Fractions 8.1.1

Fractions

Introduction

This package provides the Fraction type, used for representing rational numbers. It offers a comprehensive set of features for:

• Creating fractions from various data types (integers, decimals, strings, etc.)
• Performing common mathematical operations like addition, subtraction, multiplication, division, remainder, absolute value, and more.
• Rounding fractions to a specified precision.
• Converting fractions to different data types (decimals, strings, etc.).
• Formatting fractions for output using various notations (general, mixed number, decimal notation, etc.).

Creation

You can implicitly cast int, uint, long, ulong, decimal or BigInteger to Fraction:

Fraction a = 3;    // int
Fraction b = 4L;   // long
Fraction c = 3.3m; // decimal
Fraction d = new BigInteger(3);
// ..

[!Note] For compatibility reasons, the Fraction that is produced by these operators is automatically reduced to its lowest terms, which may have a significant performance impact. If the performance is a concern, you should consider using one of the constructors that supports specifying the whether the terms should be reduced or not.

You can explicitly cast double to Fraction, however doing so directly has some important caveats that you should be aware of:

var a = (Fraction)3.3;  // returns {3715469692580659/1125899906842624} which is 3.299999999999999822364316059975

You can explicitly cast from Fraction to any supported data type (int, uint, long, ulong, BigInteger, decimal, double). However, be aware that an OverflowException will be thrown, if the target data type's boundary values are exceeded.

Constructors

There a three types of constructors available:

• new Fraction (<value>) for int, uint, long, ulong, BigInteger, decimal and double (without rounding).
• new Fraction (<numerator>, <denominator>) using BigInteger for numerator and denominator.
• new Fraction (<numerator>, <denominator>, <reduce>) using BigInteger for numerator and denominator, as well as a bool specifying whether the resulting fraction should be normalized (reduced).

[!IMPORTANT] Please refer to the Working with non-normalized fractions section for more information about the possible side effects when working with non-reduced fractions.

Static creation methods

[!Note] All methods that were present in version 7.*, continue to return a Fraction that is automatically reduced to its lowest terms. Starting from version 8.0.0 these methods are now supplemented by an overload that adds an additional boolean parameter, specifying whether the terms should be reduced.

• Fraction.FromDecimal(decimal) : same as the implicit constructor.
• Fraction.FromDecimal(decimal, bool) : the boolean parameter specifies whether the returned Fraction should be reduced.
• Fraction.FromDouble(double) : same as the explicit constructor, the Fraction is constructed without rounding.
• Fraction.FromDoubleRounded(double) : the resulting Fraction would represent a close approximation of the input.
• Fraction.FromDoubleRounded(double, int) : the value is rounded to the specified number of significant digits.
• Fraction.FromDoubleRounded(double, int, bool) : the value is rounded to the specified number of significant digits, with the boolean parameter specifying whether the returned Fraction should be reduced.

[!IMPORTANT] Starting from version 8.0.0, the FromDouble(..) overloads no longer throw an ArgumentException when passed double.NaN, double.PositiveInfinity or double.NegativeInfinity. These values are instead represented as 0/0, +1/0 or -1/0.
For more information see the section about working with NaN and Infinity.

Creation from double without rounding

The double data type in C# uses a binary floating-point representation, which complies with the IEC 60559:1989 (IEEE 754) standard for binary floating-point arithmetic. This representation can't accurately represent all decimal fractions. For example, the decimal fraction 0.1 is represented as the repeating binary fraction .0001100110011.... As a result, a double value can only provide an approximate representation of the decimal number it's intended to represent.

Large values in the numerator / denominator

When you convert a double to a Fraction using the Fraction.FromDouble method, the resulting fraction is an exact representation of the double value, not the decimal number that the double is intended to approximate. This is why you can end up with large numerators and denominators.

var value = Fraction.FromDouble(0.1);
Console.WriteLine(value); // Ouputs "3602879701896397/36028797018963968" which is 0.10000000000000000555111512312578

The output fraction is an exact representation of the double value 0.1, which is actually slightly more than 0.1 due to the limitations of binary floating-point representation.

Comparing fractions created with double precision

Using a Fraction that was created using this method for strict Equality/Comparison should be avoided. For example:

var fraction1 = Fraction.FromDouble(0.1);
var fraction2 = new Fraction(1, 10);
Console.WriteLine(fraction1 == fraction2); // Outputs "False"

If you need to compare a Fraction created from double with others fractions you should either do so by using a tolerance or consider constructing the Fraction by specifying the maximum number of significant digits.

Possible rounding errors near the limits of the double precision

When a double value is very close to the limits of its precision, Fraction.FromDouble(value).ToDouble() == value might not hold true. This is because the numerator and denominator of the Fraction are both very large numbers. When these numbers are converted to double for the division operation in the Fraction.ToDouble method, they can exceed the precision limit of the double type, resulting in a loss of precision.

var value = Fraction.FromDouble(double.Epsilon);
Console.WriteLine(value.ToDouble() == double.Epsilon); // Outputs "False"

For more detailed information about the behavior of the Fraction.FromDouble method and the limitations of the double type, please refer to the XML documentation comments in the source code.

Creation from double with maximum number of significant digits

The Fraction.FromDoubleRounded(double, int) method allows you to specify the maximum number of significant digits when converting a double to a Fraction. This can help to avoid large numerators and denominators, and can make the Fraction suitable for comparison operations.

var value = Fraction.FromDoubleRounded(0.1, 15); // Returns a fraction with a maximum of 15 significant digits
Console.WriteLine(value); // Outputs "1/10"

If you care only about minimizing the size of the numerator/denominator, and do not expect to use the fraction in any strict comparison operations, then creating an approximated fraction using the Fraction.FromDoubleRounded(double) overload should offer the best performance.

Creation from double with rounding to a close approximation

You can use the Fraction.FromDoubleRounded(double) method to avoid big numbers in numerator and denominator. Example:

var value = Fraction.FromDoubleRounded(0.1);
Console.WriteLine(value); // Outputs "1/10"

However, please note that while rounding to an approximate value would mostly produce the expected result, it shouldn't be relied on for any strict comparison operations. Consider this example:

var doubleValue = 1055.05585262;
var roundedValue = Fraction.FromDoubleRounded(doubleValue);      // returns {4085925351/3872710} which is 1055.0558526199999483565771772222
var literalValue = Fraction.FromDoubleRounded(doubleValue, 15);  // returns {52752792631/50000000} which is 1055.05585262 exactly
Console.WriteLine(roundedValue.CompareTo(literalValue); // Outputs "-1" which stands for "smaller than"
Console.WriteLine(roundedValue.ToDouble() == doubleValue); // Outputs "true" as the actual difference is smaller than the precision of the doubles

Creation from string

The following string patterns can be parsed:

• [+/-]n where n is an integer. Examples: +5, -6, 1234, 0
• [+/-]n.m where n and m are integers. The decimal point symbol depends on the system's culture settings. Examples: -4.3, 0.45
• [+/-]n/[+/-]m where n and m are integers. Examples: 1/2, -4/5, +4/-3, 32/100

Example:

var value = Fraction.FromString("1,5", CultureInfo.GetCultureInfo("de-DE"))
// Returns 3/2 which is 1.5
Console.WriteLine(value);

[!TIP] You should consider the TryParse methods when reading numbers as text from user input. Furthermore it is best practice to always supply a culture information (e.g. CultureInfo.InvariantCulture). Otherwise you will sooner or later parse wrong numbers because of different decimal point symbols or included Thousands character.

Here is a table presenting the different overloads, and the default parameters that are assumed for each of them:

Conversion

You can convert a Fraction to any supported data type by calling:

• .ToInt32()
• .ToUInt32()
• .ToInt64()
• .ToUInt64()
• .ToBigInteger()
• .ToDecimal()
• .ToDouble()
• .ToString() (using current culture)
• .ToString(string) (using format string and the system's current culture)
• .ToString(string,IFormatProvider)

If the target's data type boundary values are exceeded the system will throw an OverflowException.

Example:

var rationalNumber = new Fraction(1, 3);
var value = rationalNumber.ToDecimal();
// result is 0.33333
Console.WriteLine(Math.Round(value, 5));

String format

Note: The special characters #, and 0 like in #.### are not supported. Consider converting the Fraction to decimal/double if you want to support the custom formats.

Example:

var value = new Fraction(3, 2);
// returns 1 1/2
Console.WriteLine(value.ToString("m", CultureInfo.GetCultureInfo("de-DE")));

Decimal Notation Formatter

The DecimalNotationFormatter class allows for formatting Fraction objects using the standard decimal notation, and the specified format and culture-specific format information. Unlike standard numeric types such as double and decimal, there is no limit to the represented range or precision when using DecimalNotationFormatter.

Usage

Here is a general example of how to use the DecimalNotationFormatter:

Fraction value = Fraction.FromString("123456789987654321.123456789987654321");
string formattedValue = DecimalNotationFormatter.Instance.Format("G36", value, CultureInfo.InvariantCulture);
Console.WriteLine(formattedValue); // Outputs "123456789987654321.123456789987654321"

In this example, the Format method is used to format the value of a Fraction object into a string using the 'G' (General) format with a precision specifier of 36, which formats the fraction with up to 36 significant digits.

Supported Formats

The Format method supports the following format strings. For more information about these formats, see the official .NET documentation.

Please note that the 'R' format and the custom formats are handled by casting the Fraction to double, which may result in a loss of precision.

Significant Digits After Radix Format

The 'S' format is a non-standard format that formats the fraction with significant digits after the radix and dynamically switches between decimal and scientific notation depending on the value of the fraction.

For fractions where the absolute value is greater than or equal to 0.001 and less than 10000, the 'S' format uses decimal notation. For all other values, it switches to scientific notation.

Here are a few examples:

Fraction value = new Fraction(1, 3);
Console.WriteLine(value.ToString("S")); // Outputs "0.33"

value = newFraction(1, 1000);
Console.WriteLine(value.ToString("S")); // Outputs "0.001"

value = new Fraction(1, 100000);
Console.WriteLine(value.ToString("S")); // Outputs "1E-05"

Mathematic operators

The following mathematic operations are supported:

• .Reduce() returns a normalized fraction (e.g. 2/4 → 1/2)
• .Add(Fraction) returns the sum of (a + b)
• .Subtract(Fraction) returns the difference of (a - b)
• .Multiply(Fraction) returns the product of (a * b)
• .Divide(Fraction) returns the quotient of (a / b)
• .Remainder(Fraction) returns the remainder (or left over) of (a % b)
• .Negate() returns a negated fraction (same operation as (a * -1))
• .Abs() returns the absolute value |a|
• Fraction.Pow(Fraction, int) returns a base raised to a power (a ^ exponent) (e.g. 1/10^(-1) → 10/1)
• Fraction.Round(Fraction, int, MidpointRounding) returns the fraction, which is rounded to the specified precision
• Fraction.RoundToBigInteger(Fraction, MidpointRounding) returns the fraction as rounded BigInteger

As extension method:

• FractionExt.Sqrt(this Fraction, int) returns the square root, specifying the precision after the decimal point.

Example:

 var a = new Fraction(1, 3);
 var b = new Fraction(2, 3);
 var result = a * b;
 // returns 2/9 which is 0,2222...
 Console.WriteLine(result);

Working with non-normalized fractions

[!IMPORTANT] For performance reasons, as of version 8.0.0, mathematical operations such as addition and multiplication no longer reduce the result to it's lowest terms, unless both operands are already simplified. This change in behavior may introduce unexpected results when, for example, calling ToString on a Fraction that is the result of an expression, having one or more of the values de-serialized using the default JsonFractionConverter settings (i.e. without explicit reduction).

The following rules apply:

$F ⊙ F = F$
$F ⊙ NF = NF$
$NF ⊙ F = NF$
$NF ⊙ NF = NF$

That said, the following applies for normalized fractions:

var a = new Fraction(4, 4, normalize: true); // a is 1/1
var b = new Fraction(2);    // b is 2/1 (automatically normalized)
var result = a / b;         // result is 1/2

$\frac{1}{1}/\frac{2}{1}=\frac{1}{2}$

However, for non-normalized fractions the following applies:

var a = new Fraction(4, 4, normalize: false);
var b = new Fraction(2);    // b is 2/1 (automatically normalized)
var result = a / b;         // result is 4/8

$\frac{4}{4}/\frac{2}{1}=\frac{4}{8}$

Working with NaN and Infinity

Starting from version 8.0.0, it is now possible for a Fraction to be constructed with 0 in the denominator. This is typically the result of a division by zero which, depending on the sign of the dividend, now returns Fraction.PositiveInfinity, Fraction.NegativeInfinity or Fraction.NaN.

Subsequent mathematical operations with these values follow the same rules, as implemented by the double type. For example, any number multiplied by infinity results in infinity (or negative infinity, depending on the sign), and any number divided by infinity is zero.

You can check if a Fraction represents one of the special values using the properties:

• Fraction.IsPositiveInfinity
• Fraction.IsNegativeInfinity
• Fraction.IsNaN

[!TIP] You could also check if a Fraction represents either NaN or Infinity by testing whether it's Denomintor.IsZero.

Equality operators

Fraction implements the following interfaces:

• IEquatable<Fraction>,
• IComparable,
• IComparable<Fraction>

[!IMPORTANT] Please note that the default behavior of the .Equals(Fraction) method has changed with version 8.0.0. Equals now compares the calculated value from the $numerator/denominator$ ($Equals(\frac{1}{2}, \frac{2}{4}) = true$).

In case you want to compare the numerator and denominator exactly (i.e. $Equals(\frac{1}{2}, \frac{2}{4}) = false$) then you can use the new FractionComparer.StrictEquality comparer.

That said:

var a = new Fraction(1, 2, normalize: true);
var b = new Fraction(1, 2, normalize: false);
var c = new Fraction(2, 4, normalize: false); // the fraction is not reduced

// result1 is true
var result1 = a == a;

// result2 is true
var result2 = a == b;

// result3 is true
var result3 = a == c;

// Special case:
// same behavior as with double, see https://learn.microsoft.com/en-us/dotnet/api/system.double.op_equality#remarks
// result4 is false 
var result4 = Fraction.NaN == Fraction.NaN;

Under the hood

The data type stores the numerator and denominator as BigInteger. Per default it will reduce fractions to its normalized form during creation. The result of each mathematical operation will be reduced as well. There is a special constructor to create a non-normalized fraction. Be aware that Equals relies on normalized values when comparing two different instances.

Performance considerations

We have a suite of benchmarks that test the performance of various operations in the Fractions library. These benchmarks provide valuable insights into the relative performance of different test cases. For more detailed information about these benchmarks and how to interpret them, please refer to the Fractions Benchmarks Readme in the benchmarks subfolder.

Build from source

Just run dotnet build -c release.

Required software frameworks

• .Net 8.0 SDK