for being a rational , the property that denominator hold it both numerator and denominator are co prime is
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In mathematics, a rational number is a number such as -3/7 that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Every integer is a rational number: for example, 5 = 5/1. The set of all rational numbers, often referred to as "the rationals"[citation needed], the field of rationals[citation needed] or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold {\displaystyle \mathbb {Q} }\mathbb {Q} , Unicode /ℚ);[2][3] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".
The decimal expansion of a rational number either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...).[4] Conversely, any repeating or terminating decimal represents a rational number. These statements are true not just in base 10, but also in any other integer base (for example, binary or hexadecimal).
A real number that is not rational is called irrational.[5] Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1]
Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows:
{\displaystyle \left(p_{1},q_{1}\right)\sim \left(p_{2},q_{2}\right)\iff p_{1}q_{2}=p_{2}q_{1}.}{\displaystyle \left(p_{1},q_{1}\right)\sim \left(p_{2},q_{2}\right)\iff p_{1}q_{2}=p_{2}q_{1}.}
The fraction p/q then denotes the equivalence class of (p, q).
Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.[6]
In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (for more, see Construction of the real numbers).
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