Where i is real

Komplexiti guide

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Demo Sets

Complex Numbers & Arithmetic

Cartesian, polar & exponential forms, and quotient z = a/b

Complex Equations

Rational equation, cube roots of −27 and 8i in exponential form

Loci

Circle, inscribed arc, Joukowski locus

Draggable Points

Drag foci a and b on the diagram (grid-snapped) to watch their perpendicular bisector transform live

Line Loci

Horizontal line, vertical line, and perpendicular bisector

Equations with Conjugates

Four roots in Cartesian form: two real, two complex

Conjugate Loci

Loci involving z̅ — a line, modulus with conjugate, and a non-standard locus

Inequalities

Compound arg inequality, strict circle, strict perpendicular bisector half-plane

Extrema

Circle with exact integer modulus extrema and exact arg extrema at 15° and 75°

Union of Loci

(arg(z)−π/4)(|z|−2)=0: a factored equation drawing the ray and the circle together, with phase/modulus colouring

Mixed Union

(|z|−1)(sec(z)−1)=0: a circle plus sec(z)=1's compact periodic roots and poles, drawn and listed together, with phase/modulus colouring

Poles & Holes

Rational equation with a shared factor: a hole and a genuine pole, both shown by default

Phase & Modulus Colouring

The same rational equation, with phase/modulus (domain) colouring switched on alongside its pole and hole

Gamma Function

Γ(2z−1) with poles and phase/modulus colouring

Zeta Function

ζ(z) with trivial zeros, its pole, and phase/modulus colouring

Essential Singularity

e^1/z=0, no roots but an essential singularity at z=0, with phase/modulus colouring

Rational Function

(z²−1)/(z²+1)=0, roots at ±1 and poles at ±i, with phase/modulus colouring

Möbius Transformation

(z−i)/(2z+1)=0, root at z=i and pole at z=−1/2, with phase/modulus colouring

Nested Cosine Fractal

A 4-fold nested xcos(xcos(…))=0 whose zero locus shows intricate fractal-like structure, with phase/modulus colouring - try the high-resolution snapshot button for extra detail

Cosine-Exponential Fractal

e^cos(z²)=0 has no roots (e^w is never 0) but its modulus swings between astronomically huge and tiny, producing striking black/white fractal-like fan patterns, with phase/modulus colouring

Viewing shared expressions

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How to use Komplexiti

Complex Numbers & Constants

Cartesian form: a = 1 + 2i or z = 3 - 4i

Modulus-argument: z = 2*e^(i*pi/3) or 2(cos(pi/4) + i*sin(pi/4))

Functions & Operations: abs(z), mod(z), arg(z), conj(z), re(z), im(z)

Gamma & zeta functions: Type out gamma(z) or zeta(z) letter by letter to insert Γ(z) or ζ(z)

Dynamic vectors: Define variables (e.g. w = a + b, z = a*b, z = a/b)

Polynomial Equations & Roots

Roots of unity / powers: z^3 = 1, z^4 = -16, z^5 = 1 + i

Polynomials: z^2 + 2z + 5 = 0, z^3 - 3z + 2 = 0

Display format: Click the format badge below an equation to cycle Cartesian ↔ Mod-Arg ↔ Exponential

Loci in the Complex Plane

Circle: |z - (1+i)| = 2 or abs(z - a) = 3

Perpendicular bisector: |z - 1| = |z - i| or abs(z - a) = abs(z - b)

Half-line (ray): arg(z - (1+i)) = pi/4

Cartesian lines: re(z) = 2, im(z) = -1

Spirals: arg(z) = |z| or 1/2*arg(z+i) = |z-1|

Poles, Holes & Domain Colouring

Rational equations: (z^2-1)/(z^2+1) = 0 finds roots, poles, and any removable holes automatically

Toggle markers: Click the coloured dot next to Poles, Holes, or Essential Singularities in a card to show/hide them on the diagram

Domain colouring: Click the rainbow icon on a card to shade the whole plane by phase (hue) and modulus (brightness) of that expression

Navigation & Controls

Pan diagram

Drag with mouse/finger or ← → ↑ ↓

Add a point constant: Right-click and drag on the diagram, release to drop a snapped point (desktop mouse only)

Zoom

Mouse wheel, pinch, or + -

Reset view

Click RESET AXES button

Display mode

Toggle vector arrows vs point dots

Label size & theme

Cycle size A / Toggle light & dark mode

Press Esc or click outside to close

Komplexiti - Complex Number Explorer

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