How to Multiply and Divide Complex Numbers? (+FREE Worksheet!)
Learn how to multiply and divide complex numbers into a few simple steps using the following step-by-step guide.
Multiply and Divide Complex Numbers: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Separate partsKeep real and imaginary terms in their own lanes.
- Use i squaredReplace \(i^2\) with -1 whenever it appears.
- Use conjugatesFor division, multiply by the conjugate to make the denominator real.
Worked examples
Add complex numbers
- Add real parts: 4 + 2.
- Add imaginary parts: 3i – 5i.
- Write both parts together.
Use i squared
- Multiply coefficients to get 5.
- i times i is i squared.
- Replace i squared with -1.
Try one before moving on
Multiply and Divide Complex Numbers: pop-up practice
Related Topics
Step-by-step Guide to Multiplying and Dividing Complex Numbers
- Multiplying complex numbers: \(\color{blue}{(a+bi)×(c+di)=(ac-bd)+(ad+bc)i}\)
- Dividing complex numbers: \(\color{blue}{\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2 }+\frac{-ad+bc}{c^2+ d^2}i }\)
- Imaginary number rule: \(\color{blue}{i^2=-1}\)
For education statistics and research
Multiplying and Dividing Complex Numbers – Example 1:
Solve: \(\frac{4-2i}{2+i}=\)
Solution:
Use the rule for dividing complex numbers:
\(\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2 }+\frac{-ad+bc}{c^2+ d^2 } i→\)
\(\frac{4-2i}{2+i}×\frac{2-i}{2-i}=\frac{(4)×(2)+(-2) × (1)}{2^2+ (1)^2 }+\frac{(-2)×(2)-(4) × (1)}{2^2+(1)^2 } i=\frac {8-2} {5}+ \frac{-4-4} {5}i=\frac{6}{5}-\frac{8}{5} i\)
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Multiplying and Dividing Complex Numbers – Example 2:
Solve: \((2-3i)(4-3i)\)
Solution:
Use the rule: \((a+bi) × (c+di)=(ac-bd)+(ad+bc)i\)
\(((2) × (4)-(-3)(-3))+((2) × (-3)+(-3)(4))i= (8-9)+(-6-12)i=-1-18i\)
Multiplying and Dividing Complex Numbers – Example 3:
Solve: \((2-8i)(3-5i)\)
Solution:
Use the rule: \((a+bi) × (c+di)=(ac-bd)+(ad+bc)i\)
\(((2) × (3)-(-8) × (-5))+((2) × (-5)+(-8) × (3))i=(6-40)+(-10-24)i=-34-34i\)
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Multiplying and Dividing Complex Numbers – Example 4:
Solve: \(\frac{2-3i}{2+i}=\)
Solution:
Use the rule for dividing complex numbers:
\(\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2+d^2 }+\frac{-ad+bc}{c^2+ d^2 } i→\)
\(\frac{2-3i}{2+i}×\frac{2-i}{2-i}=\frac{(2)×(2)+(-3) × (1)}{2^2+ (-1)^2 }+\frac{(-3)×(2)+(-2) × (1)}{2^2+(-1)^2 } i=\frac{4-3}{5}-\frac {-6-2}{5}i=\frac{1}{5}-\frac{8}{5} i\)
Exercises for Multiplying and Dividing Complex Numbers
Simplify.
- \(\color{blue}{(4i)(– i)(2 – 5i)}\)
- \(\color{blue}{(3 – 7i)(4 – 5i)}\)
- \(\color{blue}{(–5 + 9i)(3 + 5i)}\)
- \(\color{blue}{(7 + 3i)(7+ 8i)}\)
- \(\color{blue}{(5 + 4i)^2}\)
- \(\color{blue}{2(3i) – (5i)(– 8 + 5i)}\)
Download Multiplying and Dividing Complex Numbers Worksheet
- \(\color{blue}{8 – 20i }\)
- \(\color{blue}{-23 – 43i}\)
- \(\color{blue}{-60 + 2i}\)
- \(\color{blue}{25 + 77i}\)
- \(\color{blue}{9 + 40i}\)
- \(\color{blue}{25 + 46i}\)
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