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README
Rationals .NET
Implementation of rational number arithmetic for .NET with arbitrary precision.
var left = (Rational) 1 / 2;
var right = (Rational) 1 / 4;
var sum = left + right; // equals to: 3 / 4
Download
Binaries of the last build can be downloaded on the AppVeyor CI page of the project.
The library is also published on NuGet.org, install using:
PM> InstallPackage Rationals
The library can be used since .NET 4.0 or .NET Standard 1.3. It's also CLS compliant to allow use in VB.NET.
Features
 implicit conversions  rationals integrate seamlessly with other number types
 unlimited precision  rationals use
BigInteger
inside  canonical form  each rational can have its canonical form (irreducible fraction where denominator is always positive)
 comparison & equality
 separate whole and fractional part  any rational number can be separated into a whole part (integer quotient aka result of integer division) and fractional part (reminder of the integral division aka result of modulo operation)
 continued fraction expansion  expand rational numbers to continued fraction (sequence of coefficients), construct rational numbers from sequence of continued fraction coefficients
 rational number approximation  approximate floating point numbers (decimal, double, float) as rational numbers with customizable tolerance
 multiple formatting options 
ToString("C")
(canonical form),ToString("W")
(whole + fractional part), or normal fraction format
Documentation
The Rationals library is an alternative to BigRational found in BCL. Its implementation is quite trivial, it doesn't do any lowlevel magic to make sure it's the best performer. But it should be easy to use and has few nice features.
Constructors
There are just two constructors to rational numbers. For most scenarios, you might not need to use those constructors directly and rely on the typecasting operators.
// constructor from a whole BigInteger number
var p = new Rational(new BigInteger(3));
// constructor from a BigInteger numerator and denominator
var p = new Rational(new BigInteger(1), new BigInteger(2));
Implicit conversions from other types
Rationals are much easier created by implicit conversions. There exist implicit conversions from: int
, uint
, short
, ushort
, long
, ulong
, byte
, sbyte
, and BigInteger
.
Rational p = 5;
Explicit conversions from other types
There exist explicit operator conversions from decimal
, double
, and float
.
var p = (Rational) 0.5; // equals to: 1/2
Note that sometimes you might not get very nice rational numbers as the output.
var p = (Rational) 0.71428571428M; // equals to: 71428571428/100000000000 (or 17857142857/25000000000 simplified)
However, 0.71428571428
is almost exactly 5/7
. To actually read the number as 5/7
, you have to use the Rational.Approximate
function and provide some tolerance.
Approximation
Approximation tries to find the "simplest" rational number for given decimal/floating point number.
The library supports approximation of decimal
, double
, and float
numbers. An optional second tolerance
parameter might be given.
var p1 = Rational.Approximate(0.71428571428M); // 17857142857/25000000000
var p2 = Rational.Approximate(0.71428571428M, tolerance: 0.00000000001M); // 5/7
var q1 = Rational.Approximate(0.3333); // 3333/10000
var q2 = Rational.Approximate(0.3333, 0.0001); // 1/3
Continuous fraction expansion
Rational numbers can be expanded into Continuous fractions. The library can expand any rational number into a such a fraction, the result of this is the sequence of those coefficients.
For example, the rational number 649/200
can be represented as a continuous fraction with coefficients 3, 4, 12, 4
, which when used in the formula 3 + 1/(4 + 1/(12 + 1/4))
give the result 649/200
.
// compute rational number from continuous fraction coefficients
var p = Rational.FromContinuedFraction(new BigInteger[] { 3, 4, 12, 4 }); // 649/200
// find continuous fraction coefficients for a rational number
var coefficients = ((Rational) 10 / 7).ToContinuedFraction(); // 1, 2, 3
Reducing fractions, canonical form
The library supports reducing (simplifying) fractions. To reduce a fraction, you can use the CanonicalForm
property. That returns a rational number that's irreducible, and where also the denominator is always positive. Canonical form of zero is 0/1
.
var p1 = ((Rational) 9 / 12).CanonicalForm; // 3/4
var p2 = ((Rational) (9) / 12).CanonicalForm; // 3/4
var p3 = ((Rational) 9 / 12).CanonicalForm; // 3/4
var p4 = ((Rational) 0 / 12).CanonicalForm; // 0/1
There should always be just one canonical form of any rational number.
Whole and fractional parts
A rational number x/y
can be thought of as having a whole and fractional part a + b/c
.
For example, the 14/4
can be written as 3 + 2/4
where 3
is the whole part, and 2/4
is the fractional part.
var r1 = (Rational)14 / 4;
BigInteger a1 = r1.WholePart; // 3
Rational bc1 = r1.FractionPart; // 2/4
var r2 = (Rational)(49) / 10;
BigInteger a2 = r2.WholePart; // 5
Rational bc2 = r2.FractionPart; // 1/10
Explicit conversions to other types
Rational numbers can be explicitly converted to decimal
, double
, and float
decimal numbers. Note that the Rational
type has unlimited precision where the types to convert to are limited. As a result of that, there might be some rounding occurring or an overflow.
var p1 = (Rational) 1 / 2;
var x1 = (decimal) p1; // 0.5
var p2 = (Rational) 1 / 3;
var x2 = (double) p2; // 0.33333333333333337
Rational numbers can also be explicitly converted to whole number types int
, uint
, short
, ushort
, long
, ulong
, byte
, and sbyte
. For these, we only take the whole part of the fractional number:
var p1 = (Rational) 3 / 2;
var x1 = (int) p1; // 1
var p2 = (Rational) (3) / 2;
var x2 = (int) p2; // 2
BigInteger inspired properties
There are several other useful properties of the rational number that have similar equivalents in BigInteger:
.IsZero
returns true if the number is equal to 0.IsOne
returns true if the is equal to 1.Sign
returns anint
number (negative, positive, or zero) that indicates the sign of the number.IsPowerOfTwo
returns true if the number is a power of two
Overloaded operators
Rational numbers have all the common numeric operators overloaded so that their use in C# is very idiomatic. All of these operators should behave as expected: +
, unary 
, binary 
, *
, /
, ++
, 
, ==
, !=
, <
, >
, <=
, >=
.
var p = (Rational) 3 / 4;
var q = (Rational) 1 / 3;
var result = p / q; // 9/4
Mathematical operations
The Rational
class has a range of static methods that implement common mathematical operations. Some of these can be used through the corresponding overloaded operators.
var p = (Rational) 3 / 4;
var q = (Rational) 1 / 3;
Rational.Invert(p); // 4/3
Rational.Negate(p); // 3/4
Rational.Add(p, q); // 13/12
Rational.Subtract(p, q); // 5/12
Rational.Multiply(p, q); // 3/12
Rational.Divide(p, q); // 9/4
Rational.Pow(p, 2); // 9/16
Rational.Abs(p); // 3/4
Rational.Log10(p); // 0.12493873660829985
Rational.Log(p); // 0.28768207245178079 (base is e)
Rational.Log(p, 2); // 0.4150374992788437
Rational.Root(p, 2); // 0.8660254037844386 (square root, result is double)
Rational.RationalRoot((Rational) 9 / 16, 2); // 3/4 (square root, result is rational)
Computing magnitude
Magnitude of a number can be thought of as the exponent of 10 if the number was written in scientific notation.
 Magnitude of 0 is 0.
 Magnitude of 5 is 0.
 Magnitude of 12 is 1.
 Magnitude of 3988222 is 6.
 Magnitude of 0.2223 is 1.
 Magnitude of 0.04 is 2.
To find the magnitude of rational number, use the .Magnitude
property.
var p = (Rational) 1 / 11;
int magnitude = p.Magnitude; // 2
Significant digits
Every rational number has a Digits
property that enumerates all significant digits of the rational number. You might want to use this together with the Magnitude
property.
Keep in mind that the result of this might be infinite. For example, for the rational number 1/3
, it will return an infinite sequence of threes.
((Rational) 200).Digits; // 2
((Rational) 1/2).Digits; // 5
((Rational) 1/3).Digits.Take(10); // 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
((Rational) (213)/31).Digits.Take(10); // 6, 8, 7, 0, 9, 6, 7, 7, 4, 1
((Rational) 0).Digits; // 0
Formatting
Rational numbers can be formatted in three formats passed into the .ToString()
method:
"F"
(default, normal fraction) whole number will be formated as a whole number, e.g.
10/5
as2
.  fractional number will be formatted as it is, e.g.
9/5
as9/5
.
 whole number will be formated as a whole number, e.g.
"C"
(canonical fraction) number will be converted to canonical form, and then formatted as
"F"
.
 number will be converted to canonical form, and then formatted as
"W"
(whole + fractional part)` number will be separated as a whole and fractional part and formatted with a space between them, e.g.
9/5
as1 4/5
.
 number will be separated as a whole and fractional part and formatted with a space between them, e.g.
Parsing
The Rational
class has 4 different static methods for parsing strings: .Parse
, .TryParse
, .ParseDecimal
, and .TryParseDecimal
.
The .Parse
and .TryParse
methods accept strings in two formats:
 Fractional format (e.g.
3/4
)  Whole fractional format (e.g.
5 1/2
)
The .ParseDecimal
and .TryParseDecimal
methods try to parse the string into decimal
type and then convert it to Rational
. An optional tolerance
parameter might be given to parse nicer fractions, as it uses the Approximate
function inside.
var p1 = Rational.Parse("7/5"); // 7/5
var p2 = Rational.Parse("1 2/5"); // 7/5
var p3 = Rational.ParseDecimal("1.4"); // 7/5