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| {{redirect|Rationals||Rational (disambiguation)}}
| | If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. They found out all the possible information about bringing up your baby and save money at the same time. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br>These folders as well as files have to copied and the saved. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Websites using this content based strategy are always given top scores by Google. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. If you have any kind of questions relating to where and how to make use of [http://www.mevemo.dk/2013/11/18/hello-world/ backup plugin], you can contact us at our site. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. It's now become a great place to sell it thanks to Woo - Commerce. Make sure you have the latest versions of all your plugins are updated. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. As with a terminology, there are many methods to understand how to use the terminology. |
| {{Refimprove|date=September 2013}}
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| In [[mathematics]], a '''rational number''' is any [[number]] that can be expressed as the [[quotient]] or fraction ''p''/''q'' of two [[integer]]s, with the [[denominator]] ''q'' not equal to zero.<ref name="Rosen">{{cite book |last=Rosen |first=Kenneth |year=2007 |title=Discrete Mathematics and its Applications |edition=6th |publisher=McGraw-Hill |location=New York, NY |isbn=978-0-07-288008-3 |pages=105,158-160}}</ref> Since ''q'' may be equal to 1, every integer is a rational number. The [[set (mathematics)|set]] of all rational numbers is usually denoted by a boldface '''Q''' (or [[blackboard bold]] <math>\mathbb{Q}</math>, [[Unicode]] {{unicode|ℚ}}); it was thus named in 1895 by [[Giuseppe Peano|Peano]] after ''[[wikt:quoziente|quoziente]]'', Italian for "[[quotient]]".
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| The [[decimal expansion]] of a rational number always either terminates after a finite number of [[numerical digit|digits]] or begins to [[repeating decimal|repeat]] the same finite [[sequence]] of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for [[decimal|base 10]], but also for [[binary numeral system|binary]], [[hexadecimal]], or any other integer [[radix|base]].
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| A [[real number]] that is not rational is called [[irrational number|irrational]]. Irrational numbers include [[square root of 2|{{sqrt|2}}]], [[Pi|π]], [[E (mathematical constant)|''e'']], and [[Golden ratio|''φ'']]. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is [[countable set|countable]], and the set of real numbers is [[uncountable set|uncountable]], [[almost all]] real numbers are irrational.<ref name="Rosen"/>
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| The rational numbers can be [[Formalism (mathematics)|formally]] defined as the [[equivalence class]]es of the [[quotient set]] {{nowrap|('''Z''' × ('''Z''' \ {0})) / ~,}} where the [[cartesian product]] {{nowrap|'''Z''' × ('''Z''' \ {0})}} is the set of all [[ordered pair]]s (''m'',''n'') where ''m'' and ''n'' are [[integer]]s, ''n'' is not 0 {{nowrap|(''n'' ≠ 0)}}, and "~" is the [[equivalence relation]] defined by {{nowrap|(''m''<sub>1</sub>,''n''<sub>1</sub>) ~ (''m''<sub>2</sub>,''n''<sub>2</sub>)}} [[iff|if, and only if]], {{nowrap|''m''<sub>1</sub>''n''<sub>2</sub> − ''m''<sub>2</sub>''n''<sub>1</sub> {{=}} 0.}}
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| In [[abstract algebra]], the rational numbers together with certain operations of [[addition]] and [[multiplication]] form a [[field (mathematics)|field]]. This is the archetypical field of [[characteristic (algebra)|characteristic]] zero, and is the [[field of fractions]] for the [[ring (mathematics)|ring]] of integers. Finite [[field extension|extensions]] of '''Q''' are called [[algebraic number field]]s, and the [[algebraic closure]] of '''Q''' is the field of [[algebraic number]]s.<ref name="Gilbert">{{cite book |last1=Gilbert |first1=Jimmie |last2=Linda |first2=Gilbert |year=2005 |title=Elements of Modern Algebra |edition=6th |publisher=Thomson Brooks/Cole |location=Belmont, CA |isbn=0-534-40264-X |pages=243-244}}</ref>
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| In [[mathematical analysis]], the rational numbers form a [[dense set|dense subset]] of the real numbers. The real numbers can be constructed from the rational numbers by [[completion (metric space)|completion]], using [[Cauchy sequence]]s, [[Dedekind cut]]s, or infinite [[decimal]]s.
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| Zero divided by any other integer equals zero, therefore zero is a rational number (but [[division by zero]] is undefined).
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| ==Terminology==
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| The term ''rational'' in reference to the set '''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 (mathematics)|field]] considered is the field '''Q''' of 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 [[algebraic curve|rational algebraic curve]] is ''not'' an algebraic curve with rational coefficients.
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| ==Arithmetic==
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| {{see also|Fraction (mathematics)#Arithmetic with fractions}}
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| ===Embedding of integers===
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| Any integer ''n'' can be expressed as the rational number ''n''/1.
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| ===Equality===
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| :<math>\frac{a}{b} = \frac{c}{d}</math> if and only if <math>ad = bc.</math>
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| <!--Examples:
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| :<math>\frac{1}{3} = \frac{2}{6}</math>
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| :<math>\frac{-1}{2} = \frac{1}{-2}</math>
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| :<math>\frac{0}{1} = \frac{0}{2}</math>-->
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| ===Ordering===
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| Where both denominators are positive:
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| :<math>\frac{a}{b} < \frac{c}{d}</math> if and only if <math>ad < bc.</math>
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| If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations:
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| :<math>\frac{-a}{-b} = \frac{a}{b}</math>
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| and
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| :<math>\frac{a}{-b} = \frac{-a}{b}.</math>
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| ===Addition===
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| Two fractions are added as follows:
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| :<math>\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.</math>
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| ===Subtraction===
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| :<math>\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}.</math>
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| ===Multiplication===
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| The rule for multiplication is:
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| :<math>\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}.</math>
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| ===Division===
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| Where ''c'' ≠ 0:
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| :<math>\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}.</math>
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| Note that division is equivalent to multiplying by the [[multiplicative inverse|reciprocal]] of the divisor fraction:
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| :<math>\frac{ad}{bc} = \frac{a}{b} \times \frac{d}{c}.</math>
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| ===Inverse===
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| [[Additive inverse|Additive]] and [[multiplicative inverse]]s exist in the rational numbers:
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| :<math> - \left( \frac{a}{b} \right) = \frac{-a}{b} = \frac{a}{-b} \quad\mbox{and}\quad
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| \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \mbox{ if } a \neq 0. </math>
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| ===Exponentiation to integer power===
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| If ''n'' is a non-negative integer, then
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| :<math>\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}</math>
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| and (if ''a'' ≠ 0): | |
| :<math>\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}.</math>
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| ==Continued fraction representation==
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| {{Main|Continued fraction}}
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| A '''finite continued fraction''' is an expression such as
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| :<math>a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}},</math>
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| where ''a<sub>n</sub>'' are integers. Every rational number ''a''/''b'' has two closely related expressions as a finite continued fraction, whose [[coefficient]]s ''a<sub>n</sub>'' can be determined by applying the [[Euclidean algorithm]] to (''a'',''b'').
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| ==Formal construction==
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| [[File:RationalRepresentation.pdf|thumb|right|300px|A diagram showing a representation of the equivalent classes of pairs of integers]]
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| Mathematically we may construct the rational numbers as [[equivalence class]]es of [[ordered pair]]s of [[integer]]s (''m'',''n''), with {{nowrap|''n'' ≠ 0}}. This space of equivalence classes is the [[quotient space]] {{nowrap|('''Z''' × ('''Z''' \ {0})) / ~,}} where {{nowrap|(''m''<sub>1</sub>,''n''<sub>1</sub>) ~ (''m''<sub>2</sub>,''n''<sub>2</sub>)}} if, and only if, {{nowrap|''m''<sub>1</sub>''n''<sub>2</sub> − ''m''<sub>2</sub>''n''<sub>1</sub> {{=}} 0.}} We can define addition and multiplication of these pairs with the following rules:
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| :<math>\left(m_1, n_1\right) + \left(m_2, n_2\right) \equiv \left(m_1n_2 + n_1m_2, n_1n_2\right)</math>
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| :<math>\left(m_1, n_1\right) \times \left(m_2, n_2\right) \equiv \left(m_1m_2, n_1n_2\right)</math>
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| and, if ''m''<sub>2</sub> ≠ 0, division by | |
| :<math>\frac{\left(m_1, n_1\right)} {\left(m_2, n_2\right)} \equiv \left(m_1n_2, n_1m_2\right).</math>
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| The equivalence relation (''m''<sub>1</sub>,''n''<sub>1</sub>) ~ (''m''<sub>2</sub>,''n''<sub>2</sub>) if, and only if, {{nowrap|''m''<sub>1</sub>''n''<sub>2</sub> − ''m''<sub>2</sub>''n''<sub>1</sub> {{=}} 0}} is a [[congruence relation]], i.e. it is compatible with the addition and multiplication defined above, and we may define '''Q''' to be the [[quotient set]] {{nowrap|1=('''Z''' × ('''Z''' \ {0})) / ~,}} i.e. we identify two pairs (''m''<sub>1</sub>,''n''<sub>1</sub>) and (''m''<sub>2</sub>,''n''<sub>2</sub>) if they are equivalent in the above sense. (This construction can be carried out in any [[integral domain]]: see [[field of fractions]].) We denote by [(''m''<sub>1</sub>,''n''<sub>1</sub>)] the equivalence class containing (''m''<sub>1</sub>,''n''<sub>1</sub>). If (''m''<sub>1</sub>,''n''<sub>1</sub>) ~ (''m''<sub>2</sub>,''n''<sub>2</sub>) then, by definition, (''m''<sub>1</sub>,''n''<sub>1</sub>) belongs to [(''m''<sub>2</sub>,''n''<sub>2</sub>)] and (''m''<sub>2</sub>,''n''<sub>2</sub>) belongs to [(''m''<sub>1</sub>,''n''<sub>1</sub>)]; in this case we can write {{nowrap|[(''m''<sub>1</sub>,''n''<sub>1</sub>)] {{=}} [(''m''<sub>2</sub>,''n''<sub>2</sub>)]}}. Given any equivalence class [(''m'',''n'')] there are a countably infinite number of representation, since
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| :<math>\cdots = [(-2m,-2n)] = [(-m,-n)] = [(m,n)] = [(2m,2n)] = \cdots.</math>
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| The canonical choice for [(''m'',''n'')] is chosen so that {{nowrap|[[Greatest common divisor|gcd]](''m'',''n'') {{=}} 1}}, i.e. ''m'' and ''n'' share no common factors, i.e. ''m'' and ''n'' are [[coprime]]. For example, we would write [(1,2)] instead of [(2,4)] or [(−12,−24)], even though {{nowrap|[(1,2)] {{=}} [(2,4)] {{=}} [(−12,−24)]}}.
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| We can also define a [[total order]] on '''Q'''. Let ∧ be the [[and (logic)|''and''-symbol]] and ∨ be the [[or (logic)|''or''-symbol]]. We say that {{nowrap|1=[(''m''<sub>1</sub>,''n''<sub>1</sub>)] ≤ [(''m''<sub>2</sub>,''n''<sub>2</sub>)]}} if:
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| :<math>(n_1n_2 > 0 \ \and \ m_1n_2 \le n_1m_2) \ \or \ (n_1n_2 < 0 \ \and \ m_1n_2 \ge n_1m_2).</math>
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| The integers may be considered to be rational numbers by the [[embedding]] that maps ''m'' to [(''m'',1)].
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| ==Properties==
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| [[File:Diagonal argument.svg|thumb|right|170px|A diagram illustrating the countability of the rationals]]
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| The set '''Q''', together with the addition and multiplication operations shown above, forms a [[field (mathematics)|field]], the [[field of fractions]] of the [[integer]]s '''Z'''.
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| The rationals are the smallest field with [[characteristic (algebra)|characteristic]] zero: every other field of characteristic zero contains a copy of '''Q'''. The rational numbers are therefore the [[prime field]] for characteristic zero.
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| The [[algebraic closure]] of '''Q''', i.e. the field of roots of rational polynomials, is the [[algebraic number]]s.
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| 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]].
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| The rationals are a [[densely ordered]] set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
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| :<math>\frac{a}{b} < \frac{c}{d}</math>
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| (where <math>b,d</math> are positive), we have
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| :<math>\frac{a}{b} < \frac{ad + bc}{2bd} < \frac{c}{d}.</math>
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| Any [[totally ordered]] set which is countable, dense (in the above sense), and has no least or greatest element is [[order isomorphism|order isomorphic]] to the rational numbers.
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| ==Real numbers and topological properties==
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| The rationals are a [[dense set|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 set|finite]] expansions as [[continued fraction|regular continued fractions]].
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| By virtue of their order, the rationals carry an [[order topology]]. The rational numbers, as a subspace of the real numbers, also carry a [[subspace topology]]. The rational numbers form a [[metric space]] by using the [[absolute difference]] metric {{nowrap|''d''(''x'',''y'') {{=}} {{!}}''x'' − ''y''{{!}},}} and this yields a third topology on '''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]] [[Topological property|metrizable space]] without [[isolated point]]s. The space is also [[totally disconnected space|totally disconnected]]. The rational numbers do not form a [[completeness (topology)|complete metric space]]; the [[real numbers]] are the completion of '''Q''' under the metric {{nowrap|''d''(''x'',''y'') {{=}} {{!}}''x'' − ''y''{{!}},}} above.
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| ==''p''-adic numbers==
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| {{see also|P-adic Number}}
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| In addition to the absolute value metric mentioned above, there are other metrics which turn '''Q''' into a topological field:
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| Let ''p'' be a [[prime number]] and for any non-zero integer ''a'', let {{nowrap|{{!}}''a''{{!}}<sub>''p''</sub> {{=}} ''p''<sup>−''n''</sup>}}, where ''p<sup>n</sup>'' is the highest power of ''p'' [[divisor|dividing]] ''a''.
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| In addition set {{nowrap|{{!}}0{{!}}<sub>''p''</sub> {{=}} 0.}} For any rational number ''a''/''b'', we set {{nowrap|{{!}}''a''/''b''{{!}}<sub>''p''</sub> {{=}} {{!}}''a''{{!}}<sub>''p''</sub> / {{!}}''b''{{!}}<sub>''p''</sub>.}}
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| Then {{nowrap|''d<sub>p</sub>''(''x'',''y'') {{=}} {{!}}''x'' − ''y''{{!}}<sub>''p''</sub>}} defines a [[metric space|metric]] on '''Q'''.
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| The metric space ('''Q''',''d<sub>p</sub>'') is not complete, and its completion is the [[p-adic number|''p''-adic number field]] '''Q'''<sub>''p''</sub>. [[Ostrowski's theorem]] states that any non-trivial [[absolute value (algebra)|absolute value]] on the rational numbers '''Q''' is equivalent to either the usual real absolute value or a [[p-adic number|''p''-adic]] absolute value.
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| ==See also==
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| *[[Floating point]]
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| *[[Ford circle]]s
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| *[[Niven's theorem]]
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| *[[Rational data type]]
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| ==External links==
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| *{{springer|title=Rational number|id=p/r077620}}
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| *[http://mathworld.wolfram.com/RationalNumber.html "Rational Number" From MathWorld – A Wolfram Web Resource]
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| ==References==
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| <references/>
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| {{Number Systems}}
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| [[Category:Elementary mathematics]]
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| [[Category:Field theory]]
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| [[Category:Fractions]]
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| [[Category:Rational numbers| ]]
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| {{Link FA|lmo}}
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If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. They found out all the possible information about bringing up your baby and save money at the same time. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.
These folders as well as files have to copied and the saved. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Websites using this content based strategy are always given top scores by Google.
There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. If you have any kind of questions relating to where and how to make use of backup plugin, you can contact us at our site. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. It's now become a great place to sell it thanks to Woo - Commerce. Make sure you have the latest versions of all your plugins are updated.
As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. As with a terminology, there are many methods to understand how to use the terminology.