source: node_modules/fraction.js/README.md@ e4c61dd

# Fraction.js - ℚ in JavaScript

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[![MIT license](http://img.shields.io/badge/license-MIT-brightgreen.svg)](http://opensource.org/licenses/MIT)

Do you find the limitations of floating-point arithmetic frustrating, especially when rational and irrational numbers like π or √2 are stored within the same finite precision? This can lead to avoidable inaccuracies such as:

```javascript
1 / 98 * 98 // Results in 0.9999999999999999
```

For applications requiring higher precision or where working with fractions is preferable, consider incorporating *Fraction.js* into your project.

The library effectively addresses precision issues, as demonstrated below:

```javascript
Fraction(1).div(98).mul(98) // Returns 1
```

*Fraction.js* uses a `BigInt` representation for both the numerator and denominator, ensuring minimal performance overhead while maximizing accuracy. Its design is optimized for precision, making it an ideal choice as a foundational library for other math tools, such as [Polynomial.js](https://github.com/rawify/Polynomial.js) and [Math.js](https://github.com/josdejong/mathjs).

## Convert Decimal to Fraction

One of the core features of *Fraction.js* is its ability to seamlessly convert decimal numbers into fractions.

```javascript
let x = new Fraction(1.88);
let res = x.toFraction(true); // Returns "1 22/25" as a string
```

This is particularly useful when you need precise fraction representations instead of dealing with the limitations of floating-point arithmetic. What if you allow some error tolerance?

```javascript
let x = new Fraction(0.33333);
let res = x.simplify(0.001) // Error < 0.001
       .toFraction(); // Returns "1/3" as a string
```

## Precision

As native `BigInt` support in JavaScript becomes more common, libraries like *Fraction.js* use it to handle calculations with higher precision. This improves the speed and accuracy of math operations with large numbers, providing a better solution for tasks that need more precision than floating-point numbers can offer.

## Examples / Motivation

A simple example of using *Fraction.js* might look like this:

```javascript
var f = new Fraction("9.4'31'"); // 9.4313131313131...
f.mul([-4, 3]).mod("4.'8'"); // 4.88888888888888...
```

The result can then be displayed as:

```javascript
console.log(f.toFraction()); // -4154 / 1485
```

Additionally, you can access the internal attributes of the fraction, such as the sign (s), numerator (n), and denominator (d). Keep in mind that these values are stored as `BigInt`:

```javascript
Number(f.s) * Number(f.n) / Number(f.d) = -1 * 4154 / 1485 = -2.797306...
```

If you attempted to calculate this manually using floating-point arithmetic, you'd get something like:

```javascript
(9.4313131 * (-4 / 3)) % 4.888888 = -2.797308133...
```

While the result is reasonably close, it’s not as accurate as the fraction-based approach that *Fraction.js* provides, especially when dealing with repeating decimals or complex operations. This highlights the value of precision that the library brings.

### Laplace Probability

Here's a straightforward example of using *Fraction.js* to calculate probabilities. Let's determine the probability of rolling a specific outcome on a fair die:

- **P({3})**: The probability of rolling a 3.
- **P({1, 4})**: The probability of rolling either 1 or 4.
- **P({2, 4, 6})**: The probability of rolling 2, 4, or 6.

#### P({3}):

```javascript
var p = new Fraction([3].length, 6).toString(); // "0.1(6)"
```

#### P({1, 4}):

```javascript
var p = new Fraction([1, 4].length, 6).toString(); // "0.(3)"
```

#### P({2, 4, 6}):

```javascript
var p = new Fraction([2, 4, 6].length, 6).toString(); // "0.5"
```

### Convert degrees/minutes/seconds to precise rational representation:

57+45/60+17/3600

```javascript
var deg = 57; // 57°
var min = 45; // 45 Minutes
var sec = 17; // 17 Seconds

new Fraction(deg).add(min, 60).add(sec, 3600).toString() // -> 57.7547(2)
```


### Rational approximation of irrational numbers

To approximate a number like *sqrt(5) - 2* with a numerator and denominator, you can reformat the equation as follows: *pow(n / d + 2, 2) = 5*.

Then the following algorithm will generate the rational number besides the binary representation.

```javascript
var x = "/", s = "";

var a = new Fraction(0),
    b = new Fraction(1);
for (var n = 0; n <= 10; n++) {

  var c = a.add(b).div(2);

  console.log(n + "\t" + a + "\t" + b + "\t" + c + "\t" + x);

  if (c.add(2).pow(2).valueOf() < 5) {
    a = c;
    x = "1";
  } else {
    b = c;
    x = "0";
  }
  s+= x;
}
console.log(s)
```

The result is

```
n   a[n]        b[n]        c[n]            x[n]
0   0/1         1/1         1/2             /
1   0/1         1/2         1/4             0
2   0/1         1/4         1/8             0
3   1/8         1/4         3/16            1
4   3/16        1/4         7/32            1
5   7/32        1/4         15/64           1
6   15/64       1/4         31/128          1
7   15/64       31/128      61/256          0
8   15/64       61/256      121/512         0
9   15/64       121/512     241/1024        0
10  241/1024    121/512     483/2048        1
```

Thus the approximation after 11 iterations of the bisection method is *483 / 2048* and the binary representation is 0.00111100011 (see [WolframAlpha](http://www.wolframalpha.com/input/?i=sqrt%285%29-2+binary))

I published another example on how to approximate PI with fraction.js on my [blog](https://raw.org/article/rational-numbers-in-javascript/) (Still not the best idea to approximate irrational numbers, but it illustrates the capabilities of Fraction.js perfectly).


### Get the exact fractional part of a number

```javascript
var f = new Fraction("-6.(3416)");
console.log(f.mod(1).abs().toFraction()); // = 3416/9999
```

### Mathematical correct modulo

The behaviour on negative congruences is different to most modulo implementations in computer science. Even the *mod()* function of Fraction.js behaves in the typical way. To solve the problem of having the mathematical correct modulo with Fraction.js you could come up with this:

```javascript
var a = -1;
var b = 10.99;

console.log(new Fraction(a)
  .mod(b)); // Not correct, usual Modulo

console.log(new Fraction(a)
  .mod(b).add(b).mod(b)); // Correct! Mathematical Modulo
```

fmod() imprecision circumvented
---
It turns out that Fraction.js outperforms almost any fmod() implementation, including JavaScript itself, [php.js](http://phpjs.org/functions/fmod/), C++, Python, Java and even Wolframalpha due to the fact that numbers like 0.05, 0.1, ... are infinite decimal in base 2.

The equation *fmod(4.55, 0.05)* gives *0.04999999999999957*, wolframalpha says *1/20*. The correct answer should be **zero**, as 0.05 divides 4.55 without any remainder.


## Parser

Any function (see below) as well as the constructor of the *Fraction* class parses its input and reduce it to the smallest term.

You can pass either Arrays, Objects, Integers, Doubles or Strings.

### Arrays / Objects

```javascript
new Fraction(numerator, denominator);
new Fraction([numerator, denominator]);
new Fraction({n: numerator, d: denominator});
```

### Integers

```javascript
new Fraction(123);
```

### Doubles

```javascript
new Fraction(55.4);
```

**Note:** If you pass a double as it is, Fraction.js will perform a number analysis based on Farey Sequences. If you concern performance, cache Fraction.js objects and pass arrays/objects.

The method is really precise, but too large exact numbers, like 1234567.9991829 will result in a wrong approximation. If you want to keep the number as it is, convert it to a string, as the string parser will not perform any further observations. If you have problems with the approximation, in the file `examples/approx.js` is a different approximation algorithm, which might work better in some more specific use-cases.


### Strings

```javascript
new Fraction("123.45");
new Fraction("123/45"); // A rational number represented as two decimals, separated by a slash
new Fraction("123:45"); // A rational number represented as two decimals, separated by a colon
new Fraction("4 123/45"); // A rational number represented as a whole number and a fraction
new Fraction("123.'456'"); // Note the quotes, see below!
new Fraction("123.(456)"); // Note the brackets, see below!
new Fraction("123.45'6'"); // Note the quotes, see below!
new Fraction("123.45(6)"); // Note the brackets, see below!
```

### Two arguments

```javascript
new Fraction(3, 2); // 3/2 = 1.5
```

### Repeating decimal places

*Fraction.js* can easily handle repeating decimal places. For example *1/3* is *0.3333...*. There is only one repeating digit. As you can see in the examples above, you can pass a number like *1/3* as "0.'3'" or "0.(3)", which are synonym. There are no tests to parse something like 0.166666666 to 1/6! If you really want to handle this number, wrap around brackets on your own with the function below for example: 0.1(66666666)

Assume you want to divide 123.32 / 33.6(567). [WolframAlpha](http://www.wolframalpha.com/input/?i=123.32+%2F+%2812453%2F370%29) states that you'll get a period of 1776 digits. *Fraction.js* comes to the same result. Give it a try:

```javascript
var f = new Fraction("123.32");
console.log("Bam: " + f.div("33.6(567)"));
```

To automatically make a number like "0.123123123" to something more Fraction.js friendly like "0.(123)", I hacked this little brute force algorithm in a 10 minutes. Improvements are welcome...

```javascript
function formatDecimal(str) {

  var comma, pre, offset, pad, times, repeat;

  if (-1 === (comma = str.indexOf(".")))
    return str;

  pre = str.substr(0, comma + 1);
  str = str.substr(comma + 1);

  for (var i = 0; i < str.length; i++) {

    offset = str.substr(0, i);

    for (var j = 0; j < 5; j++) {

      pad = str.substr(i, j + 1);

      times = Math.ceil((str.length - offset.length) / pad.length);

      repeat = new Array(times + 1).join(pad); // Silly String.repeat hack

      if (0 === (offset + repeat).indexOf(str)) {
        return pre + offset + "(" + pad + ")";
      }
    }
  }
  return null;
}

var f, x = formatDecimal("13.0123123123"); // = 13.0(123)
if (x !== null) {
  f = new Fraction(x);
}
```

## Attributes


The Fraction object allows direct access to the numerator, denominator and sign attributes. It is ensured that only the sign-attribute holds sign information so that a sign comparison is only necessary against this attribute.

```javascript
var f = new Fraction('-1/2');
console.log(f.n); // Numerator: 1
console.log(f.d); // Denominator: 2
console.log(f.s); // Sign: -1
```


## Functions

### Fraction abs()

Returns the actual number without any sign information

### Fraction neg()

Returns the actual number with flipped sign in order to get the additive inverse

### Fraction add(n)

Returns the sum of the actual number and the parameter n

### Fraction sub(n)

Returns the difference of the actual number and the parameter n

### Fraction mul(n)

Returns the product of the actual number and the parameter n

### Fraction div(n)

Returns the quotient of the actual number and the parameter n

### Fraction pow(exp)

Returns the power of the actual number, raised to an possible rational exponent. If the result becomes non-rational the function returns `null`.

### Fraction log(base)

Returns the logarithm of the actual number to a given rational base. If the result becomes non-rational the function returns `null`.

### Fraction mod(n)

Returns the modulus (rest of the division) of the actual object and n (this % n). It's a much more precise [fmod()](#fmod-impreciseness-circumvented) if you like. Please note that *mod()* is just like the modulo operator of most programming languages. If you want a mathematical correct modulo, see [here](#mathematical-correct-modulo).

### Fraction mod()

Returns the modulus (rest of the division) of the actual object (numerator mod denominator)

### Fraction gcd(n)

Returns the fractional greatest common divisor

### Fraction lcm(n)

Returns the fractional least common multiple

### Fraction ceil([places=0-16])

Returns the ceiling of a rational number with Math.ceil

### Fraction floor([places=0-16])

Returns the floor of a rational number with Math.floor

### Fraction round([places=0-16])

Returns the rational number rounded with Math.round

### Fraction roundTo(multiple)

Rounds a fraction to the closest multiple of another fraction. 

### Fraction inverse()

Returns the multiplicative inverse of the actual number (n / d becomes d / n) in order to get the reciprocal

### Fraction simplify([eps=0.001])

Simplifies the rational number under a certain error threshold. Ex. `0.333` will be `1/3` with `eps=0.001`

### boolean equals(n)

Check if two rational numbers are equal

### boolean lt(n)

Check if this rational number is less than another

### boolean lte(n)

Check if this rational number is less than or equal another

### boolean gt(n)

Check if this rational number is greater than another

### boolean gte(n)

Check if this rational number is greater than or equal another

### int compare(n)

Compare two numbers.
```
result < 0: n is greater than actual number
result > 0: n is smaller than actual number
result = 0: n is equal to the actual number
```

### boolean divisible(n)

Check if two numbers are divisible (n divides this)

### double valueOf()

Returns a decimal representation of the fraction

### String toString([decimalPlaces=15])

Generates an exact string representation of the given object. For repeating decimal places, digits within repeating cycles are enclosed in parentheses, e.g., `1/3 = "0.(3)"`. For other numbers, the string will include up to the specified `decimalPlaces` significant digits, including any trailing zeros if truncation occurs. For example, `1/2` will be represented as `"0.5"`, without additional trailing zeros.

**Note:** Since both `valueOf()` and `toString()` are provided, `toString()` will only be invoked implicitly when the object is used in a string context. For instance, when using the plus operator like `"123" + new Fraction`, `valueOf()` will be called first, as JavaScript attempts to combine primitives before concatenating them, with the string type taking precedence. However, `alert(new Fraction)` or `String(new Fraction)` will behave as expected. To ensure specific behavior, explicitly call either `toString()` or `valueOf()`.

### String toLatex(showMixed=false)

Generates an exact LaTeX representation of the actual object. You can see a [live demo](https://raw.org/article/rational-numbers-in-javascript/) on my blog.

The optional boolean parameter indicates if you want to show the a mixed fraction. "1 1/3" instead of "4/3"

### String toFraction(showMixed=false)

Gets a string representation of the fraction

The optional boolean parameter indicates if you want to showa mixed fraction. "1 1/3" instead of "4/3"

### Array toContinued()

Gets an array of the fraction represented as a continued fraction. The first element always contains the whole part.

```javascript
var f = new Fraction('88/33');
var c = f.toContinued(); // [2, 1, 2]
```

### Fraction clone()

Creates a copy of the actual Fraction object


## Exceptions

If a really hard error occurs (parsing error, division by zero), *Fraction.js* throws exceptions! Please make sure you handle them correctly.


## Installation

You can install `Fraction.js` via npm:

```bash
npm install fraction.js
```

Or with yarn:

```bash
yarn add fraction.js
```

Alternatively, download or clone the repository:

```bash
git clone https://github.com/rawify/Fraction.js
```

## Usage

Include the `fraction.min.js` file in your project:

```html
<script src="path/to/fraction.min.js"></script>
<script>
  var x = new Fraction("13/4");
</script>
```

Or in a Node.js project:

```javascript
const Fraction = require('fraction.js');
```

or 

```javascript
import Fraction from 'fraction.js';
```


## Coding Style

As every library I publish, Fraction.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:

```bash
npm install
npm run build
```

## Run a test

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

```bash
npm run test
```

## Copyright and Licensing

Copyright (c) 2025, [Robert Eisele](https://raw.org/)
Licensed under the MIT license.