Rational number

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Graphical representation of fractions whose divider is 4. These four fractions are rational numbers.

Them rational numbers are all numbers that can be represented as the quotient of two integers or, more precisely, an integer and a positive natural; that is, a common fraction $a/b$ with numerator $a$ and denominator $b$ different from zero. The term "rational" refers to a fraction or part of a whole. The set of rational numbers is denoted by Q (or $mathbb{Q}$ in blackboard that derives from "quotient" (from Latin Quotiensadapted as 'quotient' to several European languages. This set of numbers includes integers ( $mathbb{Z}$ ) and fractional numbers and is a subset of real numbers ( $mathbb{R}$ ).

The decimal writing of a rational number is either a finite decimal number or a semiperiodic number. This is true not only for numbers written in base 10 (decimal system); it is also in binary, hexadecimal or any other integer base. Conversely, any number that admits of finite or periodic expansion (in any integer base) is a rational number.

A real number that is not rational is called an irrational number; the decimal expression of irrational numbers, unlike rational ones, is infinite aperiodic.

In a strict sense, rational number is the set of all fractions equivalent to a given; of all, it is taken as Canon representative from such a rational number to the irreducible fraction. The equivalent fractions of each other – rational number– are a kind of equivalence, the result of the application of an equivalence ratio $mathbb{Z}$ .

History

The Egyptians calculated the resolution of practical problems using fractions whose denominators are positive integers; they are the first rational numbers used to represent the "parts of a whole", through the concept of the reciprocal of an integer.

Ancient Greek mathematicians considered two quantities to be commensurable if it was possible to find a third such that the first two were multiples of the last, that is, it was possible to find a unit common for which the two magnitudes have an integer measure. The Pythagorean principle that every number is a quotient of integers, expressed in this way that any two magnitudes must be commensurable, therefore rational numbers.

Etymologically, the fact that these numbers are called rational corresponds to that they are the reason of two integers, word whose root comes from the Latin ratio, and this in turn of the Greek λόγος (reason), which is as the mathematicians of ancient Greece called these numbers. Notation $mathbb{Q}$ employed to name the set of rational numbers comes from the Italian word quoziente, derived from the work of Giuseppe Peano in 1895.

Arithmetic of rational numbers

See also: Fraction § Arithmetic with fractions

Equivalence relations and ordering

Integer immersion

Any integer n can be expressed as the rational number n/1 because that is frequently written ${displaystyle scriptstyle mathbb {Z} subset mathbb {Q} }$ (technically, it is said that the rationals contain an isomorphic ring of the whole numbers).

Equivalence

If true:

{displaystyle {frac {a}{b}}={frac {c}{d}}quad longleftrightarrow quad ad=bc}

Order

When both denominators are positive:

<math alttext="{displaystyle {frac {a}{b}}<{frac {c}{d}}quad longleftrightarrow quad adab.cd ad.bc{displaystyle {frac {a}{b}}}{frac}{d}{d}quad longleftrightarrow quad adrivbc}<img alt="{displaystyle {frac {a}{b}}<{frac {c}{d}}quad longleftrightarrow quad ad

If any of the denominators is negative, the fractions must first be converted into equivalent ones with positive denominators, following the equations:

{displaystyle {frac {-a}{-b}}={frac {a}{b}}}

and

{displaystyle {frac {a}{-b}}={frac {-a}{b}}}

Rational Operations

The operations of addition, subtraction, multiplication, and division are called rational operations.

Sum

The sum or addition of two rational numbers is defined as the operation that makes their sum correspond to any pair of rational numbers:

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

Subtraction

The operation that makes its difference correspond to any pair of rational numbers is called subtraction or difference and is considered the inverse operation of addition.

{displaystyle {frac {c}{d}}-{frac {a}{b}}={frac {c}{d}}+left(-{frac {a}{b}}right)}

Multiplication

The multiplication or product of two rational numbers:

{displaystyle {frac {a}{b}}cdot {frac {c}{d}}={frac {acdot c}{bcdot d}}}

Division

The division or quotient of two rationales is defined r between s other than 0, to the product ${displaystyle rtimes s^{-1}}$ . In another notation,

{displaystyle {frac {a}{b}}div {frac {c}{d}}={frac {a}{b}}cdot {frac {d}{c}}}

It is a fully defined operation, but it is assumed to be an inverse operation of multiplication that solves the equation s x=r, s≠0.

Inverses

Additive and multiplicative inverses exist in rational numbers:

{displaystyle -left({frac {a}{b}}right)={frac {-a}{b}}={frac {a}{-b}}quad {mbox{y}}quad left({frac {a}{b}}right)^{-1}={frac {b}{a}}{mbox{ si }}aneq 0.}

Decimal Writing

Rational number in decimal base

See also: Decimal representation, Number of decimal and Number of periodic decimal.

Every real number admits an unlimited decimal representation, this representation is unique if infinite sequences of 9 are excluded (such as the periodic 0.9). Using the decimal representation, any rational number can be expressed as a finite (exact) or periodic decimal number and vice versa. In this way, the decimal value of a rational number is simply the result of dividing the numerator by the denominator.

Rational numbers are characterized by having a decimal writing that can only be of three types:

Exacta: the decimal part has a finite number of figures. By not being significant, the zeros on the right of the decimal separator can be omitted, resulting in a "finite" or "terminal" expression. For example:

frac 8 5 = 1,6

Pure Periodical: the whole decimal part is repeated indefinitely. Example:

begin{array}{rcl}cfrac 1 7&=&0,142857142857dots\&=&0,overline{142857}end{array}

Mixed periodic: not all the decimal part is repeated. Example:

begin{array}{rcl}cfrac 1 {60}&=&0,01666dots\&=&0,01overline{6}end{array}

In the same way, the representation of a rational number in a positional numeral system in bases other than ten is applied.

Rational number in other bases

In a rational base positional numbering system, irreducible fractions whose denominator contains prime factors other than those that factor the base have no finite representation.

For example, on base 10, a rational will have finite development if and only if the denominator of its irreducible fraction is of the way ${displaystyle 2^{n}cdot 5^{p}}$ ( $n$ and $p$ whole), as well as on a duodecimal basis, the representation of all those fractions whose denominator contains prime factors other than 2 and 3.

Formal construction

See also: Domain of integrity and Body of quotients.

Formal construction of rational numbers as ordered pairs.

The set of rational numbers can be built from the set of fractions whose numerator and whose denominator are whole numbers. The set of rational numbers is not directly identifiable with the set of fractions, because sometimes a rational number can be represented by more than one fraction, for example:

$2,5 = frac{25}{10} = frac{10}{4} = frac{5}{2}$

In order to define rational numbers, it must be defined when two different fractions are equivalent and therefore represent the same rational number.

Formally, every rational number can be represented as the equivalence class of an ordered pair of integers (a,b), with b≠ 0, with the following equivalence relation:

{displaystyle left(a,bright)sim (c,d){text{ si y solo si }}ad=bc}

where the class equivalence space is the quotient space ${displaystyle (mathbb {Z} times mathbb {Z} setminus left{0right})/sim }$ . Summary and multiplication operations are defined as

{displaystyle {begin{aligned}left(a,bright)+(c,d)&=(ad+bc,bd)\left(a,bright)times (c,d)&=(ac,bd)end{aligned}}}

It is verified that the two defined operations are compatible with the equivalence ratio, indicating in a way that $mathbb{Q}$ can be defined as the quotient set ${displaystyle (mathbb {Z} times mathbb {Z} setminus left{0right})/sim }$ with the equivalence ratio described above.

Take into account that the defined operations are nothing more than the formalization of the usual operations between fractions:

{displaystyle {begin{aligned}{frac {a}{b}}+{frac {c}{d}}&={frac {ad+bc}{bd}}\{frac {a}{b}}cdot {frac {c}{d}}&={frac {ac}{bd}}end{aligned}}}

It denotes as [()a,b) to the kind of equivalencies that corresponds to the different representations of the same rational number ${displaystyle {tfrac {a}{b}}={tfrac {ka}{kb}}}$ , with kIt's a fraction. I mean:

{displaystyle [(a,b)]={cdots(-2a,-2b),(-a,-b),(a,b),(2a,2b),cdots }.}

It is taken as canonical representative the pair (a,b) such that mcd(a,b)= 1. Any other pair can be used in the case of operations. For example, ${displaystyle [(1,2)]={(1,2),(2,4)(3,6)...}}$ is the kind of equivalence of the rational number ${displaystyle {tfrac {1}{2}}}$ .

With previous operations, $mathbb{Q}$ is a body, where the class (0.1) plays the role of zero, and the class (1.1) of one. The opposite element of the class (a,b) is the class (-a,b). Besides, if a0, class (a,b) is different from zero, then (a,b) is invertible (inverse multiplier) and its reverse corresponds to the class (b,a).

You can also define a total order in $mathbb{Q}$ as follows:

0{text{ y }}adleq bc){text{ ó }}(bd(a,b)\leq \leq (c,d)Yes and only if(bd▪0andad\leq \leq bc)or(bd.0andad\geq \geq bc){displaystyle (a,b)leq (c,d){text{ if and only if }(bd literal0{text{ and }adleq bc){text{ or }}}(bd precedent {text{ and }}adgeq bc)}} 0{text{ y }}adleq bc){text{ ó }}(bd

The set of rational numbers can also be constructed as the field of quotients of the integers, that is,

{displaystyle mathbb {Q} =mathrm {Frac} (mathbb {Z}).}

Properties

Algebraics

The set of rational numbers $mathbb{Q}$ equipped with the sum and product operations meets the commutative, associative and distributive properties, i.e.:

{displaystyle {frac {a}{b}}+{frac {c}{d}}={frac {c}{d}}+{frac {a}{b}}}

(switching)

{displaystyle left({frac {a}{b}}+{frac {c}{d}}right)+{frac {e}{f}}={frac {a}{b}}+left({frac {c}{d}}+{frac {e}{f}}right)}

(associative)

{displaystyle {frac {a}{b}}times left({frac {c}{d}}+{frac {e}{f}}right)={frac {a}{b}}times {frac {c}{d}}+{frac {a}{b}}times {frac {e}{f}}}

(distributive).

There are neutral elements for the sum and product. For the sum, the zero, denoted by 0, as ${displaystyle {tfrac {a}{b}}+0={tfrac {a}{b}}}$ for any ${displaystyle {tfrac {a}{b}}}$ . For the product is the 1, which can be represented by ${displaystyle {tfrac {n}{n}}=1}$ , with n different from 0, since ${displaystyle {tfrac {a}{b}}times 1={tfrac {a}{b}}}$ .

It has symmetrical elements for the operations of sum and product. Thus, the symmetrical element regarding the sum for any rational number ${displaystyle {tfrac {a}{b}}}$ That's it. ${displaystyle -{tfrac {a}{b}}}$ , called opposite elementsince ${displaystyle {tfrac {a}{b}}+(-{tfrac {a}{b}})=0}$ . The same occurs in the case of the symmetrical element regarding the product, for any rational number ${displaystyle q={tfrac {a}{b}}}$ , other than 0, exists ${displaystyle q^{-1}={tfrac {b}{a}}}$ , called inverse multiplier such as ${displaystyle qtimes q^{-1}={tfrac {a}{b}}times {tfrac {b}{a}}=1}$ .

The whole $Q$ with the operations of addition and multiplication defined above, it forms a commutative body, the body of quotients of the integers $Z$ .

The rationals are the least body with null characteristic. Any other null feature body contains a copy of $Q$ .

The algebraic closing $Q$ It's the set of algebraic numbers.

Rationales form a unique factoring domain as any rational different from zero can be broken down into the form: $q = u p_1^{alpha_1}dots p_n^{alpha_n}$ where $p_iin mathbb{N}$ are integer primes, $alpha_iin mathbb{Z}$ (being some of them negative if q is not whole) and $uin{1,-1}$ . For example $260/693= 2^2 3^{-2}5^1 7^{-1}11^{-1}13^1,$ .

Conjunctistas

Diagram used in the demonstration that rationals are numberable (Georg Cantor).

The set of rational numbers is numberable, i.e. there is a bijection between $N$ and $Q$ (they have the same number of items). The set of real numbers is not numberable (part non-namely of the real, they constitute the irrational numbers.

Topological

The whole $Q$ form a dense subset of real numbers $R$ by construction itself $R$ (arquimediana property): any real number has arbitrarily rational near.
They possess a finite expansion as a continuous fraction regular.
With the topology of the order, they form a topological ring, or a partially ordained group; they have an induced topology; they also form a metric space with the metric $d(x,y) = |x -y|$ .
The rationals are an example of space that is not locally compact.
They are characterized topologically by being the only numberable meterable space without isolated points (it is also totally discontinuous). Rational numbers do not form a complete metric space.

P-adic number

Main article: P-Adic Number

Sea $p$ a prime number and for all integer not null $a$ I mean, $|a|_p=p^{-n}$ where $p^n$ is the greatest power $p$ which divides to $a$ .

Yeah. $|0|_p=0$ and for each rational number $frac{a}{b}$ ${displaystyle left|{frac {a}{b}}right|_{p}={frac {|a|_{p}}{|b|_{p}}}}$ then multiplier function $d_pleft(x, yright) = |x - y|_p$ defines a metric over $Q$ .

The metric space $left(Q,d_pright)$ is not complete, its completeness is the body of the p-addic numbers $Q_p$ . The Ostrowski theorem ensures that all non-trivial absolute value over $Q$ is equivalent to either the usual absolute value, or the absolute value p-Adic.

This in algebraic representations and not in arithmetic representations.

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