Simulic
English

Complex numbers on the Argand diagram – operations, polar form and De Moivre's theorem

MathematicsAlgebraAges 15–16

Loading…

Use with my class ✨ Customize with AI Report a problem

Drag two complex numbers z and w on the Argand diagram and see z + w as vector addition, zw as a rotation by arg w with scaling by |w|, z/w, the conjugate and the distance |z − w|, each shown in Cartesian, modulus–argument and Euler form re^(iθ). The De Moivre screen draws the powers zⁿ on a spiral and the n nth roots of a number as a regular polygon. The Polynomial roots screen shows complex conjugate roots appearing as the parabola lifts off the x-axis (Δ < 0), plus the roots of a cubic.

Lesson: Complex numbers: Cartesian form, the Argand diagram, modulus and argument, polar and Euler form, De Moivre's theorem, nth roots, complex roots of quadratic and cubic equations

What it shows

A complex number z = a + bi, with i² = −1, is a point on the Argand diagram, or a vector from the origin. Its modulus |z| is the length of that vector and its argument arg z is the angle from the positive real axis, which gives the polar form r(cos θ + i sin θ) = re^(iθ). Adding complex numbers adds vectors; multiplying multiplies the moduli and adds the arguments, so it rotates and scales. De Moivre's theorem extends this to powers and roots, and polynomials with real coefficients have their non-real roots in conjugate pairs.

How to use

On Operations, drag the dots z and w or move the Re and Im sliders, choose an Operation and press Animate to watch z move to the result. The Set w buttons show what multiplying by i, −1 or 2 does. On De Moivre & roots, choose Powers zⁿ or nth roots of c, set n and drag the point. On Polynomial roots, move a, b and c, drag the vertex, or press Sweep c.

Parameters you can change

  • Screen Operations, De Moivre & roots, Polynomial roots
  • Real part of z -5–5
  • Imaginary part of z -5–5
  • Real part of w -5–5
  • Imaginary part of w -5–5
  • Operation (Operations screen) z + w (add), z − w (subtract), z · w (multiply), z / w (divide), z̄ (conjugate), |z − w| and midpoint
  • Angle unit Degrees, Radians
  • Snap to a 0.5 grid when dragging
  • Mode (De Moivre screen) Powers zⁿ, nth roots of c
  • Power or root index n 1–12
  • Modulus of z (powers) or c (roots) 0.2–4
  • Argument of z or c -180–180 °
  • Polynomial degree (roots screen) Quadratic, Cubic
  • Coefficient a -3–3
  • Coefficient b -6–6
  • Coefficient c -6–10
  • Coefficient d (cubic) -6–6
  • Show root paths as the constant term changes

Questions to explore

  1. Where does a point go if you multiply it by i four times in a row, and why?
  2. Why do the nth roots of 1 always form a regular polygon whose vertices add up to zero?
  3. As the parabola y = x² − 2x + c is raised, which part of its complex roots changes and which stays the same?

Related simulations