simple-squiggle

README.md (15812B)

---

 # Fraction.js - ℚ in JavaScript
 
 [![NPM Package](https://nodei.co/npm-dl/fraction.js.png?months=6&height=1)](https://npmjs.org/package/fraction.js)
 
 [![Build Status](https://travis-ci.org/infusion/Fraction.js.svg?branch=master)](https://travis-ci.org/infusion/Fraction.js)
 [![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, have a look at *Fraction.js*:
 
 ```javascript
 var Fraction = require('fraction.js');
 
 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 in mind and outperforms any other implementation, as you can see [here](http://jsperf.com/convert-a-rational-number-to-a-babylonian-fractions/28). This basic data-type 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)
 ```
 
 Rounding a fraction to the closest tape measure value
 === 
 
 A tape measure is usually divided in parts of `1/16`. Rounding a given fraction to the closest value on a tape measure can be determined by
 
 ```javascript
 function closestTapeMeasure(frac) {
 
     /*
     k/16 ≤ a/b < (k+1)/16
     ⇔ k ≤ 16*a/b < (k+1)
     ⇔ k = floor(16*a/b)
     */
     return new Fraction(Math.round(16 * Fraction(frac).valueOf()), 16);
 }
 // closestTapeMeasure("1/3") // 5/16
 ```
 
 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()); // Will print 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 will. 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 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 one of the following commands:
 
 ```
 bower install fraction.js
 ```
 or
 
 ```
 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 require.js
 ===
 ```html
 <script src="require.js"></script>
 <script>
 requirejs(['fraction.js'],
 function(Fraction) {
     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) 2014-2019, [Robert Eisele](https://www.xarg.org/)
 Dual licensed under the MIT or GPL Version 2 licenses.