             11+6i       -7+2i

(b)  CA = (1+2i)(4+i)+(i)(6+4i)              (1+2i)(-2+3i)+(i)(-3i)            =      -2+15i          -5-i

            (3-i)(4+i)+(8)(6+4i)                (3-i)(-2+3i)+(8)(-3i)                     61+31i       -3-13i

           (4+2i)(4+i)+(1-i)(6+4i)            (4+2i)(-2+3i)+(1-i)(-3i)                   24+10i       -17+5i

(c)  (2+i)B  =   (2+i)(2-i)         (2+i)(3-4i)        =       5             10-5i

                   (2+i)(5+2i)        (2+i)(-7+5i)             8+9i         -19+3i

By Keqin Hu