Complex Numbers
Numbers a + bi where i² = −1.
Key idea
Numbers a + bi where i² = −1.
In depth
Complex numbers a + bi extend the reals by introducing i with i² = −1. This makes every quadratic solvable. They are plotted on a plane with real and imaginary axes.
Worked example
√(−9) = 3i. And x² + 4 = 0 has roots ±2i.
Formula
Real-world uses
- Analysing alternating current circuits, signal processing and quantum states
Common mistake
Calling i a variable. It is a fixed number with i² = −1.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Complex Numbers?
Numbers a + bi where i² = −1. It is a topic at the advanced level in Algebra.
What is the formula for Complex Numbers?
The key formula is i² = −1.
Where is Complex Numbers used in real life?
Real-world examples: Analysing alternating current circuits, signal processing and quantum states.
What should I learn before Complex Numbers?
Make sure you are comfortable with Fundamental Theorem of Algebra, the previous topic in Algebra.
What is a common mistake with Complex Numbers?
Calling i a variable. It is a fixed number with i² = −1.
How can I practise Complex Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Complex Numbers?
Move on to Complex Arithmetic next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.