Powers of i
Each multiplication by i is a quarter turn anticlockwise, so the powers repeat every 4.
Station 01 · Arithmetic of complex numbers
Meet i
Define i by i² = −1, see how the number system grows from ℕ to ℂ, and solve quadratics whose discriminant is negative.
- i² = −1
- x² + k = 0
- ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
- Δ < 0 quadratics
- Re(z), Im(z)
Which set does it live in?
Tap the smallest set that contains the number.
0 / 0
ℂcomplex
ℝreal
ℚrational
ℤintegers
ℕnatural
Quadratics with a negative discriminant
Enter real coefficients for ax² + bx + c = 0. When Δ < 0 the roots are a conjugate pair, mirror images in the real axis.
Station 02 · Geometric representation
The Argand plane
Drag z anywhere. Watch its modulus, argument, principal argument and polar form update, along with its conjugate and reciprocal.
- plot z = a + ib
- |z|, arg z, Arg z ∈ (−π, π]
- z = r(cos θ + i sin θ)
- z̄, z z̄ = |z|²
- Re z = (z + z̄)/2
- 1/z = z̄/|z|²
Plot from polar form
Type a modulus and an argument, such as 2 and 2pi/3, or sqrt2 and -pi/4.
Reading z
Conjugate identities
Station 03 · Arithmetic and identities
Add, subtract, multiply, divide
Drag z₁ and z₂, or type them. Addition is vector addition; multiplication multiplies moduli and adds arguments.
- + − × ÷ with and without digital tools
- dividing with the conjugate
- |z₁z₂| = |z₁||z₂|
- arg(z₁z₂) = arg z₁ + arg z₂
- conjugate identities
- triangle inequality
- equality of complex numbers
Identities, checked live
| Identity | Left | Right |
|---|
Cartesian working
Polar view
Station 04 · Solving equations with complex numbers
Square roots and equations
Find the two square roots of a complex number, solve quadratics with complex coefficients, and build polynomials to see the conjugate root theorem at work.
- square roots of a + ib
- quadratics with complex a, b, c
- conjugate root theorem
- real quadratic factors
The square roots of w
or drag w on the plane
Quadratics with complex coefficients
Solve az² + bz + c = 0. Coefficients can be any complex expressions.
Conjugate root builder
Tap the plane to place roots of a monic polynomial P(z). Tap a root to remove it. With real coefficients forced on, every non-real root brings its conjugate.
Proof of the conjugate root theorem
Station 05 · Powers of complex numbers
De Moivre's theorem
Raising z = r(cos θ + i sin θ) to the power n raises the modulus to the power n and multiplies the argument by n. The powers trace a spiral.
- (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
- positive and negative integer n
- proof by induction
- deriving trig identities
Computing zn
Derive a trig identity
Multiple angle n =
Proof by mathematical induction
Reveal one line at a time. Ask students to predict each line before you show it.
Station 06 · Roots of complex numbers
The n-th roots
The n solutions of zn = w sit evenly spaced on a circle, forming a regular n-gon. Change n, pick a preset, or choose Custom and drag w.
- n-th roots of ±1
- roots on the unit circle
- n-th roots of any w
- sum of roots = 0
- real quadratic factors
Solving zn = w
The roots
| k | polar | Cartesian |
|---|
Properties
Station 07 · Complex numbers as vectors
Rotation, dilation and vectors
Multiplying by w rotates every point about the origin by arg w and scales it by |w|. Rotating a vector by 90° is multiplying by i.
- complex numbers as vectors
- rotation and dilation
- × i is a 90° rotation
- proving geometric results
Transform a triangle by z ↦ wz
Drag A, B, C and w.
Constructions with vectors
Station 08 · Describing lines, curves and regions
Loci and regions
Combine conditions on z to shade a region. Solid boundaries are included, dashed boundaries are not, and open circles mark excluded points.
- Re(z) = c, Im(z) = k
- |z − z₁| = |z − z₂|
- |z − z₁| = r
- arg(z − z₁) = θ
- regions
- arcs (circle geometry)
Drag P to check which conditions it satisfies.
Type values like
1+i, -2i, pi/3.Which conditions describe the shaded region or curve?
0 / 0
Station 09 · Practice
Quick-fire practice
Fresh questions every time, with full worked solutions. Choose the topics, then type exact or decimal answers.
Type: 3-2i · sqrt3+i · 3pi/4 · -1/2 · 4sqrt2
Worked solution
Answer the question, or tap Show solution.