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Multiplying Complex Numbers

 

The process of multiplying complex numbers is very similar when we multiply two binomials using the FOIL Method. The only difference is the introduction of the expression

sqrt(-1)=i

But also, if we square both sides of this equation we get

i^2=-1

This brings us to the conclusion that these two equations mean exactly the same thing. Don't forget this, you'll need this later!

sqrt(-1) = i is equivalent to the equation i^2=-1

 

Let's go over some examples!


 

Example 1: Find the product of the complex numbers (2+5i)(4-3i).

 

Multiply the two binomials using FOIL or any method you prefer. You will have the opportunity to combine like terms. Then replace any instance of i^2 by -1. After doing so, you may need to combine similar terms again especially the new numbers that arise from substituting of i^2 by -1.

 

(2+5i)(4-3i)=17+14i

 


Example 2: Find the product of the complex number 5-2i and its conjugate.

The conjugate of the complex number is the same number but you switch or change the sign of the imaginary component, that is, just the b*i term.

 

Given a complex number Its conjugate
a+bi a-bi
a-bi a+bi

 

Therefore, the conjugate of the given complex number 5-2i must be 5+2i. We can now multiply them to get the product.

(5-2i)(5+2i)=29

 


 

Example 3: Simplify the square of the complex number (-1-i)^2.

The exponent of 2 implies that we are going to multiply the expression inside the parenthesis by itself. Be careful with the signs when you multiply each term. Remember, negative times negative is positive, etc.

(-1-i)^2=2i

 


 

Example 4: Find the product of (3i-4i^2)(-2i+3i^2).

Notice that each term has the expression "imaginary number "i"" attached to it. Don't panic when you see something like this in your homework. Treat it just like the previous problems.

 

  • Start with FOIL
(3i-4i^2)(-2i+3i^2)=-6i^2+9i^3-12i^4
  • Combine like terms
(3i-4i^2)(-2i+3i^2)=-6i^2+17i^3-12i^4
  • We know that i^2=-1. Factor the imaginary terms using this.
(3i-4i^2)(-2i+3i^2)=-6i^2+17(i^2*i)-12(i^2*i^2)
  • Replace i^2 by -1.
(3i-4i^2)(-2i+3i^2)=-6(-1)+17(-1*i)-12(-1*-1)
  • Simplify to get the final answer.
(3i-4i^2)(-2i+3i^2)=-6-17i

 


 

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About This Topic: Multiplying Complex Numbers

 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

 

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