Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠ p q {\displaystyle {\tfrac {p}{q}}} ⁠ of two integers, a numerator p and a nonzero denominator q. For example, ⁠ 3 7 {\displaystyle {\tfrac {3}{7}}} ⁠ is a rational number, as is every integer (for example, − 5 = − 5 1 {\displaystyle -5={\tfrac {-5}{1}}} ). The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface Q, or blackboard bold ⁠ Q . {\displaystyle \mathbb {Q} .} ⁠ A rational number is a real number. The real numbers that are rational are those whose decimal expansion 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...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases). A real number that is not rational is called irrational. Irrational numbers include the square root of 2 (⁠ 2 {\displaystyle {\sqrt {2}}} ⁠), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. The field of rational numbers is the unique field that contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field. A field has characteristic...