Index: node_modules/fraction.js/README.md
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+# 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.
+
+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/rawify/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
+
+You can install `Fraction.js` via npm:
+
+```bash
+npm install fraction.js
+```
+
+Or with yarn:
+
+```bash
+yarn add fraction.js
+```
+
+Alternatively, download or clone the repository:
+
+```bash
+git clone https://github.com/rawify/Fraction.js
+```
+
+## Usage
+
+Include the `fraction.min.js` file in your project:
+
+```html
+
+
+```
+
+Or in a Node.js project:
+
+```javascript
+const Fraction = require('fraction.js');
+```
+
+or
+
+```javascript
+import Fraction from '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.
+
+## 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.