Imaginary Numbers
A free Algebra II lesson from the “Complex Numbers” unit, with a worked example and practice problems including step-by-step solutions.
Imaginary numbers extend the number system so square roots of negative numbers can be handled. The imaginary unit i is defined by i^2 = -1, so sqrt(-1) = i. This matters because quadratic equations with negative discriminants still have solutions, but those solutions are complex rather than real. When practicing, separate the negative sign from the radicand, rewrite sqrt(-1) as i, and simplify any remaining square root. A common mistake is saying that square roots of negative numbers have no solution; in the complex number system, they do.
What you'll learn
- Use i as sqrt(-1)
- Simplify square roots of negative numbers
- Identify powers of i
Worked example
Problem. Simplify sqrt(-36).
- Rewrite sqrt(-36) as sqrt(36) x sqrt(-1).
- sqrt(36) = 6.
- sqrt(-1) = i, so the result is 6i.
Answer: 6i
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(-25).
Show solution
- sqrt(25) = 5 and sqrt(-1) = i.
Answer: 5i
2. What is i^2?
Show solution
- By definition, i^2 = -1.
Answer: -1
3. Simplify sqrt(-48).
Show solution
- sqrt(-48) = i sqrt(48).
- sqrt(48) = 4sqrt(3).
Answer: 4isqrt(3)
4. Which value equals i^4?
Choices: 1 · -1 · i · -i
Show solution
- i^2 = -1, so i^4 = 1.
Answer: 1
5. Find the discriminant of x^2 + 4x + 13 = 0.
Show solution
- Use b^2 - 4ac.
- 4^2 - 4(1)(13) = 16 - 52.
- The discriminant is -36, which points toward complex solutions.
Answer: -36
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