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In mathematics, a rational number is any number that can be expressed as the quotient or fraction (mathematics) ''p''/''q'' of two integers, ''p'' and ''q'', with the denominator, ''q'', not equal to zero. Since ''q'' may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode (unicode:ℚ)); it was thus denoted in 1895 by Peano after ''quoziente'', Italian for "quotient". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal). A real number that is not rational is called irrational. Irrational numbers include , 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.〔 The rational numbers can be formally defined as the equivalence classes of the quotient set where the cartesian product is the set of all ordered pairs (''m'',''n'') where ''m'' and ''n'' are integers, ''n'' is not 0 , and "~" is the equivalence relation defined by if, and only if, In abstract algebra, the rational numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. 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. Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undefined). ==Terminology== The term ''rational'' in reference to the set Q refers to the fact that a rational number represents a ''ratio'' of two integers. In mathematics, the adjective ''rational'' often means that the underlying field considered is the field Q of rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals". However, rational function does ''not'' mean the underlying field is the rational numbers, and a rational algebraic curve is ''not'' an algebraic curve with rational coefficients. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rational number」の詳細全文を読む スポンサード リンク
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