section_path: Fractions in abstract mathematics
section_name: Fractions in abstract mathematics
paragraph_text: Fractions in abstract mathematics
  
  In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an ordered pair (a, b) {\displaystyle (a,b)} of integers a {\displaystyle a} and b ≠ 0, {\displaystyle b\neq 0,} for which the operations addition, subtraction, multiplication, and division are defined as follows:
  
  These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined as:
  
  Furthermore, the relation, specified as
  
  is an equivalence relation of fractions. Each fraction from one equivalence class may be considered as a representative for the whole class, and each whole class may be considered as one abstract fraction. This equivalence is preserved by the above defined operations, i.e., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions
  
  and similarly for the other operations.
  
  In the case of fractions of integers, the fractions a / b with a and b coprime and b > 0 are often taken as uniquely determined representatives for their equivalent fractions, which are considered to be the same rational number. This way the fractions of integers make up the field of the rational numbers.
  
  More generally, a and b may be elements of any integral domain R, in which case a fraction is an element of the field of fractions of R. For example, polynomials in one indeterminate, with coefficients from some integral domain D, are themselves an integral domain, call it P. So for a and b elements of P, the generated field of fractions is the field of rational fractions (also known as the field of rational functions).