Lesson Multiplication and division of complex numbers in complex plane

Multiplication and Division of complex numbers in the complex plane

In this lesson you will learn about the geometric interpretation for multiplication and division of complex numbers.

Let me remind you that the set of complex numbers and arithmetical operations over them were introduced in the lesson Complex numbers and arithmetical operations in this module. The geometric presentation of complex numbers in the complex plane was introduced in the lesson Complex plane in this module.
We refer to this lesson for the trigonometric form of complex numbers.

Multiplication of complex numbers in the complex plane

Let us consider complex numbers u=a+bi and v=c+di. These complex numbers are pictured in Figure 1 by vectors OM and ON. Let ,        are trigonometric forms of these complex numbers in the complex plane XOY (see Figure 1). So, the complex number u has the modulus r and the argument (angle XOM in Figure 1), while the complex number v has the modulus s and the argument (angle XON in Figure 1). Perform multiplication of the complex numbers u and v using the multiplication definition of complex numbers (lesson Complex numbers and arithmetical operations). You get .

Figure 1. Multiplication of complex numbers in the Complex Plane

There are the following formulas in Trigonometry:

, and
.

Hence,

.

This is the multiplication formula for complex numbers in the trigonometric form.
Very simple and nice, isn't?

Point P and the vector XOP in Figure 1 (shown in blue) depict the product of complex numbers u and v. The length of the vector XOP is equal to rs (the modulus of the product uv) and the angle XOP is equal to the sum of angles XOM and XON (the argument of the product uv).

Examples
1) In Figure 1
    .    The modulus and the argument = 26.6°.
    . The modulus and the argument = 45°.
    .               The modulus and the argument = 71.6°.

2) Calculate .
    The modulus of is equal and the argument is equal to =45°.
    So, the modulus of is equal to 2 and the argument is equal to 90°.
    The answer is: .

Summary

The geometric interpretation of multiplication of complex numbers is read as follows:
the result of multiplication of two complex numbers with modulus r and s and arguments and is the complex number with the modulus rs and the argument : .

Division of complex numbers in the complex plane

Let us consider division of the complex number u=a+bi by the complex number v=c+di.

These complex numbers are pictured in Figure 2 by vectors OM and ON and have trigonometric forms and . The complex number u has the modulus r and the argument (angle XOM in Figure 2), while the complex number v has the modulus s and the argument (angle XON in Figure 2).

As you may just know (for example, from the lesson Complex numbers and arithmetical operations over them in this module), to divide the complex number u=a+bi by the complex number v=c+di is the same as to multiply u by the inverse complex number of v.

The inverse of the complex number is the complex number (see the same lesson we referred right above). Note that the inverse complex number has the inverse modulus and the opposite argument value. Now you can apply the rule of multiplication of complex numbers in the trigonometric form, which was derived in the previous section of this lesson. By doing so, you get =       ==       =.

Figure 2. Division of complex numbers in the Complex Plane

This is the division formula for complex numbers in the trigonometric form.

Point P and the vector XOP in Figure 2 (shown in blue) depict the quotient of complex numbers u and v. The length of the vector XOP is equal to r/s (the modulus of the quotient u/v) and the angle XOP is equal to the difference of angles XOM and XON (the argument of the quotient u/v).

Example 3
Use the same complex numbers u and v as in Example 1 above:
    .    The modulus and the argument = 26.6°.
    . The modulus and the argument = 45°.
    .             The modulus and the argument = -18.4°.

Summary

The geometric interpretation of division of complex numbers is read as follows:
the result of division of the complex number with the modulus r and the argument by the complex number with the modulus s and the argument is the complex number with the modulus r/s and the argument : .



For your convenience, below is the list of my relevant lessons on complex numbers in this site in the logical order. They all are under the current topic Complex numbers in the section Algebra II.
Complex numbers and arithmetical operations over them
Complex plane
Addition and subtraction of complex numbers in complex plane
Multiplication and division of complex numbers in complex plane                          (this lesson)
Raising a complex number to an integer power
How to take a root of a complex number
Solution of the quadratic equation with real coefficients on complex domain
How to take a square root of a complex number
Solution of the quadratic equation with complex coefficients on complex domain