Learn how to multiply two complex numbers. For example, multiply (1+2i)⋅(3+i).
A complex number is any number that can be written as a+bi, where i is the imaginary unit and a and b are real numbers.
When multiplying complex numbers, it's useful to remember that the properties we use when performing arithmetic with real numbers work similarly for complex numbers.
Sometimes, thinking of i as a variable, like x, is helpful. Then, with just a few adjustments at the end, we can multiply just as we'd expect. Let's take a closer look at this by walking through several examples.
Multiplying a real number by a complex number
Multiply −4(13+5i). Write the resulting number in the form of a+bi.
If your instinct tells you to distribute the −4, your instinct would be right! Let's do that!
And that's it! We used the distributive property to multiply a real number by a complex number. Let's try something a little more complicated.
Multiplying a pure imaginary number by a complex number
Multiply 2i(3−8i). Write the resulting number in the form of a+bi.
Again, let's start by distributing the 2i to each term in the parentheses.
At this point, the answer is not of the form a+bi since it contains i2.
However, we know that i2=−1. Let's substitute and see where that gets us.
Using the commutative property, we can write the answer as 16+6i, and so we have that 2i(3−8i)=16+6i.
Check your understanding
Excellent! We're now ready to step it up even more! What follows is the more typical case that you'll see when you're asked to multiply complex numbers.
Multiplying two complex numbers
Multiply (1+4i)(5+i). Write the resulting number in the form of a+bi.
In this example, some find it very helpful to think of i as a variable.
In fact, the process of multiplying these two complex numbers is very similar to multiplying two binomials! Multiply each term in the first number by each term in the second number.