Complex Numbers
A free College Algebra lesson from the “Complex Numbers” unit, with a worked example and practice problems including step-by-step solutions.
Complex numbers combine a real part and an imaginary part in the form a + bi. The imaginary unit i is defined by i^2 = -1, which allows square roots of negative numbers to be expressed and simplified. Complex numbers matter because many polynomial and quadratic equations have solutions that are not real. When practicing, combine real parts with real parts and imaginary parts with imaginary parts, and use i^2 = -1 when multiplying. A common mistake is treating i like an ordinary variable instead of using its special power cycle.
What you'll learn
- Use i as sqrt(-1)
- Simplify imaginary radicals
- Add and subtract complex numbers
Worked example
Problem. Simplify sqrt(-64).
- sqrt(-64) = sqrt(64) x sqrt(-1).
- sqrt(64) = 8.
- sqrt(-1) = i.
Answer: 8i
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 sqrt(-36).
Show solution
- sqrt(36) = 6.
Answer: 6i
2. What is i^2?
Show solution
- By definition, i^2 = -1.
Answer: -1
3. Simplify (3 + 2i) + (5 - 7i).
Show solution
- Combine real and imaginary parts.
Answer: 8 - 5i
4. Simplify i^2.
Show solution
- By definition, i^2 = -1.
Answer: -1
5. Simplify sqrt(-49).
Show solution
- sqrt(-49) = sqrt(49)sqrt(-1).
Answer: 7i
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