Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as 'the rationals', the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold Q {displaystyle mathbb {Q} } , Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for 'quotient'. In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as 'the rationals', the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold Q {displaystyle mathbb {Q} } , Unicode ℚ); it was thus denoted in 1895 by Giuseppe 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 √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. Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) such that q ≠ 0, for the equivalence relation defined by (p1, q1) ~ (p2, q2) if, and only if p1q2 = p2q1. With this formal definition, the fraction p/q becomes the standard notation for 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. 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. 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, 'rational' is often used as a noun abbreviating 'rational number'. The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (that is a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term 'polynomial over the rationals' is generally preferred, for avoiding confusion with 'rational expression' and 'rational function' (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions. Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers, and b > 0. This is often called the canonical form. Starting from a rational number a/b, its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.