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Complex Numbers

Visual Intuition — Geometry & Motion

⊕ Particle Motion
01 /

The Argand Plane

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Explore z = x + iy

z = 3 + 2i
|z| = 3.606
θ = 33.69°
A complex number is simply a point in 2D space. The real axis runs left-right, the imaginary axis runs up-down. Every point you know from geometry has a complex number partner.
02 /

Addition = Vector Addition

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z₁ + z₂ [+ z₃]

z₁
Real 2.0
Imag 1.0
z₂
Real 1.0
Imag 2.0
Sum = z₁ + z₂ OUTPUT
Real —
Imag —
Sum = —
Place z₂'s tail at z₁'s tip. The resultant vector is the sum — identical to how forces add in physics.
03 /

Subtraction = Add the Negative

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z₁ − z₂ [− z₃]

z₁
Real 3.0
Imag 2.0
z₂
Real 1.0
Imag 3.0
Result = z₁ − z₂ OUTPUT
Real —
Imag —
Result = —
4 steps shown on canvas: ① z₁ in cyan. ② z₂ gets flipped 180° to become −z₂ (pink, from origin). ③ −z₂ is moved to the tip of z₁ (dashed pink). ④ The green Result is where you land — exactly z₁ + (−z₂).
04 /

Multiplication = Rotation + Scaling

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z₁ × z₂ [× z₃]

z₁
Real 2.0
Imag 1.0
z₂
Real 1.0
Imag 1.0
Z = z₁ × z₂ OUTPUT
|Z| —
∠Z —
Z = —
|Product| = —
∠Product = —
Multiplying by z₂ rotates z₁ by θ₂ and scales it by r₂. Multiplying by i (r=1, θ=90°) is a pure 90° rotation — no scaling.
05 /

Division = Rotation + Scaling (Inverse)

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z₁ ÷ z₂ [÷ z₃]

z₁
Real 4.0
Imag 2.0
z₂
Real 1.0
Imag 1.0
|Quotient| = —
∠Quotient = —
Quotient = z₁ ÷ z₂ OUTPUT
Real —
Imag —
Result = —
Dividing by z₂ un-rotates by θ₂ and shrinks by r₂. Use conjugate: z₁÷z₂ = z₁·z̄₂÷|z₂|². The result magnitude is r₁÷r₂ and angle is θ₁−θ₂.
06 /

Multiplying by i — The 90° Rotation Rule

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×i¹
+90°
—
CCW quarter turn
×i²
+180°
—
Half turn = −1 rule
×i³
+270°
—
CW quarter turn
×i⁴
+360°
—
Full turn — back to z

Choose starting z

z = 3 + 1i
z·i = —
z·i² = —
z·i³ = —
z·i⁴ = —
i¹=i  ·  i²=−1  ·  i³=−i  ·  i⁴=1
i is a 90° rotation operator. Each multiplication by i spins the number counterclockwise by exactly 90°, keeping its magnitude unchanged. After 4 steps you return to the start — because 4 × 90° = 360°. This is why i² = −1: start at +1 on the real axis, rotate 90° twice, land on −1.
07 /

Polar Form

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z = r·eiθ = r(cosθ + i·sinθ)

x = r·cosθ = 2.50
y = r·sinθ = 0.00
z = 2.50 + 0.00i
eiθ = 1.000 + 0.000i
e−iθ = 1.000 − 0.000i
eiθ · e−iθ = 1.000
Polar form separates how far (r) from which direction (θ) — exactly like vectors in physics. Euler's formula links this to e, cos, and sin in a single elegant identity.
08 /

Complex Conjugate = Reflection

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z̄ = x − iy

z = 3 + 2i
z̄ = 3 − 2i
z·z̄ = 13 (= |z|²)
The conjugate flips the imaginary part. Geometrically it's a mirror across the real axis. Multiplying z by its conjugate always gives a real number — the squared magnitude.
09 /

Powers = Repeated Rotation

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z^n for n = 0 → N (negative N = division)

z^n: magnitude rⁿ
angle: n·θ
period: 12 steps
Each power applies another rotation of θ. When r=1, the points trace a perfect circle. When θ divides 360° evenly, you get the roots of unity. Negative N runs the sequence backwards: each step divides by z instead of multiplying — rotating by −θ and scaling by 1/r.
10 /

Number Line & Argand Plane

REAL NUMBER 3
MULTIPLY BY i
z
3
z²
9
z·i
—
z·i²
—
z·i³
—
z·i⁴
—
Two worlds, one picture. The number line (top) is 1D — your number and its square live there. The Argand plane (below) is 2D — multiplying by i lifts your number off the real axis, rotating it 90° each time. After 4 rotations you return home.
11 /

De Moivre's Theorem

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DE MOIVRE'S THEOREM
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
or equivalently: (r·eiθ)ⁿ = rⁿ · einθ
Modulus r 1.00
Angle θ (degrees) 45°
Power n 3
▷ Animation
SELECT TO ANIMATE:
Speed 1.0×
z = —
r = —   θ = —
zⁿ = —
rⁿ = —   nθ = —
VERIFICATION
cos nθ + i·sin nθ = —
Drag the point on the circle. The gold vector is z, the green vector is zⁿ. The angle multiplies by n and the modulus raises to the power n.
12 /

Complex Roots — The Hidden Dimension

FLOOR AXIS →
Re(z)
real part of z — can be any real number
FLOOR AXIS ↑
Im(z)
imaginary part of z — can be any real number
VERTICAL ↑
Re(c·zⁿ) − Re(W)
value of polynomial — zero when z is a root
How to read this: The floor (Re × Im) is the complex plane — every point on it is a complex number z = Re(z) + i·Im(z). The surface above it shows Re(zⁿ) − Re(W). Where the surface hits zero (the floor), that's a root. The golden curve is the slice along the real axis (Im=0), i.e. xⁿ − Re(W) — a normal real polynomial curve. The glowing spikes mark all n roots.
Argand Plane · zⁿ = W
Re(z) axis
Im(z) axis
root circle r = |W|^(1/n)
roots (zⁿ = W)

c·zⁿ = W  →  find all z

Coefficient c
zⁿ = W
|W/c| = —
arg(W/c) = —
circle r = |W/c|1/n = —
ROOTS  z_k = |W/c|^(1/n) · e^{i(φ+2πk)/n}
Presets
What are the axes?
Re(z) and Im(z) are the two parts of z — they span the floor (the complex plane).

The vertical axis is Re(c·zⁿ) − Re(W). Roots are where the surface hits the floor.

Complex c rotates the whole surface — the roots rotate by −arg(c)/n and their radius scales by |c|−1/n.
13 /

Conjugate Root Pairs

For any polynomial with real coefficients, complex roots always appear in conjugate pairs: if z = a + bi is a root, so is z̄ = a − bi. They are mirror images across the real axis. Drag the sliders to explore.

Polynomial Coefficients

Roots

Why? For real coefficients, p(z̄) = p(z) conjugated. If p(z) = 0, then p(z̄) = 0 too — guaranteed. The real axis is an axis of symmetry for the root set. Real roots lie on the axis; complex roots come in reflected pairs above and below.
14 /

Polynomial Surface & Roots

X AXIS (floor)
Re(z)
real part of input z
Z AXIS (floor)
Im(z)
imaginary part — pure imaginary
Y AXIS (height)
|P(z)|
output magnitude — real or imaginary
The geometric meaning of a root: Every point on the XZ floor is a complex number z = Re(z) + i·Im(z) (the Argand plane lying flat). The surface above shows |P(z)|. When P(z) = 0, the output is exactly the origin (0 + 0i) — so the surface touches the floor. Those floor-touch points are the roots. Roots appear on the Argand circle diagram below. Drag to rotate · scroll to zoom.
DRAG TO ROTATE · SCROLL TO ZOOM
Argand Plane — root locations
Computed roots (P(z) = 0)

    Polynomial P(z)

    Coefficient type
    Presets
    Coefficients
    Why the Argand circle? The mini-map shows the XZ plane from above (top-down view). The gold dashed circle is the unit circle — roots of unity lie exactly on it. Each coloured dot is where the 3D surface touches the floor (P(z) = 0), meaning that complex input z maps to the origin of the output plane. The route to zero is shown on the Argand plane because that's where the input lives.
    P(z) = 0
    ⟹ output = 0 + 0i
    ⟹ Re(output) = 0, Im(output) = 0
    ⟹ origin of output complex plane
    ⟹ surface height |P(z)| = 0
    ⟹ surface touches the XZ floor
    15 /

    Euler's Formula & e−(t+iθ)

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    e−(t+iθ) = e−t(cos θ − i sin θ)

    e−(t+iθ) = e⁰·(cos 0° − i·sin 0°)
    = 1.000 + 0.000i
    |z| = 1.000 = e−t
    arg = 0.00°
    Decomposition
    Re = e−tcos θ
    1.000
    Im = −e−tsin θ
    0.000
    e−t (radius)
    1.000
    e−t → 0 as t→∞
    t=0: no decay
    e−(t+iθ) = e−t · e−iθ
    The real part t shrinks the radius exponentially — pure decay. The imaginary part iθ rotates — pure spin. Together they trace an inward spiral. Set t=0 for the unit circle.
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