Complex Numbers
Complex numbers add an imaginary unit i where i² = −1.
About this topic
Complex numbers extend the reals by introducing i, defined by i² = −1, so that equations like x² + 1 = 0 have solutions. A complex number is written a + bi, with real part a and imaginary part b. You add and subtract by combining like parts, and multiply as with binomials, replacing i² with −1.
For example (2 + 3i) + (1 − i) = 3 + 2i, and (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i. The conjugate of a + bi is a − bi, and multiplying by it gives the real number a² + b², which is how you divide: (1 + i)/(2 − i) is multiplied top and bottom by 2 + i. Powers of i cycle through i, −1, −i, 1.
Despite the name, imaginary numbers are very real tools. Electrical engineers use them for alternating current, physicists for quantum mechanics and waves, and they are the basis of fractal images and signal processing in phones.
The word imaginary is misleading; these numbers are as legitimate as negatives once defined. A common error is writing i² as 1 or treating √(−4) as undefined; it is 2i.
Key ideas
- Form a + bi
- Add real and imaginary parts separately
- Conjugates multiply to a real number
Quick check
Use cases: where Complex Numbers helps
- Homework: review the key idea behind Complex Numbers before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Complex Numbers without notes.
- Visual learning: play with the interactive lab to see how Complex Numbers changes when you adjust its inputs.
- Teaching a friend: explain Complex Numbers in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Complex Numbers?
Complex numbers add an imaginary unit i where i² = −1.
What will I learn in Complex Numbers?
Form a + bi. Add real and imaginary parts separately. Conjugates multiply to a real number.
Is Complex Numbers part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Complex Numbers lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Complex Numbers?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.