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-# Fraction.js - ℚ in JavaScript
-
-[![NPM Package](https://img.shields.io/npm/v/fraction.js.svg?style=flat)](https://npmjs.org/package/fraction.js "View this project on npm")
-[![MIT license](http://img.shields.io/badge/license-MIT-brightgreen.svg)](http://opensource.org/licenses/MIT)
-
-
-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:
-
-```javascript
-1 / 98 * 98 // = 0.9999999999999999
-```
-
-If you need more precision or just want a fraction as a result, just include *Fraction.js*:
-
-```javascript
-var Fraction = require('fraction.js');
-// or
-import Fraction from 'fraction.js';
-```
-
-and give it a trial:
-
-```javascript
-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](https://github.com/infusion/Polynomial.js) and [Math.js](https://github.com/josdejong/mathjs).
-
-Convert decimal to fraction
-===
-The simplest job for fraction.js is to get a fraction out of a decimal:
-```javascript
-var x = new Fraction(1.88);
-var res = x.toFraction(true); // String "1 22/25"
-```
-
-Examples / Motivation
-===
-A simple example might be
-
-```javascript
-var f = new Fraction("9.4'31'"); // 9.4313131313131...
-f.mul([-4, 3]).mod("4.'8'"); // 4.88888888888888...
-```
-The result is
-
-```javascript
-console.log(f.toFraction()); // -4154 / 1485
-```
-You could of course also access the sign (s), numerator (n) and denominator (d) on your own:
-```javascript
-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:
-
-```javascript
-(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}):
-```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
-===
-
-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.
-
-```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) < 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](http://www.xarg.org/2014/03/precise-calculations-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()); // 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:
-
-```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() impreciseness 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 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 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](http://www.xarg.org/2014/03/precise-calculations-in-javascript/) 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.
-
-```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:
-
-```
-npm install fraction.js
-```
-
-Using Fraction.js with the browser
-===
-```html
-<script src="fraction.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.
-
-
-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](https://raw.org/)
-Licensed under the MIT license.