Properties
The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers Z.
The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime field for characteristic zero.
The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the algebraic numbers.
The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.
The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
(where are positive), we have
Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.
Read more about this topic: Rational Number
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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