. . . . . . . . "En math\u00E9matiques, les hyperop\u00E9rations (ou hyperop\u00E9rateurs) constituent une suite infinie d'op\u00E9rations qui prolonge logiquement la suite des op\u00E9rations arithm\u00E9tiques \u00E9l\u00E9mentaires suivantes : 1. \n* addition (n = 1) : 2. \n* multiplication (n = 2) : 3. \n* exponentiation (n = 3) : Reuben Goodstein proposa de baptiser les op\u00E9rations au-del\u00E0 de l'exponentiation en utilisant des pr\u00E9fixes grecs : t\u00E9tration (n = 4), pentation (n = 5), hexation (n = 6), etc. L'hyperop\u00E9ration \u00E0 l'ordre n peut se noter \u00E0 l'aide d'une fl\u00E8che de Knuth au rang n \u2013 2. ."@fr . . . "En matematiko, hiperoperatoro estas funkcio de tri argumentoj, a\u016D familio de la hiper-n funkcioj de du argumentoj: (Vidu supren-sagan notacion de Knuth kaj \u0109enitan sagan notacion de Conway.)"@eo . "Hiperoperatoro"@eo . . . "42907"^^ . . . . . "In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written as using n \u2212 2 arrows in Knuth's up-arrow notation.Each hyperoperation may be understood recursively in terms of the previous one by: It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function: This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes's number and googolplexplex (e.g. is much larger than Skewes's number and googolplexplex), but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3). This recursion rule is common to many variants of hyperoperations."@en . . . "\uC218\uD559\uC5D0\uC11C \uD558\uC774\uD37C \uC5F0\uC0B0 \uC218\uC5F4(Hyperoperation sequence)\uC740 \uD558\uC774\uD37C \uC5F0\uC0B0\uC774\uB77C \uBD88\uB9AC\uB294 \uB367\uC148, \uACF1\uC148, \uAC70\uB4ED\uC81C\uACF1\uC73C\uB85C \uC2DC\uC791\uD558\uB294 \uC774\uD56D\uC5F0\uC0B0 \uC218\uC5F4\uC774\uB2E4. \uC774 \uC218\uC5F4\uC758 n\uBC88\uC9F8 \uD558\uC774\uD37C \uC5F0\uC0B0\uC740 n\uC758 \uADF8\uB9AC\uC2A4\uC5B4 \uC811\uB450\uC0AC\uC5D0 \uC811\uBBF8\uC0AC -ation\uC744 \uBD99\uC778 \uB2E8\uC5B4\uB85C \uBD88\uB9AC\uBA70, \uCEE4\uB204\uC2A4 \uC717\uD654\uC0B4\uD45C \uD45C\uAE30\uBC95\uC5D0\uC11C (n-2)\uAC1C\uC758 \uD654\uC0B4\uD45C\uB85C \uD45C\uAE30\uD560 \uC218 \uC788\uB2E4."@ko . "\uC218\uD559\uC5D0\uC11C \uD558\uC774\uD37C \uC5F0\uC0B0 \uC218\uC5F4(Hyperoperation sequence)\uC740 \uD558\uC774\uD37C \uC5F0\uC0B0\uC774\uB77C \uBD88\uB9AC\uB294 \uB367\uC148, \uACF1\uC148, \uAC70\uB4ED\uC81C\uACF1\uC73C\uB85C \uC2DC\uC791\uD558\uB294 \uC774\uD56D\uC5F0\uC0B0 \uC218\uC5F4\uC774\uB2E4. \uC774 \uC218\uC5F4\uC758 n\uBC88\uC9F8 \uD558\uC774\uD37C \uC5F0\uC0B0\uC740 n\uC758 \uADF8\uB9AC\uC2A4\uC5B4 \uC811\uB450\uC0AC\uC5D0 \uC811\uBBF8\uC0AC -ation\uC744 \uBD99\uC778 \uB2E8\uC5B4\uB85C \uBD88\uB9AC\uBA70, \uCEE4\uB204\uC2A4 \uC717\uD654\uC0B4\uD45C \uD45C\uAE30\uBC95\uC5D0\uC11C (n-2)\uAC1C\uC758 \uD654\uC0B4\uD45C\uB85C \uD45C\uAE30\uD560 \uC218 \uC788\uB2E4."@ko . . . . "Hiperoperaci\u00F3n"@es . . . . . "\u0413\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440 \u2014 \u043E\u0431\u043E\u0431\u0449\u0435\u043D\u0438\u0435 \u0442\u0440\u0430\u0434\u0438\u0446\u0438\u043E\u043D\u043D\u044B\u0445 \u0430\u0440\u0438\u0444\u043C\u0435\u0442\u0438\u0447\u0435\u0441\u043A\u0438\u0445 \u043E\u043F\u0435\u0440\u0430\u0446\u0438\u0439 \u2014 \u0441\u043B\u043E\u0436\u0435\u043D\u0438\u044F, \u0443\u043C\u043D\u043E\u0436\u0435\u043D\u0438\u044F \u0438 \u0432\u043E\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0432 \u0441\u0442\u0435\u043F\u0435\u043D\u044C, \u0440\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u043C\u044B\u0445 \u043A\u0430\u043A \u0433\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u044B 1-\u0433\u043E, 2-\u0433\u043E \u0438 3-\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043D\u043D\u043E, \u2014 \u043D\u0430 \u0432\u044B\u0441\u0448\u0438\u0435 \u043F\u043E\u0440\u044F\u0434\u043A\u0438 (\u0442\u0435\u0442\u0440\u0430\u0446\u0438\u044F, \u043F\u0435\u043D\u0442\u0430\u0446\u0438\u044F \u0438 \u0442\u0430\u043A \u0434\u0430\u043B\u0435\u0435)."@ru . "En math\u00E9matiques, les hyperop\u00E9rations (ou hyperop\u00E9rateurs) constituent une suite infinie d'op\u00E9rations qui prolonge logiquement la suite des op\u00E9rations arithm\u00E9tiques \u00E9l\u00E9mentaires suivantes : 1. \n* addition (n = 1) : 2. \n* multiplication (n = 2) : 3. \n* exponentiation (n = 3) : Reuben Goodstein proposa de baptiser les op\u00E9rations au-del\u00E0 de l'exponentiation en utilisant des pr\u00E9fixes grecs : t\u00E9tration (n = 4), pentation (n = 5), hexation (n = 6), etc. L'hyperop\u00E9ration \u00E0 l'ordre n peut se noter \u00E0 l'aide d'une fl\u00E8che de Knuth au rang n \u2013 2. . La fl\u00EAche de Knuth au rang m est d\u00E9finie r\u00E9cursivement par : et Elle peut aussi se d\u00E9finir \u00E0 l'aide de la r\u00E8gle : . Chacune cro\u00EEt plus vite que la pr\u00E9c\u00E9dente. Des suites similaires ont historiquement port\u00E9 diverses appellations, telles que la fonction d'Ackermann (\u00E0 3 arguments), la hi\u00E9rarchie d'Ackermann, la hi\u00E9rarchie de Grzegorczyk (plus g\u00E9n\u00E9rale), la version de Goodstein de la fonction d'Ackermann, hyper-n."@fr . . . . . . . . . . . . . . . . . . . . . . "\u8D85\u904B\u7B97\u5E8F\u5217\u662F\u6570\u5B66\u4E2D\u4E00\u79CD\u4E8C\u5143\u8FD0\u7B97\u7684\u5E8F\u5217\uFF0C\u524D\u4E09\u9879\u5206\u522B\u4E3A\u52A0\u6CD5\u3001\u4E58\u6CD5\u3001\u5E42\uFF0C\u4E00\u822C\u4F86\u8AAA\uFF0C\u9664\u4E86\u5E8F\u5217\u4E2D\u7B2C\u4E00\u9805\u7684\u52A0\u6CD5\u904B\u7B97\u4E4B\u5916\uFF0C\u5E8F\u5217\u4E2D\u6BCF\u4E00\u9805\u7684\u904B\u7B97\u90FD\u662F\u91CD\u8907\u7684\u524D\u4E00\u9805\u7684\u904B\u7B97\uFF08\u4F8B\u5982\u4E58\u6CD5\u662F\u91CD\u8907\u7684\u52A0\u6CD5\uFF1A\uFF0C\u51AA\u662F\u91CD\u8907\u7684\u4E58\u6CD5\uFF1A\uFF09\u3002\u8FD9\u4E9B\u8FD0\u7B97\u901A\u79F0\u4E3A\u8D85\u8FD0\u7B97\uFF08\u6216\u7A31\u70BAhyper\u904B\u7B97\u7B26\uFF09\u3002\u5E8F\u5217\u4E2D\u7684\u7B2Cn\u9879\u79F0\u4E3A\u8D85-n\u8FD0\u7B97\u6216\u7B2Cn\u7D1A\u7684\u8D85\u904B\u7B97\uFF0C\u5176\u7B26\u865F\u70BA[n]\u3002\u82F1\u6587\u5247\u7531\u547D\u540D\uFF0C\u7576n\u22654\u6642\uFF0C\u7531n\u7684\u5E0C\u814A\u8BED\u524D\u7F00\u52A0\u4E0A\u540E\u7F00-ation\u7EC4\u6210\uFF08\u4F8B\u5982\u8D85-4\u8FD0\u7B97\u79F0\u4E3Atetration\uFF0C\u8D85-5\u8FD0\u7B97\u79F0\u4E3Apentation\uFF09\u3002\u7576n\u22653 \u6642\uFF0C\u4F7F\u7528\u9AD8\u5FB7\u7EB3\u7BAD\u53F7\u8868\u793A\u6CD5\u53EF\u5C06\u8D85-n\u8FD0\u7B97\u7684\u7B26\u865F\u8868\u793A\u4E3A(n-2)\u4E2A\u7BAD\u5934\u3002 \u8D85\u8FD0\u7B97\u53EF\u901A\u8FC7\u9012\u5F52\u8FDB\u884C\u5B9A\u4E49\uFF0C\u5C0D\u65BC\u6240\u6709\u6B63\u6574\u6578a\uFF0C\u6B63\u6574\u6578b\u548C\u6B63\u6574\u6578n\uFF1A \u9664\u8FD9\u4E00\u6700\u5E38\u89C1\u7684\u5B9A\u4E49\u4E4B\u5916\uFF0C\u8D85\u8FD0\u7B97\u8FD8\u6709\u5176\u4ED6\u7684\u53D8\u4F53\u3002\uFF08\uFF09"@zh . . . . . . . . . . . "\u30CF\u30A4\u30D1\u30FC\u6F14\u7B97\u5B50\uFF08\u30CF\u30A4\u30D1\u30FC\u3048\u3093\u3056\u3093\u3057\u3001hyper operator\uFF09\u306F\u3001\u52A0\u7B97\u3001\u4E57\u7B97\u3001\u51AA\u4E57\u3092\u4E00\u822C\u5316\u3057\u305F\u6F14\u7B97\u306E\u305F\u3081\u306E\u6F14\u7B97\u5B50\u3067\u3042\u308B\u3002"@ja . . . . . . . "\u0413\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0301\u0442\u043E\u0440 \u2014 \u043E\u0431\u043E\u0431\u0449\u0435\u043D\u0438\u0435 \u0442\u0440\u0430\u0434\u0438\u0446\u0438\u043E\u043D\u043D\u044B\u0445 \u0430\u0440\u0438\u0444\u043C\u0435\u0442\u0438\u0447\u0435\u0441\u043A\u0438\u0445 \u043E\u043F\u0435\u0440\u0430\u0446\u0438\u0439 \u2014 \u0441\u043B\u043E\u0436\u0435\u043D\u0438\u044F, \u0443\u043C\u043D\u043E\u0436\u0435\u043D\u0438\u044F \u0438 \u0432\u043E\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0432 \u0441\u0442\u0435\u043F\u0435\u043D\u044C, \u0440\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u043C\u044B\u0445 \u043A\u0430\u043A \u0433\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u044B 1-\u0433\u043E, 2-\u0433\u043E \u0438 3-\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043D\u043D\u043E, \u2014 \u043D\u0430 \u0432\u044B\u0441\u0448\u0438\u0435 \u043F\u043E\u0440\u044F\u0434\u043A\u0438 (\u0442\u0435\u0442\u0440\u0430\u0446\u0438\u044F, \u043F\u0435\u043D\u0442\u0430\u0446\u0438\u044F \u0438 \u0442\u0430\u043A \u0434\u0430\u043B\u0435\u0435). \u0412 \u0441\u0438\u043B\u0443 \u043D\u0435\u043A\u043E\u043C\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0441\u0442\u0438 (\u0432 \u043E\u0431\u0449\u0435\u043C \u0441\u043B\u0443\u0447\u0430\u0435) \u0433\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u0438\u043C\u0435\u0435\u0442 \u0434\u0432\u0435 \u043E\u0431\u0440\u0430\u0442\u043D\u044B\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u2014 \u0433\u0438\u043F\u0435\u0440\u043A\u043E\u0440\u0435\u043D\u044C \u0438 \u0433\u0438\u043F\u0435\u0440\u043B\u043E\u0433\u0430\u0440\u0438\u0444\u043C. \u0413\u0438\u043F\u0435\u0440\u043A\u043E\u0440\u0435\u043D\u044C \u0438 \u0433\u0438\u043F\u0435\u0440\u043B\u043E\u0433\u0430\u0440\u0438\u0444\u043C \u0441\u043B\u043E\u0436\u0435\u043D\u0438\u044F \u0438 \u0443\u043C\u043D\u043E\u0436\u0435\u043D\u0438\u044F \u0441\u043E\u0432\u043F\u0430\u0434\u0430\u044E\u0442, \u043E\u0431\u0440\u0430\u0437\u0443\u044F \u0432\u044B\u0447\u0438\u0442\u0430\u043D\u0438\u0435 \u0438 \u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0435\u043D\u043D\u043E, \u043D\u043E \u0443\u0436\u0435 \u0434\u043B\u044F \u0432\u043E\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0432 \u0441\u0442\u0435\u043F\u0435\u043D\u044C \u043E\u0431\u0440\u0430\u0442\u043D\u044B\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u0441\u0442\u0430\u043D\u043E\u0432\u044F\u0442\u0441\u044F \u0440\u0430\u0437\u043B\u0438\u0447\u043D\u044B\u043C\u0438 (\u043A\u043E\u0440\u0435\u043D\u044C \u0438 \u043B\u043E\u0433\u0430\u0440\u0438\u0444\u043C). \u041E\u0431\u0440\u0430\u0442\u043D\u044B\u0435 \u043E\u043F\u0435\u0440\u0430\u0446\u0438\u0438 \u043E\u0431\u043E\u0431\u0449\u0430\u044E\u0442\u0441\u044F \u0434\u043B\u044F \u0433\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0430 \u043B\u044E\u0431\u043E\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430."@ru . . "En matematiko, hiperoperatoro estas funkcio de tri argumentoj, a\u016D familio de la hiper-n funkcioj de du argumentoj: (Vidu supren-sagan notacion de Knuth kaj \u0109enitan sagan notacion de Conway.)"@eo . . "\u0413\u0456\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440"@uk . . "Hyperop\u00E9ration"@fr . "1110399282"^^ . "En matem\u00E0tiques, la successi\u00F3 d'hiperoperaci\u00F3 \u00E9s una successi\u00F3 infinita d'operacions aritm\u00E8tiques (anomenades hiperoperacions en aquest context) que comen\u00E7a amb una operaci\u00F3 un\u00E0ria (la amb n = 0). La successi\u00F3 continua amb les operacions bin\u00E0ries d'addici\u00F3 (n = 1), multiplicaci\u00F3 (n = 2) i potenciaci\u00F3 (n = 3). Es pot entendre recursivament cada hiperoperaci\u00F3 en termes de l'anterior per: i es pot escriure utilitzant n-2 fletxes de la . Tamb\u00E9 es pot definir segons la regla de recursivitat de la definici\u00F3 en la versi\u00F3 de fletxa de Knuth de la funci\u00F3 d'Ackermann: o"@ca . "\u8D85\u8FD0\u7B97"@zh . . "Em matem\u00E1tica, a seq\u00FCencia de hiperopera\u00E7\u00F5es \u00E9 uma seq\u00FCencia de opera\u00E7\u00F5es bin\u00E1rias que iniciam com a adi\u00E7\u00E3o, multiplica\u00E7\u00E3o e exponencia\u00E7\u00E3o, chamadas hiperopera\u00E7\u00F5es em geral. O n-\u00E9simo membro desta seq\u00FCencia foi nomeado por seguindo o prefixo grego de n acrescido do sufixo -\u00E7\u00E3o (como em tetra\u00E7\u00E3o, ) e pode ser escrito usando setas na Nota\u00E7\u00E3o de Knuth. Cada hiperopera\u00E7\u00E3o \u00E9 definida recursivamente em termos da anterior, como \u00E9 o caso com a nota\u00E7\u00E3o de seta para cima de Knuth. A parte da defini\u00E7\u00E3o que faz isso \u00E9 a regra recursiva da fun\u00E7\u00E3o de Ackermann: que \u00E9 comum a muitas variantes de hiperopera\u00E7\u00F5es (ver )."@pt . "En matem\u00E1ticas, la sucesi\u00F3n de hiperoperaciones\u200Bes una sucesi\u00F3n infinita de operaciones aritm\u00E9ticas (llamadas hiperoperaciones)\u200B\u200B\u200B que se inicia con la operaci\u00F3n unaria sucesor (n = 0), siguiendo con las operaciones binarias de adici\u00F3n (n = 1), multiplicaci\u00F3n (n = 2), y potenciaci\u00F3n (n = 3), despu\u00E9s de lo cual la sucesi\u00F3n contin\u00FAa con m\u00E1s operaciones binarias, que se extienden m\u00E1s all\u00E1 de la potenciaci\u00F3n, mediante la asociatividad por derecha. Para las operaciones m\u00E1s all\u00E1 de la potenciaci\u00F3n, el n-\u00E9simo miembro de esta sucesi\u00F3n es nombrado por Rub\u00E9n Goodstein despu\u00E9s del prefijo griego de n con el sufijo -ci\u00F3n (como tetraci\u00F3n (n = 4), pentaci\u00F3n (n = 5), hexaci\u00F3n (n = 6), etc.)\u200B y puede ser escrito mediante el uso de n \u2212 2 flechas en la notaci\u00F3n flecha de Knuth.Cada hiperoperaci\u00F3n puede ser entendida de forma recursiva en t\u00E9rminos de la anterior por: (m \u2265 0) Esto tambi\u00E9n puede ser definido de acuerdo a la regla de recursividad con parte de la definici\u00F3n, como en la versi\u00F3n flecha hacia arriba de Knuth de la funci\u00F3n de Ackermann: (m \u2265 -1) Esta puede ser usada f\u00E1cilmente para mostrar n\u00FAmeros mucho m\u00E1s grandes que las que la notaci\u00F3n cient\u00EDfica puede, tales como el n\u00FAmero de Skewes y el googolplex, pero hay algunos n\u00FAmeros que incluso ellos no pueden mostrar f\u00E1cilmente, tales como el n\u00FAmero de Graham y \u00C1RBOL(3). Esta repetici\u00F3n de la regla es com\u00FAn a muchas variantes de hiperoperaciones (ver )."@es . "Hiperoperaci\u00F3"@ca . . "\u0413\u0456\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u2014 \u043D\u0435\u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u0430 \u043F\u043E\u0441\u043B\u0456\u0434\u043E\u0432\u043D\u0456\u0441\u0442\u044C \u0430\u0440\u0438\u0444\u043C\u0435\u0442\u0438\u0447\u043D\u0438\u0445 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0439, \u0449\u043E \u043F\u043E\u0447\u0438\u043D\u0430\u0454\u0442\u044C\u0441\u044F \u0437 \u0443\u043D\u0430\u0440\u043D\u043E\u0457 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0457 \u043D\u0430\u0441\u0442\u0443\u043F\u043D\u0438\u0439 \u0435\u043B\u0435\u043C\u0435\u043D\u0442, \u0430 \u0434\u0430\u043B\u0456 \u0431\u0456\u043D\u0430\u0440\u043D\u0456 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0457 \u0434\u043E\u0434\u0430\u0432\u0430\u043D\u043D\u044F, \u043C\u043D\u043E\u0436\u0435\u043D\u043D\u044F, \u043F\u0456\u0434\u043D\u0435\u0441\u0435\u043D\u043D\u044F \u0434\u043E \u0441\u0442\u0435\u043F\u0435\u043D\u044F, \u0442\u0435\u0442\u0440\u0430\u0446\u0456\u044F, \u043F\u0435\u043D\u0442\u0430\u0446\u0456\u044F, \u2026 \u0411\u0443\u0432 \u0437\u0430\u043F\u0440\u043E\u043F\u043E\u043D\u043E\u0432\u0430\u043D\u0438\u0439 \u0430\u043D\u0433\u043B\u0456\u0439\u0441\u044C\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C ."@uk . . . . . . . "Em matem\u00E1tica, a seq\u00FCencia de hiperopera\u00E7\u00F5es \u00E9 uma seq\u00FCencia de opera\u00E7\u00F5es bin\u00E1rias que iniciam com a adi\u00E7\u00E3o, multiplica\u00E7\u00E3o e exponencia\u00E7\u00E3o, chamadas hiperopera\u00E7\u00F5es em geral. O n-\u00E9simo membro desta seq\u00FCencia foi nomeado por seguindo o prefixo grego de n acrescido do sufixo -\u00E7\u00E3o (como em tetra\u00E7\u00E3o, ) e pode ser escrito usando setas na Nota\u00E7\u00E3o de Knuth. Cada hiperopera\u00E7\u00E3o \u00E9 definida recursivamente em termos da anterior, como \u00E9 o caso com a nota\u00E7\u00E3o de seta para cima de Knuth. A parte da defini\u00E7\u00E3o que faz isso \u00E9 a regra recursiva da fun\u00E7\u00E3o de Ackermann:"@pt . "Der Hyper-Operator ist eine Familie von mathematischen Operatoren. Konkret ist der erste Operator die einstellige Verkn\u00FCpfung, dann kommt die Addition, die Multiplikation, die Potenzierung usw. Der Hyper-Operator dient zur kurzen Darstellung gro\u00DFer Zahlen wie Potenzt\u00FCrmen. Es gibt verschiedene Schreibweisen:"@de . . . . "Hiperopera\u00E7\u00E3o"@pt . . . . . . "Der Hyper-Operator ist eine Familie von mathematischen Operatoren. Konkret ist der erste Operator die einstellige Verkn\u00FCpfung, dann kommt die Addition, die Multiplikation, die Potenzierung usw. Der Hyper-Operator dient zur kurzen Darstellung gro\u00DFer Zahlen wie Potenzt\u00FCrmen. Es gibt verschiedene Schreibweisen:"@de . "\u30CF\u30A4\u30D1\u30FC\u6F14\u7B97\u5B50\uFF08\u30CF\u30A4\u30D1\u30FC\u3048\u3093\u3056\u3093\u3057\u3001hyper operator\uFF09\u306F\u3001\u52A0\u7B97\u3001\u4E57\u7B97\u3001\u51AA\u4E57\u3092\u4E00\u822C\u5316\u3057\u305F\u6F14\u7B97\u306E\u305F\u3081\u306E\u6F14\u7B97\u5B50\u3067\u3042\u308B\u3002"@ja . . . . . . "\u8D85\u904B\u7B97\u5E8F\u5217\u662F\u6570\u5B66\u4E2D\u4E00\u79CD\u4E8C\u5143\u8FD0\u7B97\u7684\u5E8F\u5217\uFF0C\u524D\u4E09\u9879\u5206\u522B\u4E3A\u52A0\u6CD5\u3001\u4E58\u6CD5\u3001\u5E42\uFF0C\u4E00\u822C\u4F86\u8AAA\uFF0C\u9664\u4E86\u5E8F\u5217\u4E2D\u7B2C\u4E00\u9805\u7684\u52A0\u6CD5\u904B\u7B97\u4E4B\u5916\uFF0C\u5E8F\u5217\u4E2D\u6BCF\u4E00\u9805\u7684\u904B\u7B97\u90FD\u662F\u91CD\u8907\u7684\u524D\u4E00\u9805\u7684\u904B\u7B97\uFF08\u4F8B\u5982\u4E58\u6CD5\u662F\u91CD\u8907\u7684\u52A0\u6CD5\uFF1A\uFF0C\u51AA\u662F\u91CD\u8907\u7684\u4E58\u6CD5\uFF1A\uFF09\u3002\u8FD9\u4E9B\u8FD0\u7B97\u901A\u79F0\u4E3A\u8D85\u8FD0\u7B97\uFF08\u6216\u7A31\u70BAhyper\u904B\u7B97\u7B26\uFF09\u3002\u5E8F\u5217\u4E2D\u7684\u7B2Cn\u9879\u79F0\u4E3A\u8D85-n\u8FD0\u7B97\u6216\u7B2Cn\u7D1A\u7684\u8D85\u904B\u7B97\uFF0C\u5176\u7B26\u865F\u70BA[n]\u3002\u82F1\u6587\u5247\u7531\u547D\u540D\uFF0C\u7576n\u22654\u6642\uFF0C\u7531n\u7684\u5E0C\u814A\u8BED\u524D\u7F00\u52A0\u4E0A\u540E\u7F00-ation\u7EC4\u6210\uFF08\u4F8B\u5982\u8D85-4\u8FD0\u7B97\u79F0\u4E3Atetration\uFF0C\u8D85-5\u8FD0\u7B97\u79F0\u4E3Apentation\uFF09\u3002\u7576n\u22653 \u6642\uFF0C\u4F7F\u7528\u9AD8\u5FB7\u7EB3\u7BAD\u53F7\u8868\u793A\u6CD5\u53EF\u5C06\u8D85-n\u8FD0\u7B97\u7684\u7B26\u865F\u8868\u793A\u4E3A(n-2)\u4E2A\u7BAD\u5934\u3002 \u8D85\u8FD0\u7B97\u53EF\u901A\u8FC7\u9012\u5F52\u8FDB\u884C\u5B9A\u4E49\uFF0C\u5C0D\u65BC\u6240\u6709\u6B63\u6574\u6578a\uFF0C\u6B63\u6574\u6578b\u548C\u6B63\u6574\u6578n\uFF1A \u9664\u8FD9\u4E00\u6700\u5E38\u89C1\u7684\u5B9A\u4E49\u4E4B\u5916\uFF0C\u8D85\u8FD0\u7B97\u8FD8\u6709\u5176\u4ED6\u7684\u53D8\u4F53\u3002\uFF08\uFF09"@zh . "\u0413\u0456\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u2014 \u043D\u0435\u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u0430 \u043F\u043E\u0441\u043B\u0456\u0434\u043E\u0432\u043D\u0456\u0441\u0442\u044C \u0430\u0440\u0438\u0444\u043C\u0435\u0442\u0438\u0447\u043D\u0438\u0445 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0439, \u0449\u043E \u043F\u043E\u0447\u0438\u043D\u0430\u0454\u0442\u044C\u0441\u044F \u0437 \u0443\u043D\u0430\u0440\u043D\u043E\u0457 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0457 \u043D\u0430\u0441\u0442\u0443\u043F\u043D\u0438\u0439 \u0435\u043B\u0435\u043C\u0435\u043D\u0442, \u0430 \u0434\u0430\u043B\u0456 \u0431\u0456\u043D\u0430\u0440\u043D\u0456 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0457 \u0434\u043E\u0434\u0430\u0432\u0430\u043D\u043D\u044F, \u043C\u043D\u043E\u0436\u0435\u043D\u043D\u044F, \u043F\u0456\u0434\u043D\u0435\u0441\u0435\u043D\u043D\u044F \u0434\u043E \u0441\u0442\u0435\u043F\u0435\u043D\u044F, \u0442\u0435\u0442\u0440\u0430\u0446\u0456\u044F, \u043F\u0435\u043D\u0442\u0430\u0446\u0456\u044F, \u2026 \u0411\u0443\u0432 \u0437\u0430\u043F\u0440\u043E\u043F\u043E\u043D\u043E\u0432\u0430\u043D\u0438\u0439 \u0430\u043D\u0433\u043B\u0456\u0439\u0441\u044C\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C ."@uk . . . . "\uD558\uC774\uD37C \uC5F0\uC0B0"@ko . . . . "Hyper-Operator"@de . . . . . . . "En matem\u00E1ticas, la sucesi\u00F3n de hiperoperaciones\u200Bes una sucesi\u00F3n infinita de operaciones aritm\u00E9ticas (llamadas hiperoperaciones)\u200B\u200B\u200B que se inicia con la operaci\u00F3n unaria sucesor (n = 0), siguiendo con las operaciones binarias de adici\u00F3n (n = 1), multiplicaci\u00F3n (n = 2), y potenciaci\u00F3n (n = 3), despu\u00E9s de lo cual la sucesi\u00F3n contin\u00FAa con m\u00E1s operaciones binarias, que se extienden m\u00E1s all\u00E1 de la potenciaci\u00F3n, mediante la asociatividad por derecha. Para las operaciones m\u00E1s all\u00E1 de la potenciaci\u00F3n, el n-\u00E9simo miembro de esta sucesi\u00F3n es nombrado por Rub\u00E9n Goodstein despu\u00E9s del prefijo griego de n con el sufijo -ci\u00F3n (como tetraci\u00F3n (n = 4), pentaci\u00F3n (n = 5), hexaci\u00F3n (n = 6), etc.)\u200B y puede ser escrito mediante el uso de n \u2212 2 flechas en la notaci\u00F3n flecha de Knuth.Cada hiperoperaci\u00F3n puede ser"@es . . . . . . . . . . . "Hyperoperation"@en . . . "In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3). It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function: This recursion rule is common to many variants of hyperoperations."@en . . . . . . . "\u30CF\u30A4\u30D1\u30FC\u6F14\u7B97\u5B50"@ja . "\u0413\u0438\u043F\u0435\u0440\u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440"@ru . . . . "33731923"^^ . "En matem\u00E0tiques, la successi\u00F3 d'hiperoperaci\u00F3 \u00E9s una successi\u00F3 infinita d'operacions aritm\u00E8tiques (anomenades hiperoperacions en aquest context) que comen\u00E7a amb una operaci\u00F3 un\u00E0ria (la amb n = 0). La successi\u00F3 continua amb les operacions bin\u00E0ries d'addici\u00F3 (n = 1), multiplicaci\u00F3 (n = 2) i potenciaci\u00F3 (n = 3). Despr\u00E9s d'aix\u00F2, la successi\u00F3 continua amb altres operacions bin\u00E0ries que s'estenen m\u00E9s enll\u00E0 de la potenciaci\u00F3, utilitzant l'. Per a les operacions m\u00E9s enll\u00E0 de la potenciaci\u00F3, el n-\u00E8ssim membre d'aquesta successi\u00F3 rep el nom creat per Reuben Goodstein, a partir del prefix grec de n i afegint el sufix -ci\u00F3 (com per exemple, (n = 4), pentaci\u00F3 (n = 5), (n = 6) , etc.). Es pot entendre recursivament cada hiperoperaci\u00F3 en termes de l'anterior per: i es pot escriure utilitzant n-2 fletxes de la . Tamb\u00E9 es pot definir segons la regla de recursivitat de la definici\u00F3 en la versi\u00F3 de fletxa de Knuth de la funci\u00F3 d'Ackermann: o Es pot utilitzar per mostrar f\u00E0cilment nombres molt m\u00E9s grans dels que es poden representar amb una notaci\u00F3 cient\u00EDfica (com ara el nombre de Skewes i el googolplex). Per exemple, \u00E9s molt m\u00E9s gran que el nombre de Skewes i el googolplex. Per\u00F2 hi ha alguns n\u00FAmeros que fins i tot no es poden mostrar f\u00E0cilment, com ara el i . Aquesta regla de recursi\u00F3 \u00E9s comuna a moltes variants d'hiperoperacions."@ca . . . .