Stop using (across, up) — use (how far, which way): On an Argand diagram, a + bi is the point (a, b). Instead of its sideways and upward steps, describe it by how far it is from the origin and which direction it points.
Two numbers fix the point
- How far = the modulus r = |z| = √(a² + b²).
- Which way = the argument θ = the angle from the positive real axis (anticlockwise positive).
Three ways to write it: Then z = r(cos θ + i sin θ), written shorthand as r cis θ, and identically as r eiθ (Euler form). All three describe the same point — just different costumes.
IB-style question — Cartesian to polar
A signal is modelled by the complex number z = 1 + √3 i.
Write z in the form r eiθ, where r > 0 and −π < θ ≤ π.
Step by step
- Modulus first — Pythagoras on the real and imaginary parts.
- Argument — both parts are positive, so z is in the first quadrant; take arctan of (imaginary ÷ real).
- Assemble the exponential form.
Final answer
z = 2 eiπ/3 (equivalently 2 cis π/3). It has size 2 and points 60° above the real axis.
IB-style question — watch the quadrant
Find the modulus and argument of z = −2 + 2i.
Step by step
- Modulus.
- z = (−2, 2) is in the SECOND quadrant, so the argument is between π/2 and π. The reference angle is arctan(2/2) = π/4, so θ = π − π/4.
Final answer
|z| = 2√2, arg z = 3π/4. Always sketch the point first so the quadrant — and hence the sign of θ — is right.
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Multiplying = stretch and rotate: In polar form, multiplying becomes effortless: each multiplication is a turn-and-stretch.
To multiply
- multiply the moduli
- add the arguments
To divide
- divide the moduli
- subtract the arguments
Picture it: Multiplying by a number of size r and angle θ scales the diagram by r and rotates it by θ. That single idea is why complex numbers model rotations, AC phase shifts and spirals.
IB-style question — product in polar form
Let z₁ = 2 cis(π/6) and z₂ = 3 cis(π/4).
Find z₁z₂ in the form r cis θ.
Step by step
- Multiply the moduli.
- Add the arguments (common denominator 12).
- Assemble.
Final answer
z₁z₂ = 6 cis(5π/12). No expanding brackets — just multiply sizes, add angles.
IB-style question — quotient in polar form
With z₁ = 2 cis(π/6) and z₂ = 3 cis(π/4), find z₁ ÷ z₂.
Step by step
- Divide the moduli.
- Subtract the arguments.
Final answer
z₁/z₂ = (2/3) cis(−π/12). A negative argument just means the result points below the real axis.
IB-style question — multiplying as a rotation
The point z = 3 + 4i is shown on an Argand diagram.
(a) Find iz.
(b) Describe the geometric effect of multiplying z by i.
Step by step
- (a) Multiply out, using i² = −1.
- (b) Note i = 1\,\text{cis}\,90^\circ (modulus 1, argument 90°).
- Multiplying multiplies the moduli and ADDS the arguments, so the size is unchanged and the angle grows by 90°.
Final answer
(a) iz = −4 + 3i. (b) A 90° anticlockwise rotation about the origin (no size change). In general, multiplying by r eiθ rotates by θ and enlarges by r.