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Multiplying Complex Numbers

Multiplying Complex Numbers

Author: Colleen Atakpu

In this lesson, students will learn how to multiply complex numbers with the imaginary unit i.

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Video Transcription

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Let's look at our objectives for today. We'll start by reviewing imaginary and complex numbers. We'll then look at how we multiply imaginary and complex numbers. And finally we'll do some examples multiplying complex numbers using FOIL.

Let's start by reviewing imaginary and complex numbers. The square root of a negative number is non-real or an imaginary number. We define the imaginary unit i as i is equal to the square root of negative 1. If we were to square both sides of this equation, we would have i squared on the left side, and negative 1 on the right side. So we also know that i squared is equal to negative 1.

A complex number is a value in the form a plus bi, where a and b are real numbers, and i is the imaginary unit. We use complex numbers in fields such as engineering and physics.

Now let's do an example multiplying an imaginary number by a complex number. We want to multiply 2i times 5 plus 3i. We'll multiply these numbers together using distribution. So we first multiply 2i times 5. This gives us 10i. We then multiply 2i times 3i. 2 times 3 gives us 6 and i times i gives us i squared. We've then simplify by remembering that i squared is equal to negative 1. So this gives us 10i plus 6 times negative 1. Multiplying 6 times negative 1 gives us 10i minus 6. And then we write this complex number in standard form with the real part first, negative 6, and the imaginary part second, 10i. So our final answer is negative 6 plus 10i.

Finally, let's do an example multiplying complex numbers together using FOIL. We want to multiply the complex number 8 minus 4i by the complex number 2 plus 6i. We're going to use the FOIL method because of the addition or subtraction that occurs between the real and imaginary parts of complex numbers. So multiplying complex numbers together is similar to multiplying binomials together.

Using FOIL, we start by multiplying 8 times 2, which gives us 16. We then multiply 8 times 6i which is 48i. Negative 4i times 2 gives us negative 8i. And negative 4i times 6i gives us negative 24i squared. We recall that i squared is equal to negative 1, so we can substitute negative 1 in for i squared. We then multiply negative 24 times negative 1, which gives us a negative 24.

To simplify our expression, we know that the real parts are like terms, so we can combine 16 minus negative 24 to give us a positive 40. We also know that our imaginary parts are like terms so we can combine 48i minus 8i which gives us 40i. So our final answer is 40 plus 40i.

Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. We know that from the definition of the imaginary unit i, i squared is equal to negative 1. A complex number is a value in the form a plus b times i. When multiplying an imaginary and a complex number together, we use distribution. And when multiplying two complex numbers together we use the FOIL method.

So I hope that these important points and examples helped you understand a little bit more about multiplying complex numbers. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.

Formulas to Know
Imaginary Number

i equals square root of negative 1 end root

Imaginary Number (alternate)

i squared equals negative 1