1 · IB Maths
Complex numbers on the Argand plane
A complex number is a point: real part across, imaginary part up. Drag z, or use the sliders, to read its modulus and argument, and see z², the conjugate, 1/z or iz move with it.
Predict first
Start from the default z = 2 + 1.5i (|z| = 2.50). Set Show to 'z and iz'. What is |iz|?
iz = −1.5 + 2i, which has modulus √(1.5² + 2²) = 2.5. Multiplying by i rotates the point 90° anticlockwise and leaves the modulus unchanged.
Check it in the model: Choose 'z and iz (rotation by 90°)' in the Show list and compare the |iz| value with |z|.
Drag the gold point (or focus the plane and use the arrow keys). z second point
r = √(a² + b²) ; θ = arg z, −π < θ ≤ π
z² = r²(cos2θ + i sin2θ) ; z̄ = a − bi
1/z = z̄ ÷ r² ; iz = −b + ai
Try this
- Show z and z², then drag z around the circle |z| = 1 with the polar grid on. How do the arguments of z and z² compare, and what is |z²|?
- Show z and iz. Set z = 3 + 2i and read the two arguments: by how many degrees does multiplying by i rotate the point, and does |z| change?
- Show z and 1/z and put z at 0.5 + 0.5i. Where does 1/z lie, and what happens to its modulus as you drag z further from the origin?
Exam tip: In modulus–argument form z = r(cosθ + i sinθ), multiplying multiplies the moduli and adds the arguments, so z² has modulus r² and argument 2θ; give θ in (−π, π] and in the units the question asks for.
More on 1: Arithmetic and geometric series · Compound interest and depreciation · Proof by induction · Permutations and combinations · De Moivre's theorem · Systems of linear equations · Roots of a complex number · Complex loci · Matrix transformations · Sequences and series · Exponentials and logarithms · Binomial theorem