# Complex.js - ℂ in JavaScript
[](https://npmjs.org/package/complex.js "View this project on npm")
[](http://opensource.org/licenses/MIT)
Complex.js is a well tested JavaScript library to work with [complex number arithmetic](https://raw.org/book/analysis/complex-numbers/) in JavaScript. It implements every elementary complex number manipulation function and the API is intentionally similar to [Fraction.js](https://github.com/rawify/Fraction.js). Furthermore, it's the basis of [Polynomial.js](https://github.com/infusion/Polynomial.js) and [Math.js](https://github.com/josdejong/mathjs).
## Examples
```js
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`:
```js
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](https://github.com/rawify/RootFinder.js) 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
```javascript
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.
### Doubles
```javascript
new Complex(55.4);
```
### Strings
```javascript
new Complex("123.45");
new Complex("15+3i");
new Complex("i");
```
### Two arguments
```javascript
new Complex(3, 2); // 3+2i
```
## Attributes
Every complex number object exposes its real and imaginary part as attribute `re` and `im`:
```javascript
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
```javascript
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() |
## 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
---
```js
let v1 = new Complex(1, 0);
let v2 = new Complex(1, 1);
```
Scale vector
---
```js
scale(v1, factor):= v1.mul(factor)
```
Vector norm
---
```js
norm(v):= v.abs()
```
Translate vector
---
```js
translate(v1, v2):= v1.add(v2)
```
Rotate vector around center
---
```js
rotate(v, angle):= v.mul({abs: 1, arg: angle})
```
Rotate vector around a point
---
```js
rotate(v, p, angle):= v.sub(p).mul({abs: 1, arg: angle}).add(p)
```
Distance to another vector
---
```js
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
Installing complex.js is as easy as cloning this repo or use one of the following command:
```bash
npm install complex.js
```
## Using Complex.js with the browser
```html
<script src="complex.min.js"></script>
<script>
console.log(Complex("4+3i"));
</script>
```
## 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) 2024, [Robert Eisele](https://raw.org/)
Licensed under the MIT license.