lix-website/themes/lix/assets/bootstrap/node_modules/fraction.js

Fraction.js - ℚ in JavaScript

Tired of inprecise numbers represented by doubles, which have to store rational and irrational numbers like PI or sqrt(2) the same way? Obviously the following problem is preventable:

1 / 98 * 98 // = 0.9999999999999999

If you need more precision or just want a fraction as a result, just include Fraction.js:

var Fraction = require('fraction.js');
// or
import Fraction from 'fraction.js';

and give it a trial:

Fraction(1).div(98).mul(98) // = 1

Internally, numbers are represented as numerator / denominator, which adds just a little overhead. However, the library is written with performance and accuracy in mind, which makes it the perfect basis for Polynomial.js and Math.js.

Convert decimal to fraction

The simplest job for fraction.js is to get a fraction out of a decimal:

var x = new Fraction(1.88);
var res = x.toFraction(true); // String "1 22/25"

Examples / Motivation

A simple example might be

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

The result is

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

You could of course also access the sign (s), numerator (n) and denominator (d) on your own:

f.s * f.n / f.d = -1 * 4154 / 1485 = -2.797306...

If you would try to calculate it yourself, you would come up with something like:

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

Quite okay, but yea - not as accurate as it could be.

Laplace Probability

Simple example. What's the probability of throwing a 3, and 1 or 4, and 2 or 4 or 6 with a fair dice?

P({3}):

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

P({1, 4}):

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

P({2, 4, 6}):

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

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

Now it's getting messy ;d 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.

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) < 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)

I published another example on how to approximate PI with fraction.js on my blog (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

var f = new Fraction("-6.(3416)");
console.log("" + f.mod(1).abs()); // 0.(3416)

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:

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() impreciseness circumvented

It turns out that Fraction.js outperforms almost any fmod() implementation, including JavaScript itself, php.js, 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

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

Integers

new Fraction(123);

Doubles

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

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

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 states that you'll get a period of 1776 digits. Fraction.js comes to the same result. Give it a try:

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...

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.

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 mod(n)

Returns the modulus (rest of the division) of the actual object and n (this % n). It's a much more precise fmod() 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.

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 numbers are equal

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 actual object. For repeated decimal places all digits are collected within brackets, like 1/3 = "0.(3)". For all other numbers, up to decimalPlaces significant digits are collected - which includes trailing zeros if the number is getting truncated. However, 1/2 = "0.5" without trailing zeros of course.

Note: As valueOf() and toString() are provided, toString() is only called implicitly in a real string context. Using the plus-operator like "123" + new Fraction will call valueOf(), because JavaScript tries to combine two primitives first and concatenates them later, as string will be the more dominant type. alert(new Fraction) or String(new Fraction) on the other hand will do what you expect. If you really want to have control, you should call toString() or valueOf() explicitly!

String toLatex(excludeWhole=false)

Generates an exact LaTeX representation of the actual object. You can see a live demo on my blog.

The optional boolean parameter indicates if you want to exclude the whole part. "1 1/3" instead of "4/3"

String toFraction(excludeWhole=false)

Gets a string representation of the fraction

The optional boolean parameter indicates if you want to exclude the whole part. "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.

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

Installing fraction.js is as easy as cloning this repo or use the following command:

npm install fraction.js

Using Fraction.js with the browser

<script src="fraction.js"></script>
<script>
    console.log(Fraction("123/456"));
</script>

Using Fraction.js with TypeScript

import Fraction from "fraction.js";
console.log(Fraction("123/456"));

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.

Precision

Fraction.js tries to circumvent floating point errors, by having an internal representation of numerator and denominator. As it relies on JavaScript, there is also a limit. The biggest number representable is Number.MAX_SAFE_INTEGER / 1 and the smallest is -1 / Number.MAX_SAFE_INTEGER, with Number.MAX_SAFE_INTEGER=9007199254740991. If this is not enough, there is bigfraction.js shipped experimentally, which relies on BigInt and should become the new Fraction.js eventually.

Testing

If you plan to enhance the library, make sure you add test cases and all the previous tests are passing. You can test the library with

npm test

Copyright and licensing

Copyright (c) 2023, Robert Eisele Licensed under the MIT license.