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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

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).

  1. Distribute: 8 - 2i + 12i - 3i^2.
  2. Combine imaginary terms: 8 + 10i - 3i^2.
  3. Since i^2 = -1, -3i^2 = 3, so the result is 11 + 10i.

Answer: 11 + 10i

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Simplify (3 + 2i) + (5 - 7i).

Show solution
  1. Add real parts and imaginary parts.

Answer: 8 - 5i

2. Simplify (9 + 4i) - (2 + 6i).

Show solution
  1. Subtract each part.

Answer: 7 - 2i

3. Simplify i(5 + 2i).

Show solution
  1. Distribute to get 5i + 2i^2.
  2. 2i^2 = -2.

Answer: -2 + 5i

4. Simplify (3 + i)(3 - i).

Choices: 10 · 8 · 9 - i · 9 + i

Show solution
  1. This is 9 - i^2 = 10.

Answer: 10

5. Simplify i^2 + i^4.

Show solution
  1. i^2 = -1.
  2. i^4 = (i^2)^2 = 1.
  3. -1 + 1 = 0.

Answer: 0

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