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"readme": "# Fraction.js - ℚ in JavaScript\n\n[](https://npmjs.org/package/fraction.js \"View this project on npm\")\n[](http://opensource.org/licenses/MIT)\n\n\nTired 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:\n\n```javascript\n1 / 98 * 98 // = 0.9999999999999999\n```\n\nIf you need more precision or just want a fraction as a result, just include *Fraction.js*:\n\n```javascript\nvar Fraction = require('fraction.js');\n// or\nimport Fraction from 'fraction.js';\n```\n\nand give it a trial:\n\n```javascript\nFraction(1).div(98).mul(98) // = 1\n```\n\nInternally, 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).\n\nConvert decimal to fraction\n===\nThe simplest job for fraction.js is to get a fraction out of a decimal:\n```javascript\nvar x = new Fraction(1.88);\nvar res = x.toFraction(true); // String \"1 22/25\"\n```\n\nExamples / Motivation\n===\nA simple example might be\n\n```javascript\nvar f = new Fraction(\"9.4'31'\"); // 9.4313131313131...\nf.mul([-4, 3]).mod(\"4.'8'\"); // 4.88888888888888...\n```\nThe result is\n\n```javascript\nconsole.log(f.toFraction()); // -4154 / 1485\n```\nYou could of course also access the sign (s), numerator (n) and denominator (d) on your own:\n```javascript\nf.s * f.n / f.d = -1 * 4154 / 1485 = -2.797306...\n```\n\nIf you would try to calculate it yourself, you would come up with something like:\n\n```javascript\n(9.4313131 * (-4 / 3)) % 4.888888 = -2.797308133...\n```\n\nQuite okay, but yea - not as accurate as it could be.\n\n\nLaplace Probability\n===\nSimple example. What's the probability of throwing a 3, and 1 or 4, and 2 or 4 or 6 with a fair dice?\n\nP({3}):\n```javascript\nvar p = new Fraction([3].length, 6).toString(); // 0.1(6)\n```\n\nP({1, 4}):\n```javascript\nvar p = new Fraction([1, 4].length, 6).toString(); // 0.(3)\n```\n\nP({2, 4, 6}):\n```javascript\nvar p = new Fraction([2, 4, 6].length, 6).toString(); // 0.5\n```\n\nConvert degrees/minutes/seconds to precise rational representation:\n===\n\n57+45/60+17/3600\n```javascript\nvar deg = 57; // 57°\nvar min = 45; // 45 Minutes\nvar sec = 17; // 17 Seconds\n\nnew Fraction(deg).add(min, 60).add(sec, 3600).toString() // -> 57.7547(2)\n```\n\n\nRational approximation of irrational numbers\n===\n\nNow 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*.\n\nThen the following algorithm will generate the rational number besides the binary representation.\n\n```javascript\nvar x = \"/\", s = \"\";\n\nvar a = new Fraction(0),\n b = new Fraction(1);\nfor (var n = 0; n <= 10; n++) {\n\n var c = a.add(b).div(2);\n\n console.log(n + \"\\t\" + a + \"\\t\" + b + \"\\t\" + c + \"\\t\" + x);\n\n if (c.add(2).pow(2) < 5) {\n a = c;\n x = \"1\";\n } else {\n b = c;\n x = \"0\";\n }\n s+= x;\n}\nconsole.log(s)\n```\n\nThe result is\n\n```\nn a[n] b[n] c[n] x[n]\n0 0/1 1/1 1/2 /\n1 0/1 1/2 1/4 0\n2 0/1 1/4 1/8 0\n3 1/8 1/4 3/16 1\n4 3/16 1/4 7/32 1\n5 7/32 1/4 15/64 1\n6 15/64 1/4 31/128 1\n7 15/64 31/128 61/256 0\n8 15/64 61/256 121/512 0\n9 15/64 121/512 241/1024 0\n10 241/1024 121/512 483/2048 1\n```\nThus 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))\n\n\nI 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).\n\n\nGet the exact fractional part of a number\n---\n```javascript\nvar f = new Fraction(\"-6.(3416)\");\nconsole.log(\"\" + f.mod(1).abs()); // 0.(3416)\n```\n\nMathematical correct modulo\n---\nThe 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:\n\n```javascript\nvar a = -1;\nvar b = 10.99;\n\nconsole.log(new Fraction(a)\n .mod(b)); // Not correct, usual Modulo\n\nconsole.log(new Fraction(a)\n .mod(b).add(b).mod(b)); // Correct! Mathematical Modulo\n```\n\nfmod() impreciseness circumvented\n---\nIt 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.\n\nThe 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.\n\n\nParser\n===\n\nAny function (see below) as well as the constructor of the *Fraction* class parses its input and reduce it to the smallest term.\n\nYou can pass either Arrays, Objects, Integers, Doubles or Strings.\n\nArrays / Objects\n---\n```javascript\nnew Fraction(numerator, denominator);\nnew Fraction([numerator, denominator]);\nnew Fraction({n: numerator, d: denominator});\n```\n\nIntegers\n---\n```javascript\nnew Fraction(123);\n```\n\nDoubles\n---\n```javascript\nnew Fraction(55.4);\n```\n\n**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.\n\nThe 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.\n\n\nStrings\n---\n```javascript\nnew Fraction(\"123.45\");\nnew Fraction(\"123/45\"); // A rational number represented as two decimals, separated by a slash\nnew Fraction(\"123:45\"); // A rational number represented as two decimals, separated by a colon\nnew Fraction(\"4 123/45\"); // A rational number represented as a whole number and a fraction\nnew Fraction(\"123.'456'\"); // Note the quotes, see below!\nnew Fraction(\"123.(456)\"); // Note the brackets, see below!\nnew Fraction(\"123.45'6'\"); // Note the quotes, see below!\nnew Fraction(\"123.45(6)\"); // Note the brackets, see below!\n```\n\nTwo arguments\n---\n```javascript\nnew Fraction(3, 2); // 3/2 = 1.5\n```\n\nRepeating decimal places\n---\n*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)\n\nAssume 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:\n\n```javascript\nvar f = new Fraction(\"123.32\");\nconsole.log(\"Bam: \" + f.div(\"33.6(567)\"));\n```\n\nTo 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...\n\n```javascript\nfunction formatDecimal(str) {\n\n var comma, pre, offset, pad, times, repeat;\n\n if (-1 === (comma = str.indexOf(\".\")))\n return str;\n\n pre = str.substr(0, comma + 1);\n str = str.substr(comma + 1);\n\n for (var i = 0; i < str.length; i++) {\n\n offset = str.substr(0, i);\n\n for (var j = 0; j < 5; j++) {\n\n pad = str.substr(i, j + 1);\n\n times = Math.ceil((str.length - offset.length) / pad.length);\n\n repeat = new Array(times + 1).join(pad); // Silly String.repeat hack\n\n if (0 === (offset + repeat).indexOf(str)) {\n return pre + offset + \"(\" + pad + \")\";\n }\n }\n }\n return null;\n}\n\nvar f, x = formatDecimal(\"13.0123123123\"); // = 13.0(123)\nif (x !== null) {\n f = new Fraction(x);\n}\n```\n\nAttributes\n===\n\nThe 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.\n\n```javascript\nvar f = new Fraction('-1/2');\nconsole.log(f.n); // Numerator: 1\nconsole.log(f.d); // Denominator: 2\nconsole.log(f.s); // Sign: -1\n```\n\n\nFunctions\n===\n\nFraction abs()\n---\nReturns the actual number without any sign information\n\nFraction neg()\n---\nReturns the actual number with flipped sign in order to get the additive inverse\n\nFraction add(n)\n---\nReturns the sum of the actual number and the parameter n\n\nFraction sub(n)\n---\nReturns the difference of the actual number and the parameter n\n\nFraction mul(n)\n---\nReturns the product of the actual number and the parameter n\n\nFraction div(n)\n---\nReturns the quotient of the actual number and the parameter n\n\nFraction pow(exp)\n---\nReturns the power of the actual number, raised to an possible rational exponent. If the result becomes non-rational the function returns `null`.\n\nFraction mod(n)\n---\nReturns 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).\n\nFraction mod()\n---\nReturns the modulus (rest of the division) of the actual object (numerator mod denominator)\n\nFraction gcd(n)\n---\nReturns the fractional greatest common divisor\n\nFraction lcm(n)\n---\nReturns the fractional least common multiple\n\nFraction ceil([places=0-16])\n---\nReturns the ceiling of a rational number with Math.ceil\n\nFraction floor([places=0-16])\n---\nReturns the floor of a rational number with Math.floor\n\nFraction round([places=0-16])\n---\nReturns the rational number rounded with Math.round\n\nFraction roundTo(multiple)\n---\nRounds a fraction to the closest multiple of another fraction. \n\nFraction inverse()\n---\nReturns the multiplicative inverse of the actual number (n / d becomes d / n) in order to get the reciprocal\n\nFraction simplify([eps=0.001])\n---\nSimplifies the rational number under a certain error threshold. Ex. `0.333` will be `1/3` with `eps=0.001`\n\nboolean equals(n)\n---\nCheck if two numbers are equal\n\nint compare(n)\n---\nCompare two numbers.\n```\nresult < 0: n is greater than actual number\nresult > 0: n is smaller than actual number\nresult = 0: n is equal to the actual number\n```\n\nboolean divisible(n)\n---\nCheck if two numbers are divisible (n divides this)\n\ndouble valueOf()\n---\nReturns a decimal representation of the fraction\n\nString toString([decimalPlaces=15])\n---\nGenerates 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.\n\n**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!\n\nString toLatex(excludeWhole=false)\n---\nGenerates 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.\n\nThe optional boolean parameter indicates if you want to exclude the whole part. \"1 1/3\" instead of \"4/3\"\n\nString toFraction(excludeWhole=false)\n---\nGets a string representation of the fraction\n\nThe optional boolean parameter indicates if you want to exclude the whole part. \"1 1/3\" instead of \"4/3\"\n\nArray toContinued()\n---\nGets an array of the fraction represented as a continued fraction. The first element always contains the whole part.\n\n```javascript\nvar f = new Fraction('88/33');\nvar c = f.toContinued(); // [2, 1, 2]\n```\n\nFraction clone()\n---\nCreates a copy of the actual Fraction object\n\n\nExceptions\n===\nIf a really hard error occurs (parsing error, division by zero), *fraction.js* throws exceptions! Please make sure you handle them correctly.\n\n\n\nInstallation\n===\nInstalling fraction.js is as easy as cloning this repo or use the following command:\n\n```\nnpm install fraction.js\n```\n\nUsing Fraction.js with the browser\n===\n```html\n\n\n```\n\nUsing Fraction.js with TypeScript\n===\n```js\nimport Fraction from \"fraction.js\";\nconsole.log(Fraction(\"123/456\"));\n```\n\nCoding Style\n===\nAs 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.\n\n\nPrecision\n===\nFraction.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. \n\nTesting\n===\nIf 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\n\n```\nnpm test\n```\n\n\nCopyright and licensing\n===\nCopyright (c) 2023, [Robert Eisele](https://raw.org/)\nLicensed under the MIT license.\n",
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