01 Key Concepts
The Imaginary Unit
i is defined so that i^2 = -1. It lets us take the square root of negative numbers: sqrt(-9) = 3i.
Standard Form
A complex number is written a + bi, where a is the real part and b is the imaginary part.
Adding and Subtracting Complex Numbers
Combine the real parts together and the imaginary parts together separately.
Multiplying Complex Numbers
Use distribution (FOIL) just like with binomials, then simplify using i^2 = -1.
Complex 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.
02 Key Formulas
- i^2 = -1
- (a+bi) + (c+di) = (a+c) + (b+d)i
- (a+bi)(a-bi) = a^2 + b^2
03 Solved Examples
Example 1 Simplify sqrt(-16).
- Rewrite as sqrt(16) * sqrt(-1).
- sqrt(16) = 4, and sqrt(-1) = i.
Answer: 4i
Example 2 Add (3 + 2i) + (5 - 4i).
- Combine real parts: 3+5=8.
- Combine imaginary parts: 2i + (-4i) = -2i.
Answer: 8 - 2i
Example 3 Multiply (2 + 3i)(1 - i).
- Use FOIL: 2*1 + 2*(-i) + 3i*1 + 3i*(-i).
- = 2 - 2i + 3i - 3i^2.
- Since i^2=-1: -3i^2 = 3.
- Combine: (2+3) + (-2i+3i) = 5 + i.
Answer: 5 + i
04 Practice Questions
1Simplify sqrt(-25).
5i
2Add (4+3i) + (2-5i).
6-2i
3Subtract (6+2i) - (1+4i).
5-2i
4Multiply (1+i)(1-i).
2 (since 1-i^2=1-(-1)=2)
5What is the value of i^2?
-1
📄 Complex Numbers — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.