Algebra · Quick Reference

Complex Numbers Cheat Sheet

Algebra II · Lesson 10/17

In one line: complex numbers extend the number system to include the square roots of negative numbers, using the imaginary unit i — essential for solving equations that have no real solutions.

Key Ideas

1The Imaginary Unit. i is defined so that i^2 = -1. It lets us take the square root of negative numbers: sqrt(-9) = 3i.

2Standard Form. A complex number is written a + bi, where a is the real part and b is the imaginary part.

3Adding and Subtracting Complex Numbers. Combine the real parts together and the imaginary parts together separately.

4Multiplying Complex Numbers. Use distribution (FOIL) just like with binomials, then simplify using i^2 = -1.

5Complex Conjugates. The conjugate of a+bi is a-bi; multiplying a complex number by its conjugate always gives a real number, which is useful for dividing complex numbers.

Worked Examples

Simplify sqrt(-16).

Add (3 + 2i) + (5 - 4i).

8 - 2i

Multiply (2 + 3i)(1 - i).

5 + i

Formulas

i^2 = -1

(a+bi) + (c+di) = (a+c) + (b+d)i

(a+bi)(a-bi) = a^2 + b^2

Practice Yourself

Simplify sqrt(-25).

Add (4+3i) + (2-5i).

6-2i

Subtract (6+2i) - (1+4i).

5-2i