Complex.js - ℂ in JavaScript

Complex.js is a well tested JavaScript library to work with complex number arithmetic in JavaScript. It implements every elementary complex number manipulation function and the API is intentionally similar to Fraction.js. Furthermore, it's the basis of Polynomial.js and Math.js.

Examples

let Complex = require('complex.js');

let c = new Complex("99.3+8i");
c.mul({re: 3, im: 9}).div(4.9).sub(3, 2);

A classical use case for complex numbers is solving quadratic equations ax² + bx + c = 0 for all a, b, c ∈ ℝ:

function quadraticRoot(a, b, c) {
  let sqrt = Complex(b * b - 4 * a * c).sqrt()
  let x1 = Complex(-b).add(sqrt).div(2 * a)
  let x2 = Complex(-b).sub(sqrt).div(2 * a)
  return {x1, x2}
}

// quadraticRoot(1, 4, 5) -> -2 ± i

For cubic roots have a look at RootFinder which uses Complex.js.

Parser

Any function (see below) as well as the constructor of the Complex class parses its input like this.

You can pass either Objects, Doubles or Strings.

Objects

new Complex({re: real, im: imaginary});
new Complex({arg: angle, abs: radius});
new Complex({phi: angle, r: radius});
new Complex([real, imaginary]); // Vector/Array syntax

If there are other attributes on the passed object, they're not getting preserved and have to be merged manually.

Note: Object attributes have to be of type Number to avoid undefined behavior.

Doubles

new Complex(55.4);

Strings

new Complex("123.45");
new Complex("15+3i");
new Complex("i");

Two arguments

new Complex(3, 2); // 3+2i

Attributes

Every complex number object exposes its real and imaginary part as attribute re and im:

let c = new Complex(3, 2);

console.log("Real part:", c.re); // 3
console.log("Imaginary part:", c.im); // 2

Functions

Complex sign()

Returns the complex sign, defined as the complex number normalized by it's absolute value

Complex add(n)

Adds another complex number

Complex sub(n)

Subtracts another complex number

Complex mul(n)

Multiplies the number with another complex number

Complex div(n)

Divides the number by another complex number

Complex pow(exp)

Returns the number raised to the complex exponent (Note: Complex.ZERO.pow(0) = Complex.ONE by convention)

Complex sqrt()

Returns the complex square root of the number

Complex exp(n)

Returns e^n with complex exponent n.

Complex log()

Returns the natural logarithm (base E) of the actual complex number

Note: The logarithm to a different base can be calculated with z.log().div(Math.log(base)).

double abs()

Calculates the magnitude of the complex number

double arg()

Calculates the angle of the complex number

Complex inverse()

Calculates the multiplicative inverse of the complex number (1 / z)

Complex conjugate()

Calculates the conjugate of the complex number (multiplies the imaginary part with -1)

Complex neg()

Negates the number (multiplies both the real and imaginary part with -1) in order to get the additive inverse

Complex floor([places=0])

Floors the complex number parts towards zero

Complex ceil([places=0])

Ceils the complex number parts off zero

Complex round([places=0])

Rounds the complex number parts

boolean equals(n)

Checks if both numbers are exactly the same, if both numbers are infinite they are considered not equal.

boolean isNaN()

Checks if the given number is not a number

boolean isFinite()

Checks if the given number is finite

Complex clone()

Returns a new Complex instance with the same real and imaginary properties

Array toVector()

Returns a Vector of the actual complex number with two components

String toString()

Returns a string representation of the actual number. As of v1.9.0 the output is a bit more human readable

new Complex(1, 2).toString(); // 1 + 2i
new Complex(0, 1).toString(); // i
new Complex(9, 0).toString(); // 9
new Complex(1, 1).toString(); // 1 + i

double valueOf()

Returns the real part of the number if imaginary part is zero. Otherwise null

Trigonometric functions

The following trigonometric functions are defined on Complex.js:

| Trig | Arcus | Hyperbolic | Area-Hyperbolic | |------|-------|------------|------------------| | sin() | asin() | sinh() | asinh() | | cos() | acos() | cosh() | acosh() | | tan() | atan() | tanh() | atanh() | | cot() | acot() | coth() | acoth() | | sec() | asec() | sech() | asech() | | csc() | acsc() | csch() | acsch() |

Notes on branches & ranges. Inverse trig on ℂ needs a branch choice. For acot there are two real-axis conventions:

• (Chosen) Textbook range (0, π) on ℝ. Gives familiar real values (acot(1)=π/4, acot(0)=π/2, acot(-1)=3π/4). With the principal complex branch this necessarily causes a π jump across the negative real axis (e.g., near -1±i0 → -π/4, but on -1 → 3π/4). We implement this with atan2(1, x) to select the correct quadrant and handle x=0.

• Alternative range (−π/2, π/2] on ℝ. Makes acot(-1) = -π/4 and can avoid that specific jump only if the entire complex branch is changed to match. Using plain atan(1/x) silently adopts this convention, changes real outputs, is undefined at x=0, and still wouldn’t resolve the complex behavior without a broader policy change.

We therefore prefer predictable, textbook real values and robust quadrant handling—hence atan2(1, x)-accepting the implied branch cut on ℝ⁻ by design.

Geometric Equivalence

Complex numbers can also be seen as a vector in the 2D space. Here is a simple overview of basic operations and how to implement them with complex.js:

New vector

let v1 = new Complex(1, 0);
let v2 = new Complex(1, 1);

Scale vector

scale(v1, factor):= v1.mul(factor)

Vector norm

norm(v):= v.abs()

Translate vector

translate(v1, v2):= v1.add(v2)

Rotate vector around center

rotate(v, angle):= v.mul({abs: 1, arg: angle})

Rotate vector around a point

rotate(v, p, angle):= v.sub(p).mul({abs: 1, arg: angle}).add(p)

Distance to another vector

distance(v1, v2):= v1.sub(v2).abs()

Constants

Complex.ZERO

A complex zero value (south pole on the Riemann Sphere)

Complex.ONE

A complex one instance

Complex.INFINITY

A complex infinity value (north pole on the Riemann Sphere)

Complex.NAN

A complex NaN value (not on the Riemann Sphere)

Complex.I

An imaginary number i instance

Complex.PI

A complex PI instance

Complex.E

A complex euler number instance

Complex.EPSILON

A small epsilon value used for equals() comparison in order to circumvent double imprecision.

Installation

You can install Complex.js via npm:

npm install complex.js

Or with yarn:

yarn add complex.js

Alternatively, download or clone the repository:

git clone https://github.com/rawify/Complex.js

Usage

Include the complex.min.js file in your project:

<script src="path/to/complex.min.js"></script>
<script>
    console.log(Complex("4+3i"));
</script>

Or in a Node.js project:

const Complex = require('complex.js');

or

import Complex from 'complex.js';

Coding Style

As every library I publish, Complex.js is also built to be as small as possible after compressing it with Google Closure Compiler in advanced mode. Thus the coding style orientates a little on maxing-out the compression rate. Please make sure you keep this style if you plan to extend the library.

Building the library

After cloning the Git repository run:

npm install
npm run build

Run a test

Testing the source against the shipped test suite is as easy as

npm run test

Copyright and Licensing

Copyright (c) 2025, Robert Eisele Licensed under the MIT license.