In this explainer, we will learn how to multiply two complex numbers.
The algebra of complex numbers is very similar to the algebra of binomials. Hence, applying the knowledge we have of working with binomials will take us a long way when working with complex numbers. Before we look at multiplication of complex numbers in full generality, we will consider the simpler cases of a complex number multiplied by a real number and a complex number multiplied by a purely imaginary number.
Consider a complex number . If we multiply both sides of this equation by a real number , we get . Using the distributive property, we can rewrite this as much as we would expect.
Example 1: Multiplying Complex Numbers by Real Numbers
If and , find .
Answer
Substituting in the values of and into the expression, we get
Expanding the parentheses using the distributive property, we get
Finally, we can gather our like terms to find
Having considered the simplest case of multiplying a complex number by a real number, we can now consider multiplying a complex number by a purely imaginary number. Taking , we can multiply both sides of the equation by a purely imaginary number to get . Once again, we can use the distributive property to rewrite this as
Since , we can simplify this to
The formulae for multiplying complex numbers by real and imaginary numbers are not formulae that you need to learn. Instead, we should focus on becoming familiar with the algebraic techniques required to work with complex numbers in general.
Example 2: Multiplying Complex Numbers by Imaginary Numbers
What is ?
Answer
Expanding the parentheses using the distributive property, we can write
Since , we can simplify this to
Let us now consider the product of two general complex numbers and :
Using any technique for multiplying two binomials (FOIL, grid/area, or long multiplication), we can express this as
Using the fact that , and gathering our like terms, we can rewrite this as
We summarize this as follows.
The Product of Two Complex Numbers
For two complex numbers and , we define the product
Although we have stated a general form, rather than simply memorizing it, it is more important to be familiar with the techniques of multiplying complex numbers.
Now, let us consider an example where we demonstrate the multiplication of two complex numbers.
Example 3: Multiplying Complex Numbers
Multiply by .
Answer
Using FOIL, or any other technique, we can expand the parentheses as follows:
Since , we can rewrite this as
Finally, gathering our like terms, we have
In the next example, let us look at how to calculate the square of the difference of two complex numbers.
Example 4: Squares of Complex Numbers
If and , find .
Answer
For a question like this, we have a choice: do we multiply out and then substitute in the values of and , or do we do it the other way around? If we try the first method, we will find we have to calculate , , and . This is a lot of computation. However, if we first calculate , we will only need to find the square of the result, which is much more efficient. This is the technique we will use here.
Firstly, we calculate as follows:
To find the square of this number, we can express it as a product and use the technique for multiplying complex numbers. Hence,
Using FOIL or another technique for expanding parentheses, we have
Using the fact that , we can gather our like terms and rewrite this as
This example raises the question of what the general form of the square of a complex number is. Using the techniques we have been developing to multiply complex numbers, we can derive a formula for it as follows:
Expanding the parentheses, we get
Gathering our like terms, we can state the general form as follows.
The Square of a Complex Number
For a complex number ,
Even though it is very important to know the techniques needed to derive equations like this, it can also be useful to commit the general form of the square of a complex number to memory. In the next example, we will see how remembering the formula can simplify calculations.
Now, let us consider an example where we have to calculate the real part of the square of a complex number.
Example 5: Squares of Complex Numbers
Find .
Answer
We begin by calculating the square of as follows:
Expanding the parentheses, we have
Using the fact that , and gathering like terms, we have
Finally, taking the real part, we get
Alternatively, we could save ourselves some calculation by using the fact that, for a complex number , . Taking the real part, we have
Hence,
Example 6: Powers of Complex Numbers
If , express in the form .
Answer
We start by calculating :
Expanding the parentheses, we have
To calculate , we can now multiply both sides of the equation by to get Multiplying out the parentheses, we get
Clearly, working like this to calculate higher and higher powers of complex numbers could be quite laborious. However, as we learn more about complex numbers we will learn alternative techniques which will significantly simplify the process. We finish by looking at one last example where we can apply what we know about complex numbers and multiplication to solve an equation involving complex numbers.
Example 7: Solving Equations Involving Complex Numbers
Solve the equation .
Answer
On first impression, we might think that we need to divide both sides of the equation by to isolate . However, by remembering that , we could simply multiply the whole equation by or, better yet, by .
Hence, by multiplying both sides of the equation by , we get
Expanding the parentheses, and simplifying, we arrive at the solution:
It is always good practice to check our answer. To do this, we can multiply our solution by and check that we get back our original equation:
Expanding the parentheses and simplifying, we find as required.
Key Points
- To multiply complex numbers, we use the same techniques that we use to multiply binomials.
- For two complex numbers and , we define the product as
- For a complex number ,
- Although, theoretically, using these techniques we can calculate arbitrarily large powers of complex numbers, it requires a large amount of calculation.
