means completeness and balancing, from the Arabic word الجبر

New: checkout matrices and vectors made of strings, with cyclic algebra.

NOTA BENE Imagine all code examples below as written in some REPL where expected output is documented as a comment.

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Algebra OnQuaternionsAndOctonions

Table Of Contents


algebra is under development, but API should not change until version 1.0.

I am currently adding more tests and examples to achieve a stable version.

Many functionalities of previous versions are now in separated atomic packages:



With npm do

npm install algebra

or use a CDN adding this to your HTML page

<script src=""></script>

Quick start

This is a 60 seconds tutorial to get your hands dirty with algebra.

First of all, import algebra package.

const algebra = require('algebra')

Try it out

All code in the examples below should be contained into a single file, like test/quickStart.js.

view on requirebin


Use the Real numbers as scalars.

const R = algebra.Real

Every operator is implemented both as a static function and as an object method.

Static operators return raw data, while class methods return object instances.

Use static addition operator to add three numbers.

R.add(1, 2, 3) // 1 + 2 + 3 = 6

Create two real number objects: x = 2, y = -2

let x = new R(2) // x will be overwritten, see below
const y = new R(-2)

The value r is the result of x multiplied by y.

// 2 * (-2) = -4
const r = x.mul(y)

r // Scalar { data: -4 }

// x and y are not changed // 2 // -2

Raw numbers are coerced, operators can be chained when it makes sense. Of course you can reassign x, for example, x value will be 0.1: x -> x + 3 -> x * 2 -> x ^-1

// ((2 + 3) * 2)^(-1) = 0.1
x = x.add(3).mul(2).inv()

x // Scalar { data: 0.1 }

Comparison operators equal and notEqual are available, but they cannot be chained.

x.equal(0.1) // true
x.notEqual(Math.PI) // true

You can also play with Complexes.

const C = algebra.Complex

let z1 = new C([1, 2])
const z2 = new C([3, 4])

z1 = z1.mul(z2)

z1 // Scalar { data: [-5, 10] }

z1 = z1.conj().mul([2, 0]) // [-10, -20]


Create vector space of dimension 2 over Reals.

const R2 = algebra.VectorSpace(R)(2)

Create two vectors and add them.

const v1 = new R2([0, 1])
const v2 = new R2([1, -2])

// v1 -> v1 + v2 -> [0, 1] + [1, -2] = [1, -1]
v1 = v1.add(v2)

v1 // Vector { data: [1, -1] }


Create space of matrices 3 x 2 over Reals.

const R3x2 = algebra.MatrixSpace(R)(3, 2)

Create a matrix.

//       | 1 1 |
//  m1 = | 0 1 |
//       | 1 0 |
const m1 = new R3x2([1, 1,
                     0, 1,
                     1, 0])

Multiply m1 by v1, the result is a vector v3 with dimension 3. In fact we are multiplying a 3 x 2 matrix by a 2 dimensional vector, but v1 is traited as a column vector so it is like a 2 x 1 matrix.

Then, following the row by column multiplication law we have

//  3 x 2  by  2 x 1  which gives a   3 x 1
//      ↑      ↑
//      +------+----→  by removing the middle indices.
//                   | 1 1 |
//    v3 = m1 * v1 = | 0 1 | * [1 , -1] = [0, -1, 1]
//                   | 1 0 |

const v3 = m1.mul(v1) // [0, -1, 1]

Let’s try with two square matrices 2 x 2.

const R2x2 = algebra.MatrixSpace(R)(2, 2)

const m2 = new R2x2([1, 0,
                     0, 2])

const m3 = new R2x2([0, -1,
                     1, 0])

m2 = m2.mul(m3)

m2 // Matrix { data: [0, -1, 2, 0] }

Since m2 is a square matrix we can calculate its determinant.

m2.determinant // Scalar { data: 2 }


About operators

All operators are implemented as static methods and as object methods. In both cases, operands are coerced to raw data. As an example, consider addition of vectors in a plane.

const R2 = algebra.R2

const vector1 = new R2([1, 2])
const vector2 = new R2([3, 4])

The following static methods, give the same result: [4, 6].

R2.addition(vector1, [3, 4])
R2.addition([1, 2], vector2)
R2.addition(vector1, vector2)

The following object methods, give the same result: a vector instance with data [4, 6].

const vector3 = vector1.addition([3, 4])
const vector4 = vector1.addition(vector2)

R2.equal(vector3, vector4) // true

Operators can be chained and accept multiple arguments when it makes sense.

vector1.addition(vector1, vector1).equality([3, 6]) // true

Objects are immutable // still [1, 2]



Create an algebra cyclic ring, by passing its elements. The elements are provided as a string or an array, which lenght must be a prime number. This is necessary, otherwise the result would be a wild land where you can find zero divisor beasts.

Let’s create a cyclic ring containing lower case letters, numbers and the blank char. How many are they? They are 26 + 10 + 1 = 37, that is prime! We like it.

const Cyclic = algebra.Cyclic

// The elements String or Array length must be prime.
const elements = ' abcdefghijklmnopqrstuvwyxz0123456789'

const Alphanum = Cyclic(elements)

Operators derive from modular arithmetic

const a = new Alphanum('a')

Alphanum.addition('a', 'b') // 'c'

You can also create element instances, and do any kind of operations.

const x = new Alphanum('a')

const y = x.add('c', 'a', 't')
           .mul('i', 's')
           .add('o', 'n')
           .sub('t', 'h', 'e')
           .div('t', 'a', 'b', 'l', 'e') // 's'

Yes, they are scalars so you can build vector or matrix spaces on top of them.

const VectorStrings2 = algebra.VectorSpace(Alphanum)(2)
const MatrixStrings2x2 = algebra.MatrixSpace(Alphanum)(2)

const vectorOfStrings = new VectorStrings2(['o', 'k'])

const matrixOfStrings = new MatrixStrings2x2(['c', 'o',
                                              'o', 'l'])

matrixOfStrings.mul(vectorOfStrings).data // ['x', 'y']

Note that, in the particular example above, since the matrix is simmetric it commutes with the vector, hence changing the order of the operands the result is still the same.

vectorOfStrings.mul(matrixOfStrings).data // ['x', 'y']


A composition algebra is one of ℝ, ℂ, ℍ, O: Real, Complex, Quaternion, Octonion. A generic function is provided to iterate the Cayley-Dickson construction over any field.

CompositionAlgebra(field[, num])

Let’s use for example the algebra.Boole which implements Boolean Algebra by exporting an object with all the stuff needed by algebra-ring npm package.

const CompositionAlgebra = algebra.CompositionAlgebra

const Boole = algebra.Boole

const Bit = CompositionAlgebra(Boole)

Bit.contains(false) // true
Bit.contains(4) // false

const bit = new Bit(true)
Bit.addition(false).data // true

Not so exciting, let’s build something more interesting. Let’s pass a second parameter, that is used to build a Composition algebra over the given field. It is something experimental also for me, right now I am writing this but I still do not know how it will behave. My idea (idea feliz) is that

A byte is an octonion of bits

Maybe we can discover some new byte operator, taken from octonion rich algebra structure. Create an octonion algebra over the binary field, a.k.a Z2 and create the eight units.

// n must be a power of two
const Byte = CompositionAlgebra(Boole, 8)

// Use a single char const for better indentation.
const t = true
const f = false

const byte1 = new Byte([t, f, f, f, f, f, f, f])
const byte2 = new Byte([f, t, f, f, f, f, f, f])
const byte3 = new Byte([f, f, t, f, f, f, f, f])
const byte4 = new Byte([f, f, f, t, f, f, f, f])
const byte5 = new Byte([f, f, f, f, t, f, f, f])
const byte6 = new Byte([f, f, f, f, f, t, f, f])
const byte7 = new Byte([f, f, f, f, f, f, t, f])
const byte8 = new Byte([f, f, f, f, f, f, f, t])

The first one corresponds to one, while the rest are immaginary units. Every imaginary unit multiplied by itself gives -1, but since the underlying field is homomorphic to Z2, -1 corresponds to 1.

byte1.mul(byte1).data // [t, f, f, f, f, f, f, f]
byte2.mul(byte2).data // [t, f, f, f, f, f, f, f]
byte3.mul(byte3).data // [t, f, f, f, f, f, f, f]
byte4.mul(byte4).data // [t, f, f, f, f, f, f, f]
byte5.mul(byte5).data // [t, f, f, f, f, f, f, f]
byte6.mul(byte6).data // [t, f, f, f, f, f, f, f]
byte7.mul(byte7).data // [t, f, f, f, f, f, f, f]
byte8.mul(byte8).data // [t, f, f, f, f, f, f, f]

Keeping in mind that Byte space defined above is an algebra, i.e. it has composition laws well defined, you maybe already noticed that, for example byte2 could be seen as corresponding to 4, but in this strange structure we created, 4 * 4 = 2.

You can play around with this structure.

const max = byte1.add(byte2).add(byte3).add(byte4)
                 .add(byte5).add(byte6).add(byte7).add(byte8) // [t, t, t, t, t, t, t, t]


NOTA BENE The color space example in this section is still a WiP.

The scalars are the building blocks, they are the elements you can use to create vectors and matrices. They are the underneath set enriched with a ring structure which consists of two binary operators that generalize the arithmetic operations of addition and multiplication. A ring that has the commutativity property is called abelian (in honour to Abel) or also a field.

Ok, let’s make a simple example. Real numbers, with common addition and multiplication are a scalar field: see documentation below. The good new is that you can create any scalar field as long as you provide a set with two internal operations and related neutral elements that satisfy the ring axioms. That is why it will be used something maybe you did not expect could be treated as an algebra: in the examples below during this section we will play with the color space, giving a ring structure.

Let’s consider the space of html colors in the form

RGB: Red Green Blue

composed of three hexadecimal values from 00 to ff. Let’s start defining a sum operator on hexadecimals.

Credits and thanks for dec to hex and viceversa conversions goes to this gist author.

const hexSum = (hex1, hex2) => {
  const dec1 = parseInt(hex1, 16) % 256
  const dec2 = parseInt(hex2, 16) % 256

  // Sum modulo 256 and convert to hexadecimal.
  const hexResult = parseInt((dec1 + dec2) % 256, 10).toString(16)

  // Return left padded result.
  return hexResult.padStart(2, '0')

Note that it is used modulo 256 cause we need that our set is closed on this operator, it means that the sum of two colors must be another color.

To define color sum we can split a color in an array of three hexadecimals, and sum componentwise.

const splitColor = (color) => {
  const r = color.substring(0, 2)
  const g = color.substring(2, 4)
  const b = color.substring(4, 6)

  return [r, g, b]

For example, white color ffffff will be splitted in ['ff', 'ff', 'ff'].

const colorSum = (color1, color2) => {
  const [r1, g1, b1] = splitColor(color1)
  const [r2, g2, b2] = splitColor(color2)

  const r = hexSum(r1, r2)
  const g = hexSum(g1, g2)
  const b = hexSum(b1, b2)

  return [r, g, b].join('')

You can check that this sum is well defined, and for example, green plus blue equals cyan.

colorSum('00ff00', '0000ff') // '00ffff'

The neutral element respect to this operator is black (000000).

To define a scalar field we need another operation to be used as multiplication. Let’s define a multiplication on hexadecimals first.

const hexMul = (hex1, hex2) => {
  const dec1 = parseInt(hex1, 16) % 256
  const dec2 = parseInt(hex2, 16) % 256

  // Multiply, then divide by 255 and convert to hexadecimal.
  const hexResult = parseInt((dec1 * dec2) / 255, 10).toString(16)

  // Return left padded result.
  return hex.padStart(2, '0')

Then similarly to colorSum it is possible to define a colorMul that applies hexMul componentwise.

const colorMul = (color1, color2) => {
  const [r1, g1, b1] = splitColor(color1)
  const [r2, g2, b2] = splitColor(color2)

  const r = hexMul(r1, r2)
  const g = hexMul(g1, g2)
  const b = hexMul(b1, b2)

  return [r, g, b].join('')

The neutral element for this operator is white (ffffff).

We are ready to create our scalar field over RGB colors. Arguments are the same as algebra-ring.

const RGB = algebra.Scalar(
  [ '000000', 'ffffff' ],
    equality: (a, b) => a === b,
    contains: (color) => {
      const [r, g, b] = splitColor(color)

      return (parseInt(r, 16) < 256) && (parseInt(g, 16) < 256) && (parseInt(b, 16) < 256)
    addition: colorSum,
    negation: (color) => {
      const [r, g, b] = splitColor(color)

      const decR = parseInt(r, 16)
      const decG = parseInt(g, 16)
      const decB = parseInt(b, 16)

      const minusR = decR === 0 ? 0 : 256 - decR
      const minusG = decG === 0 ? 0 : 256 - decG
      const minusB = decB === 0 ? 0 : 256 - decB

      const hexMinusR = parseInt(minusR, 10).toString(16)
      const hexMinusG = parseInt(minusG, 10).toString(16)
      const hexMinusB = parseInt(minusB, 10).toString(16)

      const paddedMinusR = hexMinusR.padStart(2, '0')
      const paddedMinusG = hexMinusG.padStart(2, '0')
      const paddedMinusB = hexMinusB.padStart(2, '0')

      return `${paddedMinusR}${paddedMinusG}${paddedMinusB}`
    multiplication: colorMul,
    inversion: (color) => {
      const [r, g, b] = splitColor(color)

      const decR = parseInt(r, 16)
      const decG = parseInt(g, 16)
      const decB = parseInt(b, 16)

      const invR = parseInt(255 * 255 / decR, 10).toString(16)
      const invG = parseInt(255 * 255 / decG, 10).toString(16)
      const invB = parseInt(255 * 255 / decB, 10).toString(16)

      const paddedInvR = invR.padStart(2, '0')
      const paddedInvG = invG.padStart(2, '0')
      const paddedInvB = invB.padStart(2, '0')

      return `${paddedInvR}${paddedInvG}${paddedInvB}`

So far so good, algebra dependencies will do some checks under the hood and complain if something looks wrong. Now we can create color instances

const green = new RGB('00ff00')
const blue = new RGB('0000ff')

And as you may expect, you can do operations with them

const cyan = green.add(blue) // '00ffff'

Scalar attributes

Is the neutral element for multiplication operator. In our RGB example it corrensponds to white (ffffff). // 'ffffff'

Is the neutral element for addition operator. In our RGB example it corrensponds to black (000000) // '000000'

The data attribute holds the raw data underneath our scalar instance. // '00ff00' // '0000ff' // '00ffff'

Scalar operators

Scalar set operators

Scalar.contains(scalar1, scalar2[, scalar3, … ])

Is a static method that checks a given argument is contained in the scalar field that was defined.

RGB.contains('ffffff') // true
RGB.contains('not a color') // false

This is a class method that checks a scalar instance is contained in the given scalar field.

green.belongsTo(RGB) // true

Scalar equality

Scalar.equality(scalar1, scalar2)

Is a static method


Scalar disequality

Scalar.disequality(scalar1, scalar2)

Is a static method


Scalar addition

Scalar.addition(scalar1, scalar2[, scalar3, … ])

Is a static method

scalar1.addition(scalar2[, scalar3, … ])

Scalar subtraction

Scalar.subtraction(scalar1, scalar2[, … ])

Is a static method

scalar1.subtraction(scalar2[, scalar3, … ])

Scalar multiplication

Scalar.multiplication(scalar1, scalar2[, scalar3, … ])

Is a static method

scalar1.multiplication(scalar2[, scalar3, … ])

Scalar division

Scalar.division(scalar1, scalar2[, scalar3, … ])

Is a static method

scalar1.division(scalar2[, scalar3, … ])

Scalar negation


Is a static method


Scalar inversion


Is a static method


Scalar conjugation


Is a static method



Inherits everything from Scalar.

const Real = algebra.Real

Real.addition(1, 2) // 3

const pi = new Real(Math.PI)
const twoPi = pi.mul(2)

Real.subtraction(twoPi, 2 * Math.PI) // 0


Inherits everything from Scalar.

It is said the Gauss brain is uncommonly big and folded, much more than the Einstein brain (both are conserved and studied). Gauss was one of the biggest mathematicians and discovered many important results in many mathematic areas. One of its biggest intuitions, in my opinion, was to realize that the Complex number field is geometrically a plane. The Complex numbers are an extension on the Real numbers, they have a real part and an imaginary part. The imaginary numbers, as named by Descartes later, were discovered by italian mathematicians Cardano, Bombelli among others as a trick to solve third order equations.

Complex numbers are a goldmine for mathematics, they are incredibly rich of deepest beauty: just as a divulgative example, take a look to the Mandelbrot set, but please trust me, this is nothing compared to the divine nature of Complex numbers.

Mandelbrot Set

The first thing I noticed when I started to study the Complex numbers is conjugation. Every Complex number has its conjugate, that is its simmetric counterparte respect to the Real numbers line.

const Complex = algebra.Complex

const complex1 = new Complex([1, 2])

complex1.conjugation() // Complex { data: [1, -2] }


Inherits everything from Scalar.


Inherits everything from Scalar.

Common spaces


The real line.

It is in alias of Real.

const R = algebra.R


The real plane.

const R2 = algebra.R2

It is in alias of VectorSpace(Real)(2).


The real space.

const R3 = algebra.R3

It is in alias of VectorSpace(Real)(3).


Real square matrices of rank 2.

const R2x2 = algebra.R2x2

It is in alias of MatrixSpace(Real)(2).


The complex numbers.

It is in alias of Complex.

const C = algebra.C


Complex square matrices of rank 2.

const C2x2 = algebra.C2x2

It is in alias of MatrixSpace(Complex)(2).


Usually it is used the H in honour of Sir Hamilton.

It is in alias of Quaternion.

const H = algebra.H


A Vector extends the concept of number, since it is defined as a tuple of numbers. For example, the Cartesian plane is a set where every point has two coordinates, the famous (x, y) that is in fact a vector of dimension 2. A Scalar itself can be identified with a vector of dimension 1.

We have already seen an implementation of the plain: R2.

If you want to find the position of an airplain, you need latitute, longitude but also altitude, hence three coordinates. That is a 3-ple, a tuple with three numbers, a vector of dimension 3.

An implementation of the vector space of dimension 3 over reals is given by R3.


Vector dimension

Strictly speaking, dimension of a Vector is the number of its elements.


It is a static class attribute.

R2.dimension // 2
R3.dimension // 3

It is also defined as a static instance attribute.

const vector = new R2([1, 1])

vector.dimension // 2

Vector norm

The norm, at the end, is the square of the vector length: the good old Pythagorean theorem. It is usually defined as the sum of the squares of the coordinates. Anyway, it must be a function that, given an element, returns a positive real number. For example in Complex numbers it is defined as the multiplication of an element and its conjugate: it works as a well defined norm. It is a really important property since it shapes a metric space. In the Euclidean topology it gives us the common sense of space, but it is also important in other spaces, like a functional space. In fact a norm gives us a distance defined as its square root, thus it defines a metric space and hence a topology: a lot of good stuff.


Is a static operator that returns the square of the lenght of the vector.

R2.norm([3, 4]).data // 25


This implements a static attribute that returns the square of the length of the vector instance.

const vector = new R2([1, 2]) // 5

Vector addition

Vector.addition(vector1, vector2)
R2.addition([2, 1], [1, 2]) // [3, 3]
const vector1 = new R2([2, 1])
const vector2 = new R2([2, 2])

const vector3 = vector1.addition(vector2)

vector3 // Vector { data: [4, 3] }

Vector cross product

It is defined only in dimension three. See Cross product on wikipedia.

Vector.crossProduct(vector1, vector2)
R3.crossProduct([3, -3, 1], [4, 9, 2]) // [-15, 2, 39]
const vector1 = new R3([3, -3, 1])
const vector2 = new R3([4, 9, 2])

const vector3 = vector1.crossProduct(vector2)

vector3 // Vector { data: [-15, 2, 39] }


MatrixSpace(Scalar)(numRows[, numCols])




Matrix multiplication

Matrix.multiplication(matrix1, matrix2)

Matrix inversion

It is defined only for square matrices which determinant is not zero.


Matrix determinant

It is defined only for square matrices.


Matrix adjoint