Interfaces for Generic Operations

At the core of the generic arithmetic framework is a set of interfaces that define arithmetic operations on generic types. A type that implements one or more of these interfaces is an Arithmetic. An arithmetic performs arithmetic operations on a specific operand type. The relationship between an airthmetic type and its operand type is declared using a custom attribute.

The arithmetic operations are grouped according to the algebraic structure of the operand type.

Addition and Subtraction: Groups

A group is a set that has an addition operator. It usually also has a subtraction operator. The IGroupOperations<T> interface defines the operations on instances of a group. It depends on IEqualityComparer<T>IComparer<T>. In addition, it defines the following properties and methods:

Multiplication: Rings

A ring is a group that has a multiplication operator. It usually also has a unit element. The IRingOperations<T> interface defines the operations on instances of a ring. It depends on IGroupOperations<T>. In addition, it defines the following properties and methods:

Division : Fields

Division presents somewhat of a challenge because it is not always well defined. For example, the value of 7/2 is not an integer, but it is a rational number. In technical terms, the set of integers is not closed under division, but the set of rational numbers is. In other words, the set of rational numbers is the closure of the integers under division. It is this relationship that is expressed by the IDivisionOperations<T, TClosure> interface.

This interface has two type parameters. The first is the operand type. The second is the type of the closure. It depends on IRingOperations<T>. In addition, it defines the following methods:

A ring that is closed under division is called a field. Examples of fields are: the rational numbers, the real numbers, and integers modulo a prime number.

Operations on fields are defined by the IFieldOperations<T> interface. It depends on IDivisionOperations<T, TClosure> with the closure type equal to the operand type. It defines no additional members.

Integer Division: Euclidean Rings

A Euclidean ring is a ring that has a well-defined division with remainder. The name derives from the fact that division with remainder is the core of Euclid's algorithm to compute the greatest common divisor of two numbers.

The quotient, q and the remainder, r of the division a / b are uniquely defined such that a = qb + r with 0 <= |r| < b. There are other ways to uniquely define q and r, which can be specified by the rounding mode.

Division with remainder is defined by the IEuclideanRingOperations<T> interface. It depends on IRingOperations<T>. In addition, it defines the following methods:

Non-Integers: Rounding

The IFractionalOperations<T> interface defines methods for rounding numbers. It depends on IFieldOperations<T>. In addition, it defines the following methods:

Real numbers

The IRealOperations<T> interface defines properties and methods involving real numbers, including a large number of transcendental functions like the ones defined for Double in the Math static class. It depends on IFractionalOperations<T>. In addition, it defines the following methods:

IEEE arithmetic

The IEEE-754 standard defines three special values: positive infinity, negative infinity, and Not-a-Number (NaN). Although these values originally applied to single and double precision floating-point numbers, they can be used more generally.

The Not-a-Number value in particular deserves special attention. It is used when the result of an operation is undefined. For example: 0/0 is undefined. NaN's compare not equal to all values, including other NaN's.

The IIeeeOperations<T> interface defines the properties and methods that support IEEE arithmetic. It defines the following methods: