Define the unit imaginary number, j, by the following equation:
j2 = -1
Therefore, j = the square root of -1
j is an imaginary number of magnitude 1.
Then define a general complex number, z, such that:
z = x + jy is a complex number
x = the real part of the complex number z
y = the imaginary part of the complex number z
The complex conjugate of, z, denoted by, z*, is obtained by changing the sign of the imaginary part of z. Thus:
z* = x - jy
Complex numbers can be plotted as two dimensional vectors in the complex x-y plane. The x-axis is the real axis and the y-axis is the imaginary axis. The expression for the complex number, z, given above is said to be in rectangular form. When plotted in the complex plane, this complex number represents a vector from the origin to a point that is x units to the right and y units above the origin. This vector will have a length or magnitude, c, which is the hypotenuse of the right triangle with legs x and y. Therefore:
c2 = x2 + y2
The direction of the vector is given by the angle, theta, between the positive real axis (x-axis) and the positive direction of the vector. It can be found as:
q (theta) = tan-1 (y/x) on a calculator the tan-1 (y/x) function only gives a two quadrant result between -90o and +90o for values in the second and third quadrants you must adjust the result unless your calculator has a four quadrant function such as an, ATAN2(x,y), function used in most programming languages.
The complex number can be represented in polar form as a vector of length c and an angle theta, q.
c is the magnitude of the complex number or vector and theta is the angle of the complex vector.
Arithmetic on the complex numbers in rectangular form can be performed as follows:
Let z1 = x1 + jy1 and z2 = x2 + jy2
Then addition and subtraction are given by;
z1 ± z2 = (x1 ± x2) + j(y1 ± y2)
Multiplication can be performed as:
z1 · z2 = (x1 + jy1)·(x2 + jy2) = (x1· x2 - y1· y2) + j(x1· y2 + y1· x2)
Multiplication is easier to perform in polar form as shown in a later section.
Let z1 = z = x + jy and z2 = z* = x - jy then:
z · z* = (x + jy)·(x - jy) = (x · x - y ·(-y)) + j(x · (-y) + y · x) = x2 + y2 = c2
Therefore the product of a complex number and its complex conjugate is equal to the square of the magnitude of the complex number.
z · z* = |z|2 = c2, This is a real number or scalar.
Division performed in rectangular form is more complicated than multiplication. The numerator and denominator must both be multiplied by the complex conjugate of the denominator. This results in a new denominator that is real and equal to the square of the magnitude of the original denominator.
Dividing both the real part and the imaginary part of the numerator by the denominator leaves the rectangular components of the quotient.
Converting the vector from rectangular to polar form:
In polar form the complex numbers can be represented in two ways.
The second form is derived from Euler's Identity:
Going back to the right triangle with legs, x. and, y, and hypotenuse, c, with angle theta, q, between c and x. The polar to rectangular conversion becomes:
Applying Euler's Identity to the second polar form above converts it back to rectangular form:
Converting to polar form can simplify the operations of multiplication and division of complex numbers, especially if your calculator has the rectangular to polar and polar to rectangular conversion functions built in.
Other useful identities:
