Complex numbers are written in polar (or trigonometric) form. The multiplications and divisions of complex numbers in polar forms are explained through examples and reinforced through questions with detailed solutions.
Complex Numbers in Polar Form
Let us represent the complex number where in the complex plane which is a system of rectangular axes, such that the real part is the coordinate on the horizontal axis and the imaginary part the coordinate on the vertical axis as shown below.
where
, is called the modulus of
, such that , is called argument of .
The reference angle to the argument is an acute angle made by the horizontal axis and the terminal side of and is defined as follows:
If and are known, then and are given by
The complex number may be expressed in polar form involving and as follows
Example 1
a) Plot the complex number on the complex plane and write it in polar form.
Solution to Example 1
See graph below of plot of on complex plane.
We need to find the reference angle in order to find angle .
The real part of is positive and its imaginary part is negative, hence the terminal side of the argument is in quadrant IV (see plot of above).
We now calculate as follows:
(see plot of )
in polar form is given by
Example 2
Write complex number in standard form.
Solution to Example 2
Multiplication and Division of Complex Numbers in Polar Forms
Let and be complex numbers in polar form. Using the sum and difference trigonometric formula for sine and cosine, it can be shown that the product and quotient of these two complex numbers is given by
Example 3
Given and
Find and
Solution to Example 3
Questions
1) Write the following complex numbers in polar forms.
.
2) Use the results in part 1 above to evaluate the following expressions in polar form.
.
Solutions to the Above Questions
1)
Below are the plots of the complex numbers in questions a), b), c) and d). These plots help in explaining how is the argument found for each of these complex numbers that are either real (positive or negative) or pure imaginary (positive or negative imaginary part).
The plot of on the complex plane gives a point at with the terminal side of the corresponding argument (angle) on the vertical positive axis and hence . (see plot above for argument and modulus)
in polar form is:
The plot of on the complex plane gives a point with the terminal side of the corresponding argument (angle) on the horizontal positive axis and hence . (see plot above for argument and modulus)
in polar form is:
(see plot above for argument and modulus)
(see plot above for argument and modulus)
Find the reference angle defined as the angle between the horizontal axis and the terminal side of the argument .
The real part and the imaginary parts are positive and therefore the terminal side of the argument is in quadrant I. Hence
Plot of with more explanations is shown below.
The reference angle defined as the angle between the horizontal axis and the terminal side of the argument .
The real part and the imaginary parts are both negative and therefore the terminal side of the argument is in quadrant III. Hence
Find the reference angle
The real part is negative and the imaginary part is positive and therefore the terminal side of the argument is in quadrant II. Hence
.
2)
and have already been expressed in polar form, we therefore use the formula for the product of two complex numbers in polar for given above.
Substitute , and by their polar form found above and use both formulas for the product in the numerator and the division