Complex Plane
Complex numbers are points on a plane with real and imaginary axes.
About this topic
The complex plane displays a + bi as the point (a, b), with the real part on the horizontal axis and the imaginary part on the vertical axis. The modulus, written with bars around a + bi, equals √(a² + b²) and is its distance from the origin.
Addition of complex numbers becomes vector addition, and multiplication becomes a combination of stretching and rotating. For 3 + 4i, the modulus is 5, found by the Pythagorean theorem. Multiplying a number by i rotates it 90 degrees counterclockwise, which explains the cycle of powers of i. The conjugate is a reflection across the real axis.
This picture makes complex arithmetic visual and leads to polar form, where a number is described by its distance and angle. Engineers use it to describe phase and amplitude of waves, and the Mandelbrot set, a famous fractal, is drawn on this plane.
Remember that the vertical axis shows only the coefficient of i, not the i itself, so 2 − 5i is plotted at (2, −5). Complex numbers cannot be ordered as greater or less like the reals, but their sizes can be compared by modulus.
Key ideas
- Real part on x, imaginary part on y
- Modulus is distance from origin
- Multiplying rotates and scales
Quick check
Use cases: where Complex Plane helps
- Homework: review the key idea behind Complex Plane before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Complex Plane without notes.
- Visual learning: play with the interactive lab to see how Complex Plane changes when you adjust its inputs.
- Teaching a friend: explain Complex Plane 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 Plane?
Complex numbers are points on a plane with real and imaginary axes.
What will I learn in Complex Plane?
Real part on x, imaginary part on y. Modulus is distance from origin. Multiplying rotates and scales.
Is Complex Plane 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 Plane 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 Plane?
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.