How to Multiply and Divide Complex Numbers? (+FREE Worksheet!)
How to multiply and divide complex numbers, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
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 to 5i)})
- (color{blue}{(3 to 7i)(4 to 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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