Concept
Rational number
Overview
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 zero if and only if it contains the rational numbers as a subfield. Finite extensions of
Q
{\displaystyle \mathbb {Q} }
are called algebraic number fields, and the algebraic closure of
Q
{\displaystyle \mathbb {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 (see Construction of the real numbers).
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Rational number is.
In union with itself, it does not by its nature produce anything imperfect or foreign. All its changes are defined; it either produces itself or the Decad, either the homogeneous or the perfect.
Plutarch, Selected Essays of Plutarch, Vol. II.
States what Rational number is.
Mathematicians have distinguished different degrees of continuity, and have confined the word "continuous," for technical purposes, to series having a certain high degree of continuity. But for philosophical purposes, all that is important in continuity is introduced by the lowest degree of continuity, which is called "compactness." A series is called "compact" when no two terms are consecutive, but between any two…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Rational number is.
Related ideas
- Finite set
- Golden ratio
- Hexadecimal
- Integer
- Ancient Greek mathematics
- Mathematics
- Real number
- Natural number
- Number
- Ordered pair
- Set (mathematics)
- Topology
- Total order
- Square root of 2
- Dedekind cut
- Mathematical analysis
- Complex number
- Countable set
- Decimal
- Equivalence relation
- Ordered field
- Congruence relation
Opposing perspectives
Place in the tradition
Part of Logicism.