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Step 13: Algebra 2

Complex numbers

Step 13 · Algebra 2 topics

Use the imaginary unit i, where i² = −1, to write square roots of negative numbers, add, multiply and divide complex numbers, and find complex solutions of quadratic equations.

The imaginary unit

No real number squares to −1. So mathematicians invented one: the imaginary unit i, with i² = −1. Then the square root of any negative number can be written with i.

Example: Square roots of negatives

Simplify √−49 and √−12.

  1. √−49 = √49 · √−1 = 7i.
  2. √−12 = √4 · √3 · i = 2i√3.

Answer7i and 2i√3

The powers of i repeat in a cycle of four.

i¹i²i³i⁴i⁵
i−1−i1i

Example: A high power

What is i²³?

  1. 23 ÷ 4 = 5 remainder 3.
  2. So i²³ is the same as i³, which is −i.

Answer−i

Adding and multiplying complex numbers

Add and subtract complex numbers by combining the real parts and the imaginary parts. Multiply them like binomials, then replace i² with −1.

Example: Adding

Simplify (3 + 4i) + (2 − 7i).

  1. Real parts: 3 + 2 = 5.
  2. Imaginary parts: 4i − 7i is −3i.

Answer5 − 3i

Example: Multiplying

Simplify (2 + 3i)(4 − i).

  1. 8 − 2i + 12i − 3i².
  2. −3i² is −3 × (−1), which is 3.
  3. 8 + 3 + 10i.

Answer11 + 10i

Example: The conjugate

Multiply (5 + 2i)(5 − 2i).

  1. 25 − 10i + 10i − 4i².
  2. 25 + 4 = 29.

Answer29, a real number

Example: Dividing

Simplify 101+3i\frac{10}{1 + 3i}.

  1. Multiply the top and bottom by the conjugate 1 − 3i. The bottom becomes 1 + 9 = 10.
  2. The top becomes 10 − 30i, and dividing by 10 gives 1 − 3i.

Answer1 − 3i

Complex solutions of quadratics

When the discriminant b² − 4ac is negative, the quadratic formula gives two complex solutions, a + bi and a − bi. They are conjugates of each other.

Example: Complex roots

Solve x² + 4x + 13 = 0.

  1. b² − 4ac = 16 − 52 = −36.
  2. x=−4±−362=−4±6i2x = \frac{-4 \pm \sqrt{-36}}{2} = \frac{-4 \pm 6i}{2}

Answerx = −2 + 3i or x = −2 − 3i

y = x² + 4x + 13 never meets the x-axis, so its solutions are not real.

Example: Square roots

Solve x² + 25 = 0.

  1. x² = −25.
  2. x = ±5i.

Answerx = 5i or x = −5i

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Simplify √−81.

    Show answer

    9i

  2. What is i¹⁰?

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    −1

  3. Simplify (6 − 2i) − (1 + 5i).

    Show answer

    5 − 7i

  4. Simplify (3 + i)(2 + 4i).

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    2 + 14i

  5. Simplify (4 − 3i)(4 + 3i).

    Show answer

    25

  6. Solve x² + 2x + 5 = 0.

    Show answer

    x = −1 + 2i or x = −1 − 2i

What next?