Fraction is a rational number library written in Java
Tired of imprecise 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:
1 / 98 * 98 // = 0.9999999999999999If you need more precision or just want a fraction as a result, have a look at Fraction.java:
import io.github.mrrefactoring.Fraction;
new Fraction(1).div(98).mul(98) // = 1Internally, 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.
The simplest job for fraction.js is to get a fraction out of a decimal:
Fraction x = new Fraction(1.88);
Fraction res = x.toFraction(true); // String "1 22/25"A simple example might be
Fraction f = new Fraction("9.4'31'"); // 9.4313131313131...
f.mul(new int[]{-4, 3}).mod("4.'8'"); // 4.88888888888888...The result is
System.out.println(f.toFraction()); // -4154 / 1485You could of course also access the sign (s), numerator (n) and denominator (d) on your own:
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:
(9.4313131 * (-4 / 3)) % 4.888888 = -2.797308133...Quite okay, but yea - not as accurate as it could be.
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}):
Fraction p = new Fraction(new int[]{3}.length, 6).toString(); // 0.1(6)P({1, 4}):
Fraction p = new Fraction(new int[]{1, 4}.length, 6).toString(); // 0.(3)P({2, 4, 6}):
Fraction p = new Fraction(new int[]{2, 4, 6}.length, 6).toString(); // 0.557+45/60+17/3600
int deg = 57; // 57°
int min = 45; // 45 Minutes
int sec = 17; // 17 Seconds
new Fraction(deg).add(min, 60).add(sec, 3600).toString() // -> 57.7547(2)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.
String x = "/";
String s = "";
Fraction a = new Fraction(0);
Fraction b = new Fraction(1);
for (int n = 0; n <= 10; n++) {
Fraction c = new Fraction(a).add(b).div(2);
System.out.println(n + "\t" + a.toFraction() + "\t" + b.toFraction() + "\t" + c.toFraction() + "\t" + x);
if (c.add(2).pow(2).compareTo(5) < 0) {
a = c;
x = "1";
} else {
b = c;
x = "0";
}
s += x;
}
System.out.println(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)
Fraction f = new Fraction("-6.(3416)");
System.out.println(f.mod(1).abs().toString()); // Will print 0.(3416)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:
int a = -1;
double b = 10.99;
System.out.println(new Fraction(a)
.mod(b).toString()); // Not correct, usual Modulo
System.out.println(new Fraction(a)
.mod(b).add(b).mod(b).toString()); // Correct! Mathematical ModuloIt turns out that Fraction.java outperforms almost any fmod() implementation, including Java itself, php.js, C++, Python, JavaScript 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.
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, Bytes, Shorts, Integers, Longs, Floats, Doubles or Strings.
new Fraction(numerator, denominator);
new Fraction(new Number[]{numerator, denominator});
new Fraction(new byte[]{numerator, denominator});
new Fraction(new short[]{numerator, denominator});
new Fraction(new int[]{numerator, denominator});
new Fraction(new long[]{numerator, denominator});
new Fraction(new float[]{numerator, denominator});
new Fraction(new double[]{numerator, denominator});new Fraction(123);new Fraction(55.4);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!new Fraction(3, 2); // 3/2 = 1.5The 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 comparision is only necessary against this attribute.
Fraction f = new Fraction("-1/2");
System.out.println(f.n); // Numerator: 1
System.out.println(f.d); // Denominator: 2
System.out.println(f.s); // Sign: -1Returns the actual number without any sign information
Returns the actual number with flipped sign in order to get the additive inverse
Returns the sum of the actual number and the parameter n
Returns the difference of the actual number and the parameter n
Returns the product of the actual number and the parameter n
Returns the quotient of the actual number and the parameter n
Returns the power of the actual number, raised to an integer exponent. Note: Rational exponents are planned, but would slow down the function a lot, because of a kinda slow root finding algorithm, whether the result will become irrational. So for now, only integer exponents are implemented.
Returns the modulus (rest of the division) of the actual object and n (this % n). It's a much more precise fmod() 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.
Returns the modulus (rest of the division) of the actual object (numerator mod denominator)
Returns the fractional greatest common divisor
Returns the fractional least common multiple
Returns the ceiling of a rational number (rounded up)
Returns the floor of a rational number (rounded down)
Returns the rational number rounded (normal round)
Returns the multiplicative inverse of the actual number (n / d becomes d / n) in order to get the reciprocal
Simplifies the rational number under a certain error threshold. Ex. 0.333 will be 1/3 with eps=0.001
Check if two numbers are equal
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
Check if two numbers are divisible (n divides this)
Returns a decimal representation of the fraction
Returns a decimal representation of the fraction (Equivalent valueOf())
Generates an exact string representation of the actual object, including repeating decimal places of any length.
Generates an exact LaTeX representation of the actual object.
The optional boolean parameter indicates if you want to exclude the whole part. "1 1/3" instead of "4/3"
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"
Gets an array of the fraction represented as a continued fraction. The first element always contains the whole part.
Fraction f = new Fraction("88/33");
long[] c = f.toContinued(); // [2, 1, 2]Get an array of fraction. [nominator, denominator, sign]
Fraction f = new Fraction(1, 2);
long[] c = f.toArray(); // [1, 2, 1]Creates a copy of the actual Fraction object
The library should work without configuring anything. However, there is one global option:
Fraction.REDUCE = <true|false>It tells Fraction.java whether to reduce the fraction or not.
// Normal behavior
Fraction f = new Fraction(3, 6);
System.out.println(f.toFraction()); // 1/2
// Disable fraction reduction
Fraction.REDUCE = false;
Fraction g = new Fraction(3, 6);
System.out.println(g.toFraction()); // 3/6
// Back to normal behavior
Fraction.REDUCE = true;
Fraction h = new Fraction(g);
System.out.println(h.toFraction()); // 1/2If a really hard error occurs (parsing error, division by zero), Fraction.java throws exceptions! Please make sure you handle them correctly.
Installing Fraction.java is as easy as cloning this repo or download lib from builds folder
Fraction.java tries to circumvent floating point errors, by having an internal representation of numerator and denominator. As it relies on Java, there is also a limit. The biggest number representable is |Long.MAX_VALUE / 1| and the smallest is |1 / Long.MAX_VALUE|, with Long.MAX_VALUE=9007199254740991.
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
gradle check
Copyright (c) 2018, Tupikin Vladislav Dual licensed under the MIT or GPL Version 2 licenses.