Complex Numbers

Definition of a complex number
A complex number z is defined to be z = (x,y), where x and y are both real numbers.
The ordered pair (x,y) can be equated with x + iy, where i = (-1)1/2 (the square root of -1).

Let z1 = (x1,y1) and z2 = (x2,y2)

Definition for addition

    z1 + z2 = (x1,y1) + (x2,y2)
      = (x1 + iy1) + (x2 + iy2)
      = (x1 + x2) + i(y1 + y2)
      = (x1 + x2, y1 + y2)

Definition for subtraction

    z1 - z2 = (x1,y1) - (x2,y2)
      = (x1 - x2, y1 - y2)
Definition for multiplication
    z1z2 = (x1,y1)(x2,y2)
      = (x1x2 - y1y2, x1y2 + x2y1)
Definition for division
    z1 / z2 = (x1,y1) / (x2,y2)
      = ( (x1x2 + y1y2) / (x22 + y22) , -(x1y2 + x2y1) / (x22 + y22)

Properties of complex numbers

    1. Comutative law for addition: z1 + z2 = z2 + z1
    2. Associative law for addition: z1 + (z2 + z3) = (z1 + z2) + z3
    3. Additive identity: There is a complex number w such that z + w = z. The ordered pair (0,0) satisifies that equation.
    4. Additive inverse: Given any complex number z, there is a complex number e such that z + e = (0,0). If z = (x, y), then e = <-x, -y).
    5. Commutative law for multiplication: z1z2 = z2z1
    6. Assocative law for multiplication: z1(z2z3) = (z1z2)z3
    7. Multiplcative identity: There is a complex number z such that zz = z for all complex numbers z. z = (1,0) satisfies that equation.
    8. Multiplicative inverses: Given any complex number z other than (0,0), there is a complex number z-1 such that zz-1 = (1,0). i.e. z-1 = (1,0) / z
    9. The distributive law: z1(z2 + z3) = z1z2 + z1z3
Some standard operations on complex numbers
Suppose z = (x, y).
    The real part of z, denoted by Re(z), is the real number x.
    The imaginary part of z, denoted by Im(z), is the real number y.
    The conjugate of z denoted by drawing a bar over z, is (x, -y). 
    _
z = (x, -y)

Examples

Modulus
The modulus or absolute value of the complex number z = x + iy is given by the equation:
    |z| = (x2 + y2)1/2

Polar coordinates
In addition to the representation of complex numbers by Cartesian coordinates, polar representation of z also exists. Let r be the modulus or z. Also, let q be the angle that the line from the origin to the complex number z makes with the positive x axis. Then z = x + iy = (r cos q, r sin q).

Argument of z
The q is the argument of z and:
    q = arg z = arctan y/x, when x not equal to 0


Exponential form
Because eiq = (cos q , sin q) , a complex number z may also be represented in an exponential form:
    z = reiq

Examples

Introductory Exercises Moderate Exercises
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