Fraction.js - ℚ in JavaSript

Tired of inprecise numbers represented by doubles? Have a look at Fraction.js, which represents rational numbers or ratios as two integers in the form of n / d.

Examples

A simple example might be

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

The result is

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.

To approximate a number like sqrt(5) - 2 as n / d, you can reformat the equation as follows: pow(n / d + 2, 2) is 5

The formulated algorithm, which also generates the binary representation, could look like

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

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

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

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

    if (Math.pow(c.n / c.d + 2, 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.

Get the exact fractional part of a number

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

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 with Fraction.js you could come up with this:

var a = -1;
var b = 10.99;

console.log(new Fraction(a)
     .mod(b)
     .toNumber()); // Not correct

console.log(new Fraction(a)
     .mod(b).add(b).mod(b)
     .toNumber()); // Correct!

fmod() impreciseness circumvented

It turns out that Fraction.js outperforms almost any fmod() implementation, including JavaScript itself, 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 it's input and reduce it to the smallest term.

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

Arrays / Objects

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

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 approximation.

Strings

new Fraction("123.45");
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!

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);
}

Functions

Fraction abs()

Returns the actual number without any sign information

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 set(n)

Set a number n to the actual object

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 will.

Fraction reciprocal()

Returns the reciprocal of the actual number (n / d becomes d / n)

boolean equals(n)

Check if two numbers are equal

boolean divisible(n)

Check if two numbers are divisible (n divides this)

double toNumber()

Returns a decimal representation of the fraction

String toString()

Generates an exact string representation of the actual object, including repeating decimal places of any length.

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 one of the following commands:

bower install fraction.js

or

npm install 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.

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) 2014, Robert Eisele (robert@xarg.org) Dual licensed under the MIT or GPL Version 2 licenses.