Complex Number Arithmetic
A free Algebra II lesson from the “Complex Numbers” unit, with a worked example and practice problems including step-by-step solutions.
Complex numbers have the form a + bi. Combine real parts with real parts, imaginary parts with imaginary parts, and replace i^2 with -1 after multiplying.
What you'll learn
- Add and subtract complex numbers
- Multiply complex numbers
- Use i^2 = -1 when simplifying
Why it matters: Vector addition, AC impedance, and phase diagrams add and multiply complex numbers component by component. Treat the real part and the imaginary part as two parallel ledgers — combine like with like, and use i^2 = -1 when multiplying.
Worked example
Problem. Multiply (2 + 3i)(4 - i).
- Distribute: 8 - 2i + 12i - 3i^2.
- Combine imaginary terms: 8 + 10i - 3i^2.
- Since i^2 = -1, -3i^2 = 3, so the result is 11 + 10i.
Answer: 11 + 10i
Practice problems
1. Simplify (3 + 2i) + (5 - 7i).
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify (3 + 2i) + (5 - 7i).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Add real parts and imaginary parts.
- Check the result by substituting or estimating: the response should match 8 - 5i and make sense in the original problem.
Answer: 8 - 5i
2. Simplify (9 + 4i) - (2 + 6i).
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify (9 + 4i) - (2 + 6i).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Subtract each part.
- Check the result by substituting or estimating: the response should match 7 - 2i and make sense in the original problem.
Answer: 7 - 2i
3. Simplify i(5 + 2i).
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify i(5 + 2i).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Distribute to get 5i + 2i^2.
- 2i^2 = -2.
- Check the result by substituting or estimating: the response should match -2 + 5i and make sense in the original problem.
Answer: -2 + 5i
4. Simplify (3 + i)(3 - i).
Choices: 10 · 8 · 9 - i · 9 + i
Show solution
- Challenge: First identify exactly what the question is asking: Simplify (3 + i)(3 - i).
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- This is 9 - i^2 = 10.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 10
5. Simplify i^2 + i^4.
Show solution
- Review: First identify exactly what the question is asking: Simplify i^2 + i^4.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- i^2 = -1.
- i^4 = (i^2)^2 = 1.
- -1 + 1 = 0.
- Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.
Answer: 0
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