Real numbers on the horizontal axis, imaginary on the vertical. Every complex number is a point — or an arrow from the origin.
Drag the complex number
3 + 1i
real part 3 · imaginary part 1
modulus |z| = √(3² + 1²) = 3.16
argument ≈ 18.43°
Two ways to name a point
- Rectangular: a + bi — go a right, b up.
- Polar: modulus r = √(a² + b²) and argument θ = the angle from the positive real axis.
- Multiplying complex numbers multiplies the moduli and adds the arguments — that's why ×i is a 90° turn (it has modulus 1, argument 90°).
Your turn
What are the modulus and argument of 3 + 4i?
Drag the point to the negative real axis: you'll see modulus = |value| and argument = 180°. Multiplying by that point flips and scales any other complex number.
Multiplying by i rotates a point 90° counterclockwise. Multiplying by i twice (= −1) rotates 180°.