No Problem Man

# npm

Meet npm Pro: unlimited public & private packages + package-based permissions.Learn more »

## algebra

1.0.1 • Public • Published

# algebra

means completeness and balancing, from the Arabic word الجبر  With npm do

## Quick start

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

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

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

First of all, import algebra package.

### Scalars

Use the Real numbers as scalars.

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.

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

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

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

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

You can also play with Complexes.

### Vectors

Create vector space of dimension 2 over Reals.

Create two vectors and add them.

### Matrices

Create space of matrices 3 x 2 over Reals.

Create a matrix.

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

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

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

## API

All operators can be 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 Cartesian Plane.

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

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

Operators can be chained when it makes sense.

Objects are immutable

### CompositionAlgebra

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])`

• num can be 1, 2, 4 or 8

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.

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.

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.

You can play around with this structure.

### Scalar

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.

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.

We are going to create a scalar field using `BigInt` elements to implement something similar to a Rational Number. The idea is to use a couple of numbers, the first one is the numerator and the second one the denominator.

Arguments we need are the same as algebra-ring. Let's start by unities; every element is a couple of numbers, the numerator and the denominator, hence unitites are:

• zero: `[ BigInt(0), BigInt(1) ]`
• one: `[ BigInt(1), BigInt(1) ]`

We need a function that computes the Great Common Divisor.

So now we can normalize a rational number, by removing the common divisors of numerator and denominator.

So far so good, algebra dependencies will do some checks under the hood and will complain if something looks wrong.

Let's create few rational numbers.

#### `Scalar.one`

Is the neutral element for multiplication operator.

#### `Scalar.zero`

Is the neutral element for addition operator.

#### `scalar.data`

The data attribute holds the raw data underneath our scalar instance.

#### `Scalar.contains(scalar)`

Checks a given argument is contained in the scalar field that was defined.

#### `scalar1.belongsTo(Scalar)`

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

#### `Scalar.equality(scalar1, scalar2)`

Is a static method

### Real

Inherits everything from Scalar. Implements algebra of real numbers.

### Complex

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. {:.responsive}

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.

### Quaternion

Inherits everything from Scalar.

Quaternions are not commutative, usually if you invert the operands in a multiplication you get the same number in absolute value but with the sign inverted.

### Octonion

Inherits everything from Scalar.

Octonions are not associative, this is getting hard: `a * (b * c)` could be equal to the negation of `(a * b) * c`.

### Common spaces

#### R

The real line.

It is in alias of Real.

#### R2

The Cartesian Plane.

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

#### R3

The real space.

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

#### R2x2

Real square matrices of rank 2.

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

#### C

The complex numbers.

It is in alias of Complex.

#### C2x2

Complex square matrices of rank 2.

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

#### H

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

It is in alias of Quaternion.

### Vector

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.

##### `Vector.dimension`

It is a static class attribute.

##### `vector.dimension`

It is also defined as a static instance attribute.

### 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.

#### `Vector.norm()`

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

#### `vector.norm`

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

#### Vector cross product

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

### Matrix

#### Matrix inversion

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

#### Matrix determinant

It is defined only for square matrices.

MIT

## Keywords

### Install

`npm i algebra`

11

1.0.1

MIT

186 kB

31

### Homepage

g14n.info/algebra

### Repository

github.com/fibo/algebra