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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

Why it matters: Electrical signals, wave behavior, and advanced equation solving all use complex numbers to keep track of real and imaginary components. The complex plane makes a + bi feel like a coordinate rather than a mysterious symbol.

Worked example

Problem. Simplify sqrt(-64).

  1. sqrt(-64) = sqrt(64) x sqrt(-1).
  2. sqrt(64) = 8.
  3. sqrt(-1) = i.

Answer: 8i

Practice problems

1. Simplify sqrt(-36).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Simplify sqrt(-36).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. sqrt(36) = 6.
  4. Check the result by substituting or estimating: the response should match 6i and make sense in the original problem.

Answer: 6i

2. What is i^2?

Show solution
  1. Core Practice: 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 (3 + 2i) + (5 - 7i).

Show solution
  1. Challenge: First identify exactly what the question is asking: Simplify (3 + 2i) + (5 - 7i).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Combine real and imaginary parts.
  4. Check the result by substituting or estimating: the response should match 8 - 5i and make sense in the original problem.

Answer: 8 - 5i

4. Simplify i^2.

Show solution
  1. Imaginary Unit: First identify exactly what the question is asking: Simplify i^2.
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  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

5. Simplify sqrt(-49).

Show solution
  1. Square Roots: First identify exactly what the question is asking: Simplify sqrt(-49).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. sqrt(-49) = sqrt(49)sqrt(-1).
  4. Check the result by substituting or estimating: the response should match 7i and make sense in the original problem.

Answer: 7i

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