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) of integers a and b ≠ 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:
- (when c ≠ 0)
In addition, an equivalence relation is specified as follows: ~ if and only if .
These definitions agree in every case with the definitions given above; only the notation is different.
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, when a and b are polynomials in one indeterminate, the field of fractions is the field of rational fractions (also known as the field of rational functions). When a and b are integers, the field of fractions is the field of rational numbers.
Read more about this topic: Numerator (fraction)
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