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Study Guides > College Algebra

Add, Subtract, and Multiply Complex Numbers

Learning Objectives

  • Add and subtract complex numbers
  • Multiply complex numbers
Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

A General Note: Addition and Subtraction of Complex Numbers

Adding complex numbers:

(a+bi)+(c+di)=(a+c)+(b+d)i\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i

Subtracting complex numbers:

(a+bi)−(c+di)=(a−c)+(b−d)i\left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i

How To: Given two complex numbers, find the sum or difference.

  1. Identify the real and imaginary parts of each number.
  2. Add or subtract the real parts.
  3. Add or subtract the imaginary parts.

Example: Adding Complex Numbers

Add 3−4i3 - 4i and 2+5i2+5i.

Answer: We add the real parts and add the imaginary parts.

(a+bi)+(c+di)=(a+c)+(b+d)i(3−4i)+(2+5i)=(3+2)+(−4+5)i =5+i\begin{array}{l}\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i\hfill \\ \left(3 - 4i\right)+\left(2+5i\right)=\left(3+2\right)+\left(-4+5\right)i\hfill \\ \text{ }=5+i\hfill \end{array}

Try It

Subtract 2+5i2+5i from 3−4i3 - 4i.

Answer: (3−4i)−(2+5i)=1−9i\left(3 - 4i\right)-\left(2+5i\right)=1 - 9i

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

(a+bi)(c+di)=ac+adi+bci+bdi2\left(a+bi\right)\left(c+di\right)=ac+adi+bci+bd{i}^{2}

Because i2=−1{i}^{2}=-1, we have

(a+bi)(c+di)=ac+adi+bci−bd\left(a+bi\right)\left(c+di\right)=ac+adi+bci-bd

To simplify, we combine the real parts, and we combine the imaginary parts.

(a+bi)(c+di)=(ac−bd)+(ad+bc)i\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i

How To: Given two complex numbers, multiply to find the product.

  1. Use the distributive property or the FOIL method.
  2. Simplify.

Example: Multiplying a Complex Number by a Complex Number

Multiply (4+3i)(2−5i)\left(4+3i\right)\left(2 - 5i\right).

Answer:

Use (a+bi)(c+di)=(ac−bd)+(ad+bc)i\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i

(4+3i)(2−5i)=(4⋅2−3⋅(−5))+(4⋅(−5)+3⋅2)i=(8+15)+(−20+6)i=23−14i\begin{array}{c}\left(4+3i\right)\left(2 - 5i\right)=\left(4\cdot 2 - 3\cdot \left(-5\right)\right)+\left(4\cdot \left(-5\right)+3\cdot 2\right)i\hfill \\ =\left(8+15\right)+\left(-20+6\right)i\hfill \\ =23 - 14i\hfill \end{array}

Try It

Multiply (3−4i)(2+3i)\left(3 - 4i\right)\left(2+3i\right).

Answer: 18+i18+i

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