Rational number

In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A ratio is an expression which compares quantities relative to each other The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction $a / b$ , where b is not zero. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object a is called the numerator, and b the denominator. Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is the numerator

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. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without e. , they are coprime). In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than 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. An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator

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). This article gives a mathematical definition For a more accessible article see Decimal. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is 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. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics, the real numbers may be described informally in several different ways 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

Quarters

The set of all rational numbers, which constitutes a field, is denoted $\mathbb{Q}$ . In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Using the set-builder notation, $\mathbb{Q}$ (standing for "Quotient") is defined as

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

where $\mathbb{Z}$ denotes the set of integers. In Set theory and its applications to Logic, Mathematics, and Computer science, set-builder notation (sometimes simply "set notation"

1 The term rational
2 Arithmetic
3 Egyptian fractions
4 Formal construction
5 Properties
6 Real numbers and topological properties of the rationals
7 p-adic numbers
8 External links

The term rational

The term rational in reference to the set $\mathbb{Q}$ 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 $\mathbb{Q}$ of rational numbers. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This article deals with the ring of complex numbers integral over Z. 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 Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one

Arithmetic

See also: Fraction (mathematics)#Arithmetic with fractions

Two rational numbers $a / b$ and $c / d$ are equal if and only if $a d = b c$ . In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object ↔

Two fractions are added as follows

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

The rule for multiplication is

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}.$

Additive and multiplicative inverses exist in the rational numbers

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

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

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

For any positive rational number, there are infinitely many different such representations, called Egyptian fractions, as they were used by the ancient Egyptians. In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} This article is about the contemporary North African ethnic group The Egyptians also had a different notation for dyadic fractions. In Mathematics, a dyadic fraction or dyadic rational is a Rational number whose Denominator is a Power of two, i

Formal construction

Mathematically we may construct the rational numbers as equivalence classes of ordered pairs of integers $\left(a, b\right)$ , with $b$ not equal to zero. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French We can define addition and multiplication of these pairs with the following rules:

$\left(a, b\right) + \left(c, d\right) = \left(ad + bc, bd\right)$

$\left(a, b\right) \times \left(c, d\right) = \left(ac, bd\right)$

and if c ≠ 0, division by

$\frac{\left(a, b\right)} {\left(c, d\right)} = \left(ad, bc\right).$

The intuition is that $\left(a, b\right)$ stands for the number denoted by the fraction $\tfrac{a}{b}.$ 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:

$\left(a, b\right) \sim \left(c, d\right) \mbox{ if and only if } ad = bc.$

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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" See Congruence (geometry for the term as used in elementary geometry In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X 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 Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients )

We can also define a total order on Q by writing

$\left(a, b\right) \le \left(c, d\right) \mbox{ if } (bd>0\mbox{ and } ad \le bc)\mbox{ or }(bd<0\mbox{ and } ad \ge bc).$

The integers may be considered to be rational numbers by the embedding that maps $p\,$ to $[(p, 1)],\,$ where $[(a,b)]\,$ denotes the equivalence class having $(a, b)\,$ as a member. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

Properties

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 $\mathbb{Z}$ . In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of $\mathbb{Q}$ . In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's The rational numbers are therefore the prime field for characteristic zero. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

The algebraic closure of $\mathbb{Q}$ , i. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is e. the field of roots of rational polynomials, is the algebraic numbers. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or

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. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to e. the set of rational numbers is a null set. In Mathematics, a null set is a set that is negligible in some sense.

The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. In Mathematics, a Partial order &le on a set X is said to be dense (or dense-in-itself) if for all x and y in X 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 and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism

Real numbers and topological properties of the rationals

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}}

By virtue of their order, the rationals carry an order topology. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. The rational numbers also carry a subspace topology. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is The rational numbers form a metric space by using the metric d(x, y) = | x − y |, and this yields a third topology on $\mathbb{Q}$ . In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined All three topologies coincide and turn the rationals into a topological field. In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are The rational numbers are an important example of a space which is not locally compact. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks The rationals are characterized topologically as the unique countable metrizable space without isolated points. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of The space is also totally disconnected. In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that The rational numbers do not form a complete metric space; the real numbers are the completion of $\mathbb{Q}$ . In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, the real numbers may be described informally in several different ways

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 $p n$ is the highest power of $p$ dividing $a$ ;

In addition write $| 0 | p = 0$ . In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without For any rational number $\frac{a}{b}$ , we set $\left|\frac{a}{b}\right|_p = \frac{|a|_p}{|b|_p}$ .

Then $d_p\left(x, y\right) = |x - y|_p$ defines a metric on $\mathbb{Q}$ . In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

The metric space $\left(\mathbb{Q}, d_p\right)$ is not complete, and its completion is the p-adic number field $\mathbb{Q}_p$ . In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 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. Ostrowski's theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the Rational numbers Q is equivalent to either 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

External links

“Rational Number” From MathWorld — A Wolfram Web Resource

Dictionary

-noun

(mathematics) A real number that can be expressed as the ratio of two integers.

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