"ପରିମେୟ ସଂଖ୍ୟା" ପୃଷ୍ଠାର ସଂସ୍କରଣଗୁଡ଼ିକ ମଧ୍ୟରେ ତଫାତ
Content deleted Content added
Hpsatapathy (ଆଲୋଚନା | ଅବଦାନ) ଟିକେNo edit summary |
Hpsatapathy (ଆଲୋଚନା | ଅବଦାନ) ଟିକେNo edit summary |
||
୧ କ ଧାଡ଼ି:
ଗାଣିତିକ ପଦ୍ଧତିରେ ଦୁଇଟି ପୂର୍ଣ୍ଣାଙ୍କର ଭଗ୍ନାଂଶ (ଏକ ଅଣଶୂନ୍ୟ ହର) ଯଥା; 'କ'/'ଖ' ଭାବେ ପ୍ରକାଶ କରାଯାଇପାରୁଥିବା ସମସ୍ତ ସଂଖ୍ୟା ହେଉଛି '''ପରିମେୟ ସଂଖ୍ୟା''' । <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> ଏଭଳି ସ୍ଥଳେ ହର ୧ ହେଉଥିଲେ ସମସ୍ତ ପୂର୍ଣ୍ଣାଙ୍କ ମଧ୍ୟ ପରିମେୟ ସଂଖ୍ୟାର ଅନ୍ତର୍ଗତ ହେବ । ପରିମେୟ ସଂଖ୍ୟା ସମୂହର ସେଟକୁ ଗାଣିତିକ ସଂକେତ 'Q' ଦ୍ୱାରା ପ୍ରକାଶ କରାଯାଏ ।<math>\mathbb{Q}</math>, Unicode ℚ);<ref>{{cite web|last1=Rouse|first1=Margaret|title=Mathematical Symbols|url=http://searchdatacenter.techtarget.com/definition/Mathematical-Symbols|accessdate=1 April 2015}}</ref>
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 any other integer [[radix|base]] (e.g. [[binary numeral system|binary]], [[hexadecimal]]).
A [[real number]] that is not rational is called [[irrational number|irrational]]. Irrational numbers include [[square root of 2|{{math|{{sqrt|2}}}}]], [[Pi|{{pi}}]], [[E (mathematical constant)|{{math|''e''}}]], and [[Golden ratio|{{math|''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"/>
Rational numbers can be [[Formalism (mathematics)|formally]] defined as [[equivalence class]]es of pairs of integers {{math|(''p'', ''q'')}} such that {{math|''q'' ≠ 0}}, for the [[equivalence relation]] defined by {{math|(''p''<sub>1</sub>, ''q''<sub>1</sub>) ~ (''p''<sub>2</sub>, ''q''<sub>2</sub>)}} [[iff|if, and only if]] {{math|''p''<sub>1</sub>''q''<sub>2</sub> {{=}} ''p''<sub>2</sub>''q''<sub>1</sub>}}. With this formal definition, the fraction {{math|''p''/''q''}} becomes the standard notation for the equivalence class of {{math|(''p''<sub>2</sub>, ''q''<sub>2</sub>)}}.
Rational numbers together with [[addition]] and [[multiplication]] form a [[field (mathematics)|field]] which contains the [[integer]]s and is contained in any field containing the integers. In other words, the field of rational numbers is a [[prime field]], and a field has [[characteristic zero]] if and only if it contains the rational numbers as a subfield. Finite [[field extension|extensions]] of {{math|'''Q'''}} are called [[algebraic number field]]s, and the [[algebraic closure]] of {{math|'''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>
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.
== ଗାଣିତିକ ଧର୍ମ ==
| |||