Complex geometry
Objective. Use modulus, argument and Möbius transformations.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Complex geometry studies complex numbers z = x + iy through their modulus |z| = √(x²+y²) and argument arg(z) = θ, where z = |z|(cos θ + i sin θ), together with transformations like Möbius maps f(z) = (az+b)/(cz+d) that send circles and lines to circles and lines. Multiplying complex numbers multiplies moduli and adds arguments, giving a geometric meaning to complex multiplication as rotation and scaling. For z = 1+i, |z| = √2 and arg(z) = π/4, so z² has modulus 2 and argument π/2, landing on the positive imaginary axis at 2i.
Worked example
Find the modulus and argument of the complex number z = −√3 + i, locating its geometry on the Argand plane.
- 1Compute the modulus: |z| = √((−√3)² + 1²) = √(3+1) = 2.
- 2Note z lies in the second quadrant since real part is negative and imaginary part positive.
- 3The reference angle is arctan(1/√3) = π/6, so the argument is π − π/6 = 5π/6.
- 4So z = 2(cos(5π/6) + i sin(5π/6)), confirming the modulus-argument form.
Answer: |z| = 2, arg(z) = 5π/6
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students compute arg(z) using arctan(y/x) alone without checking the quadrant, giving wrong arguments for points like z = −1+i, whose true argument is 3π/4, not the arctan value of −π/4.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.