Plotting on the Argand plane
Just as a coordinate plane holds ordered pairs, the complex plane (or Argand diagram) holds complex numbers. Plot z = a + bi at the point (a, b): the horizontal axis carries the real part, the vertical axis the imaginary part. So 3 + 2i sits 3 right and 2 up; -1 - 4i sits 1 left and 4 down.
It also helps to think of z as an arrow from the origin to (a, b). Adding two complex numbers then becomes tip-to-tail vector addition — a geometric picture that makes the sum visible.
Modulus: how far from the origin
The modulus of z = a + bi, written |z|, is the length of that arrow — the distance from the origin to (a, b). By the Pythagorean theorem, |z| = sqrt(a^2 + b^2). For a real number this is just its absolute value, so |z| generalizes |x| to the plane. Notice |z|^2 = a^2 + b^2 = z·z-bar, tying the modulus to the conjugate.
z = 3 + 4i |z| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 z = -1 + i |z| = sqrt((-1)^2 + 1^2) = sqrt(2)
Argument and polar form
The argument of z, written arg(z) or θ, is the angle the arrow makes with the positive real axis, measured counterclockwise. Together r = |z| and θ pin down z completely: a = r cos θ and b = r sin θ. Substituting gives the polar form z = r(cos θ + i sin θ), often abbreviated r cis θ.
- Compute the modulus r = sqrt(a^2 + b^2).
- Find a reference angle from tan θ = b/a (use |b/a|).
- Place θ in the correct quadrant using the signs of a and b — don't trust the calculator's blind arctan.
z = 1 + i r = sqrt(1 + 1) = sqrt(2) tan θ = 1/1 = 1, and z is in quadrant I → θ = 45° = π/4 polar form: sqrt(2)(cos 45° + i sin 45°) z = -1 + i (quadrant II) r = sqrt(2), reference angle 45°, θ = 135° = 3π/4 polar form: sqrt(2)(cos 135° + i sin 135°)