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Multiply Complex Numbers

Multiply Complex Numbers

Author: Sophia Tutorial

This lesson covers multiplying complex numbers.

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What's Covered

  • Complex Numbers
  • FOIL Review
  • Multiplying Complex Numbers

Multiply Complex Numbers

Complex Numbers

A complex number is a number in the form a plus b i, containing both a real part and an imaginary part.  The imaginary part is followed by i, which is the imaginary unit, square root of negative 1 end root

When multiplying complex numbers, we follow a similar process when multiplying binomial factors we may be familiar with when studying quadratics.  The multiplication process is often referred to as FOIL, which distributes terms into the factors being multiplied.  Let's take a moment to review FOIL with real numbers before looking at examples of complex number multiplication

FOIL Review

FOIL stands for First, Outside, Inside, Last, and refers to terms that are multiplied together to form individual addends to the product.  Here is an example:

When multiplying two complex numbers, we will be following the same procedure, but will need to make an additional consideration when the imaginary unit is squared. 

Multiplying Complex Numbers

Let's multiply the complex numbers left parenthesis 2 plus 3 i right parenthesis and left parenthesis 4 plus 2 i right parenthesis:

The final step here is to simplify the last term, containing the imaginary unit squared.  Recall that the imaginary unit is square root of negative 1 end root.   When this is squared, it becomes the real number negative 1.  


To simplify i squared terms, we can remove i squared completely, but reverse the sign of its coefficient.  For example, plus 6 i squared simplifies to negative 6.  This is a real number that can be combined with other like terms.

Big Idea

To multiply complex numbers, we use the FOIL process to multiply the terms in the two complex numbers.  During this process, we simplify i squared to negative 1, which is a real number.  

Formulas to Know
Imaginary Number

i equals square root of negative 1 end root

i squared equals negative 1