Rational Number

#REDIRECT Rational number

Rational number

In mathematics, a rational number (or informally fraction (mathematics)) is a ratio or quotient of two integer, usually written as the vulgar fraction ''a''/''b'', where ''b'' is not 0 (number). Each rational number can be written in infinitely many forms, for example $3/6\; =\; 2/4\; =\; 1/2$. The simplest form is when $a$ and $b$ have no common divisors, and every non-zero rational number has exactly one simplest form of this type with positive 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). The same is true for any other integral base above 1. 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 rational is called an irrational number. In mathematics, the term "rational ''something''" means that the underlying field (mathematics) considered is the field $\backslash mathbb\{Q\}$ of rational numbers. For example, rational polynomials or rational prime ideals. The set of all rational numbers is denoted by Q, or in blackboard bold $\backslash mathbb\{Q\}$. Using the set-builder notation $\backslash mathbb\{Q\}$ is defined as such: :$\backslash mathbb\{Q\}\; =\; \backslash left\backslash \{\backslash frac\{m\}\{n\}\; :\; m\; \backslash in\; \backslash mathbb\{Z\},\; n\; \backslash in\; \backslash mathbb\{Z\},\; n\; \backslash ne\; 0\; \backslash right\backslash \}$ ==Arithmetic== :$\backslash frac\{a\}\{b\}\; +\; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{ad+bc\}\{bd\}$   :$\backslash frac\{a\}\{b\}\; \backslash cdot\; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{ac\}\{bd\}$   Two rational numbers $\backslash frac\{a\}\{b\}$ and $\backslash frac\{c\}\{d\}$ are equal iff $ad\; =\; bc$ Additive and multiplicative inverses exist in the rational numbers. :$-\; \backslash left(\; \backslash frac\{a\}\{b\}\; \backslash right)\; =\; \backslash frac\{-a\}\{b\}$   :$\backslash left(\backslash frac\{a\}\{b\}\backslash right)^\{-1\}\; =\; \backslash frac\{b\}\{a\}\; \backslash mbox\{\; if\; \}\; a\; \backslash neq\; 0$ == History == === Egyptian fractions === Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance, $\backslash frac\{5\}\{7\}\; =\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{6\}\; +\; \backslash frac\{1\}\{21\}$ For any positive rational number, there are infinitely many different such representations. These representations are called ''Egyptian fractions'', because the ancient Egyptians used them. The Egyptian hieroglyph used for this is the letter that looks like a mouth, which is transliterated R, so the above fraction would be written as R2R6R21, or, using the hieroglyphs and Bi-directional text: {| border=0 cellspacing=0 cellpadding=0 |Aa13 |style="padding-left:1em; padding-right:1em;" |D21:Z1*Z1*Z1*Z1*Z1*Z1 |D21:V20*V20*Z1 |} ½ is one of exactly three exceptions: it is written as shown in the first hieroglyph above. The two other exceptions were the two only non-unit fractions for which there were symbols: {| "align=center" |D22 |$=\; \backslash frac\{2\}\{3\}$ | style="padding-left:1em;" |D23 |$=\; \backslash frac\{3\}\{4\}$ |} The Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals. == Formal construction == Mathematically we may define them as an ordered pair of integers $\backslash left(a,\; b\backslash right)$, with $b$ not equal to zero. We can define addition and multiplication of these pairs with the following rules: : $\backslash left(a,\; b\backslash right)\; +\; \backslash left(c,\; d\backslash right)\; =\; \backslash left(ad\; +\; bc,\; bd\backslash right)$ : $\backslash left(a,\; b\backslash right)\; \backslash times\; \backslash left(c,\; d\backslash right)\; =\; \backslash left(ac,\; bd\backslash right)$ To conform to our expectation that $2/4\; =\; 1/2$, we define an equivalence relation $\backslash sim$ upon these pairs with the following rule: : $\backslash left(a,\; b\backslash right)\; \backslash sim\; \backslash left(c,\; d\backslash right)\; \backslash mbox\{\; iff\; \}\; ad\; =\; bc$ This equivalence relation 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 quotient field.) We can also define a total order on Q by writing : $\backslash left(a,\; b\backslash right)\; \backslash le\; \backslash left(c,\; d\backslash right)\; \backslash mbox\{\; iff\; \}\; ad\; \backslash le\; bc$ == Properties == The set $\backslash mathbb\{Q\}$, together with the addition and multiplication operations shown above, forms a field (mathematics), the quotient field of the integers $\backslash mathbb\{Z\}$. The rationals are the smallest field with characteristic (algebra) 0: every other field of characteristic 0 contains a copy of $\backslash mathbb\{Q\}$. The algebraic closure of $\backslash mathbb\{Q\}$, i.e. the field of roots of rational polynomials, is the algebraic numbers. 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. The rationals are a dense set: between any two rationals, there sits another one, in fact infinitely many other ones. == 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 expressions of continued fraction. By virtue of their order, the rationals carry an order topology. The rational numbers are a (dense) subset of the real numbers, and as such they also carry a topological subspace. The rational numbers form a metric space by using the metric $d\backslash left(x,\; y\backslash right)\; =\; |x\; -\; y|$, and this yields a third topology on $\backslash mathbb\{Q\}$. Fortunately, 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 space is also connectedness. The rational numbers do not form a completeness (topology); the real numbers are the completion of $\backslash mathbb\{Q\}$. == ''p''-adic numbers == In addition to the absolute value metric mentioned above, there are other metrics which turn $\backslash mathbb\{Q\}$ 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 $p$ divisor $a$; in addition write $|0|\_p\; =\; 0$. For any rational number $\backslash frac\{a\}\{b\}$, we set $\backslash left|\backslash frac\{a\}\{b\}\backslash right|\_p\; =\; \backslash frac\{|a|\_p\}\{|b|\_p\}$. Then $d\_p\backslash left(x,\; y\backslash right)\; =\; |x\; -\; y|\_p$ defines a metric space on $\backslash mathbb\{Q\}$. The metric space $\backslash left(\backslash mathbb\{Q\},\; d\_p\backslash right)$ is not complete, and its completion is the p-adic number $\backslash mathbb\{Q\}\_p$. Elementary mathematics Field theory Fractions Real numbers Set theory th:จำนวนตรรกยะ

Rational number

Removed the following as it seems to me to be too formal and technical for an encyclopedia entry. I've preserved it here for discussion. user:hawthorn ---- == Construction == Mathematically we may define them as an ordered pair of integers (''a'', ''b''), with ''b'' not equal to zero. We can define addition and multiplication upon these pairs with the following rules: :: (''a'', ''b'') + (''c'', ''d'') = (''a'' × ''d'' + ''b'' × ''c'', ''b'' × ''d'') :: (''a'', ''b'') × (''c'', ''d'') = (''a'' × ''c'', ''b'' × ''d'') To conform to our expectation that 2/4 = 1/2, we define an equivalence relation ~ upon these pairs with the following rule: :: (''a'', ''b'') ~ (''c'', ''d'') if, and only if, ''a'' × ''d'' = ''b'' × ''c''. This equivalence relation 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. We can also define a total order on Q by writing :: (''a'', ''b'') ≤ (''c'', ''d'') if, and only if, ''ad'' ≤ ''bc''. ---- Removed the following as it seems more suited to a page on p-adic numbers. Maybe someone can find it a new home. user:hawthorn ---- == Other metrics == In addition to the absolute value metric mentioned above, there are other metrics which turn 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'' divisor ''a''; in addition write |0|_''p'' = 0. For any rational number ''a''/''b'', we set |''a''/''b''|_''p'' = |''a''|_''p'' / |''b''|_''p''. Then ''d''_''p''(''x'', ''y'') = |''x'' - ''y''|_''p'' defines a metric space on Q. The metric space (Q, ''d''_''p'') is not complete, and its completion is given by the p-adic number. ---- It could be argued that the Egyptian fraction stuff deserves its own page. however I've left it for now. user:hawthorn ---- I'm restoring both. they're important. -- User:Tarquin 09:53 26 Jun 2003 (UTC) ---- Couldn't you have done it without throwing out the baby with the bath water! An encyclopedia entry on the rational numbers shouldn't have to start off in such an abstract and formal way- they are just fractions for goodness sake! Whay can't we say so right from the start. Even a non-mathematician can understand this concept. I'm in favour of keeping it as general as possible as long as possible. Move the formalism to the end. I disagree that the exised stuff is all that important. The first extract is pretty much the field of fractions construction in the special case that the ring is the ring of integers, which seems like trying to sink a tack with a sledgehammer to me. The second stuff on the p-adic metric and p-adic numbers - well it just isn't what I'd expect to find on a page on the rationals is all. user:hawthorn I agree that entries shouldn't start off in an abstract or formal way. User:Pizza Puzzle : Sorry. I was a bit hasty. You're right, we should start with a layperson-friendly overview. But after the first screen-full of text, it's fine to get technical! -- User:Tarquin 21:38 26 Jun 2003 (UTC) == nominator and denominator == Perhaps mentioning the formal names of nominator and denominator would be in order in this document, just to let people know how the numbers above and below dividor line are called. It is useful information especially to people that are non-native english speakers (like me). :The terminology ''numerator'' and ''denominator'' is explained in the article on vulgar fractions, which is prominently linked to from the present article. That doesn't mean the same information couldn't be repeated here, but then again the case could be made that vulgar fraction as a whole is redundant. --User:MarkSweep 20:01, 11 Dec 2004 (UTC)

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See other meanings of words starting from letter:

R

RA | RB | RC | RD | RE | RF | RG | RH | RI | RJ | RK | RL | RM | RN | RO | RP | RS | RT | RU | RW | RX | RY | RZ |

Words begining with Rational_number:

Rational_Number
Rational_number
Rational_number
Rational_numbers
Rational_numbers
Rational_number_field

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