Where i is real

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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, half-line, perpendicular bisector, Joukowski locus

Conjugate Loci

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

Inequalities

Non-strict circle (solid boundary), strict circle (dashed boundary), non-strict half-plane

How to use Komplexiti

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), arg(z), conj(z), re(z), im(z)

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

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

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|

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Komplexiti - Complex Number Explorer