# Fraction.js - ℚ in JavaScript
[](https://npmjs.org/package/fraction.js "View this project on npm")
[](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. Integration is straightforward:
```javascript
import Fraction from 'fraction.js';
// Alternatively
var Fraction = require('fraction.js');
```
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/infusion/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
Installing fraction.js is as easy as cloning this repo or use the following command:
```bash
npm install fraction.js
```
## Using Fraction.js with the browser
```html
<script src="fraction.min.js"></script>
<script>
console.log(Fraction("123/456"));
</script>
```
## Using Fraction.js with TypeScript
```js
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.
## 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.