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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

Why it matters: AC circuit analysis, signal processing, and quantum-state math all need a number whose square is negative. The symbol i makes that consistent, and powers of i cycle through i, -1, -i, 1 every four steps.

Worked example

Problem. Simplify sqrt(-36).

  1. Rewrite sqrt(-36) as sqrt(36) x sqrt(-1).
  2. sqrt(36) = 6.
  3. sqrt(-1) = i, so the result is 6i.

Answer: 6i

Practice problems

1. Simplify sqrt(-25).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify sqrt(-25).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. sqrt(25) = 5 and sqrt(-1) = i.
  4. Check the result by substituting or estimating: the response should match 5i and make sense in the original problem.

Answer: 5i

2. What is i^2?

Show solution
  1. Warm-up: First identify exactly what the question is asking: What is i^2?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. By definition, i^2 = -1.
  4. Check the result by substituting or estimating: the response should match -1 and make sense in the original problem.

Answer: -1

3. Simplify sqrt(-48).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Simplify sqrt(-48).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. sqrt(-48) = i sqrt(48).
  4. sqrt(48) = 4sqrt(3).
  5. Check the result by substituting or estimating: the response should match 4isqrt(3) and make sense in the original problem.

Answer: 4isqrt(3)

4. Which value equals i^4?

Choices: 1 · -1 · i · -i

Show solution
  1. Challenge: First identify exactly what the question is asking: Which value equals i^4?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. i^2 = -1, so i^4 = 1.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 1

5. Find the discriminant of x^2 + 4x + 13 = 0.

Show solution
  1. Mixed Review: First identify exactly what the question is asking: Find the discriminant of x^2 + 4x + 13 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use b^2 - 4ac.
  4. 4^2 - 4(1)(13) = 16 - 52.
  5. The discriminant is -36, which points toward complex solutions.
  6. Check the result by substituting or estimating: the response should match -36 and make sense in the original problem.

Answer: -36

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