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Encyclopedia > Rational number

In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction a / b, where b is not zero. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... The integers are commonly denoted by the above symbol. ... A cake with one quarter removed. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... For other uses, see zero or 0. ...


Each rational number can be written in infinitely many forms, such as 3 / 6 = 2 / 4 = 1 / 2, but it is said to be in simplest form when a and b have no common divisors except 1 (i.e., they are coprime). Every non-zero rational number has exactly one simplest form of this type with a positive denominator. A fraction in this simplest form is said to be an irreducible fraction, or a fraction in reduced form. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Coprime - Wikipedia /**/ @import /skins-1. ... An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction. ...


The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above one, and is also true when rational numbers are considered to be p-adic numbers rather than real numbers. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not a rational number is called an irrational number. A numeral is a symbol or group of symbols that represents a number. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...


The set of all rational numbers, which constitutes a field, is denoted mathbb{Q}. Using the set-builder notation, mathbb{Q} is defined as In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...

mathbb{Q} = left{frac{m}{n} : m in mathbb{Z}, n in mathbb{Z}, n ne 0 right},

where denotes the set of integers.

Contents

The term rational

In the mathematical world, the adjective rational often means that the underlying field considered is the field mathbb{Q} of rational numbers. For example, a rational integer is an algebraic integer which is also a rational number, which is to say, an ordinary integer, and a rational matrix is a matrix whose coefficients are 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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...


Arithmetic

See also Arithmetic with fractions for a more elaborate treatment.

Two rational numbers a / b and c / d are equal if and only if ad = bc. A cake with one quarter removed. ... This article does not cite any references or sources. ...


Two fractions are added as follows

frac{a}{b} + frac{c}{d} = frac{ad+bc}{bd}.

The rule for multiplication is

Additive and multiplicative inverses exist in the rational numbers The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... The reciprocal function: y = 1/x. ...

 - left( frac{a}{b} right) = frac{-a}{b} = frac{a}{-b} quadmbox{and}quad left(frac{a}{b}right)^{-1} = frac{b}{a} mbox{ if } a neq 0.

It follows that the quotient of two fractions is given by

frac{a}{b} div frac{c}{d} = frac{ad}{bc}.

Egyptian fractions

Main article: Egyptian fraction

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers, such as An Egyptian fraction is the sum of distinct unit fractions, such as . ... The reciprocal function: y = 1/x. ...

frac{5}{7} = frac{1}{2} + frac{1}{6} + frac{1}{21}.

For any positive rational number, there are infinitely many different such representations, called Egyptian fractions, as they were used by the ancient Egyptians. The Egyptians also had a different notation for dyadic fractions. An Egyptian fraction is the sum of distinct unit fractions, such as . ... In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a fraction has denominator a power of two, i. ...


Formal construction

Mathematically we may construct the rational numbers as equivalence classes of ordered pair of integers left(a, bright), with b not equal to zero. We can define addition and multiplication of these pairs with the following rules: In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... The integers are commonly denoted by the above symbol. ...

left(a, bright) + left(c, dright) = left(ad + bc, bdright)
left(a, bright) times left(c, dright) = left(ac, bdright)

and if c ≠ 0, division by

frac{left(a, bright)} {left(c, dright)} = left(ad, bcright).

The intuition is that left(a, bright) stands for the number denoted by the fraction To conform to our expectation that tfrac{2}{4} and tfrac{1}{2} denote the same number, we define an equivalence relation sim on these pairs with the following rule: In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...

This equivalence relation is a congruence relation: it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...


We can also define a total order on Q by writing In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

The integers may be considered to be rational numbers by the embedding that maps p, to where denotes the equivalence class having (a, b), as a member. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...


Properties

a diagram illustrating the countabililty of the rationals
a diagram illustrating the countabililty of the rationals

The set mathbb{Q}, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers . Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ... The integers are commonly denoted by the above symbol. ...


The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of mathbb{Q}. The rational numbers are therefore the prime field for characteristic zero. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...


The algebraic closure of mathbb{Q}, i.e. the field of roots of rational polynomials, is the algebraic numbers. In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...


The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set. In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ...


The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers. In mathematics, the term dense has at least three different meanings. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets. ...


Real numbers

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...


By virtue of their order, the rationals carry an order topology. The rational numbers also carry a subspace topology. The rational numbers form a metric space by using the metric d(xy) = | x − y |, and this yields a third topology on mathbb{Q}. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of mathbb{Q}. In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics the term countable set is used to describe the size of a set, e. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... Please refer to Real vs. ...


p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn mathbb{Q} into a topological field:


Let p be a prime number and for any non-zero integer a let | a | p = p n, where pn is the highest power of p dividing a; In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...


In addition write | 0 | p = 0. For any rational number frac{a}{b}, we set left|frac{a}{b}right|_p = frac{|a|_p}{|b|_p}.


Then d_pleft(x, yright) = |x - y|_p defines a metric on mathbb{Q}. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...


The metric space left(mathbb{Q}, d_pright) is not complete, and its completion is the p-adic number field mathbb{Q}_p. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers mathbb{Q} is equivalent to either the usual real absolute value or a p-adic absolute value. In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... Ostrowskis theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. ... In mathematics, an absolute value is a function which measures the size of elements in a field or integral domain. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...


See also

Irrational number In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...


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