# Rectangular and Polar Forms

Before we work with reactance, we need to acquire some basic facts about complex numbers. Any complex number may be written in the following form:

a+jb

This form is called the rectangular form of a complex number.

Real numbers can be represented by a point on a number line.  Figure C-2 below shows the number 3 as a point on a number line.

Figure C-2: Real Number Line

Complex numbers can be represented using two number lines, one for the real part and one for the imaginary part. The two number lines are at right angles to each other, and the complex number is a point on the plane containing the number lines. Figure C-3 shows how the complex number 3+j4 is plotted on this graph. The horizontal axis is the real part and the vertical axis is the imaginary part.

Figure C-3: Graphical Representation of a complex number

The rectangular form of a complex number is very useful but there is another form of a complex number that is also used in electronics. It is called the polar form of a complex number. Consider Figure C-4. If we draw a line from the origin to the point representing the complex number, that line has a length that is related to the real and imaginary parts of the complex number. That length is called the magnitude of the complex number.

Figure C-4: Rectangular and Polar Forms of 3+j4

The line that represents the magnitude of the complex number makes an angle with respect to the real axis that is called the argument or phase angle of the complex number. The magnitude and argument taken together are called the polar form of a complex number. The polar form is written like this:

M is the magnitude of the complex number and q is the argument. The Ð symbol indicates that q is an angle. It is possible to convert from rectangular to polar form by using the following equations

The following equations can be used to convert from polar form back to rectangular form:

Here is an example:

Convert 3+j4 to polar form:

M = Ö(3*3 + 4*4)=Ö(9 + 16)=Ö25 = 5

q = arctan (4/3) = arctan (1.333) = 53.1 deg

Whether we write 3+j4 or 5Ð53.1deg, we are writing down the same number.

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