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rdfs:label
Modula aritmetiko Aritmetica modulare Modulární aritmetika Aritmetika modular Модульная арифметика Modulär aritmetik 合同算術 Modular arithmetic Modulair rekenen Arytmetyka modularna 모듈러 산술 Aritmética modular Aritmètica modular Arithmétique modulaire Aritmética modular 模算數 Aritmetika modular Αριθμητική υπολοίπων Модульна арифметика حسابيات معيارية
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Modulär aritmetik, moduloräkning eller kongruensräkning är ett område inom aritmetiken, där man räknar med ett begränsat antal tal. Andra tal räknas som jämlika ("kongruenta") med ett av dessa, nämligen med det av talen som blir rest vid division med antalet tal man räknar med. Den modulära aritmetiken används bland annat inom kryptologin. har samma rest vid division med n . في الرياضيات وبالتحديد في مجال النظرية الجبرية للأعداد، الحسابيات النمطية (بالإنجليزية: modular arithmetics)‏ هي مجموعة من الطرق التي تتيح حل بعض المسائل الخاصة بالأعداد الصحيحة و من ضمنها الطبيعية. وهي ترتكز على دراسة الباقي الحاصل من القسمة الإقليدية. ترتكز الحسابيات النمطية أساسا على النظر إلى باقي قسمة الأعداد الطبيعية على عدد طبيعي معين ثابت ما، بدلا من النظر إلى هذه الأعداد ذاتها. يظهر هذا جليا في مثال حسابيات المنبه، الذي يوافق حالة n=12 : العقرب الصغير يوجد في نفس الموضع في لحظتين تفصل بينهما اثنتا عشرة ساعة، وبهذا تصير الساعة 1 كالساعة 13. En matemática, la aritmética modular es un sistema aritmético para clases de equivalencia de números enteros llamadas clases de congruencia. La aritmética modular fue introducida en 1801 por Carl Friedrich Gauss en su libro Disquisitiones Arithmeticae.​ Algunas veces se le llama, sugerentemente, aritmética del reloj, ya que los números «dan la vuelta» tras alcanzar cierto valor llamado módulo.​ 수론에서 모듈러 산술(영어: modular arithmetic) 또는 합동 산술(合同算術)은 정수의 합과 곱을 어떤 주어진 수의 나머지에 대하여 정의하는 방법이다. 정수환의 몫환 의 환 구조로 생각할 수 있다. L'aritmetica modulare (a volte detta aritmetica dell'orologio poiché su questo principio si basa il calcolo delle ore a cicli di 12 o 24) rappresenta un importante ramo della matematica. Trova applicazioni nella crittografia, nella teoria dei numeri (in particolare nella ricerca dei numeri primi) ed è alla base di molte delle più comuni operazioni aritmetiche e algebriche. L'aritmetica modulare e la notazione usuale delle congruenze vennero formalmente introdotte da Carl Friedrich Gauss nel suo trattato Disquisitiones Arithmeticae, pubblicato nel 1801. Στα μαθηματικά, η αριθμητική υπολοίπων είναι ένα σύστημα αριθμητικής για ακέραιους αριθμούς, όπου οι αριθμοί «αναδιπλώνονται» έως την επίτευξη μιας ορισμένης τιμής — συντελεστή αναδίπλωσης (αναδιπλωτής: modulo πληθυντικός moduli). Η σύγχρονη προσέγγιση για την αριθμητική υπολοίπων αναπτύχθηκε από τον Καρλ Φρίντριχ Γκάους, στο βιβλίο του , που δημοσιεύθηκε το 1801. Em matemática, aritmética modular (chamada também de aritmética do relógio) é um sistema de aritmética para inteiros, onde os números "retrocedem" quando atingem um certo valor, o módulo. O matemático suíço Euler foi o pioneiro na abordagem de congruência por volta de 1750, quando ele explicitamente introduziu a ideia de congruência módulo um número natural N. A abordagem moderna da aritmética modular foi desenvolvida por Carl Friedrich Gauss em seu livro Disquisitiones Arithmeticae, publicado em 1801. Модульна арифметика — це система арифметики цілих чисел,в якій числа «обертаються навколо» деякого значення — модуля. Найбільш відомий приклад модульної арифметики — це запис часу в 12-годинному форматі, в якому день ділиться на два 12-годинних періоди. Якщо зараз 9:00, то через 4 години на годиннику буде 1:00. Якщо просто додати, то 9 + 4 = 13, але це неправильна відповідь, тому що на годиннику по досягненні стрілки 12-ї години, замість 12:00 ми отримуємо 00:00. Тому правильна відповідь, що на годиннику буде 1:00. У сучасному вигляді модульна арифметика була розвинута Гаусом в (1801). Arytmetyka modularna, arytmetyka reszt – system liczb całkowitych, w którym liczby „zawijają się” po osiągnięciu pewnej wartości nazywanej modułem, często określanej terminem modulo (skracane mod). Pierwszy pełny wykład arytmetyki reszt przedstawił Carl Friedrich Gauss w Disquisitiones Arithmeticae („Badania arytmetyczne”, 1801). 模算數(英語:Modular arithmetic)是一個整数的算术系統,其中數字超過一定值後(稱為模)後會「捲回」到較小的數值,模算數最早是出現在卡爾·弗里德里希·高斯在1801年出版的《算术研究》一書中。 模算數常見的應用是在十二小時制,將一天分為二個以十二小時計算的單位。假設現在七點,八小時後會是三點。用一般的算術加法,會得到7 + 8 = 15,但在十二小時制中,超過十二小時會歸零,不存在「十五點」。類似的情形,若時鐘目前是十二時,二十一小時後會是九點,而不是三十三點。小時數超過十二後會再回到一,為模12的模算數系統。依照上述的定義,12和12本身同餘,也和0同餘,因此12:00的時間也可以稱為是0:00,因為模12時,12和0同餘。 Na rozdíl od běžné aritmetiky je modulární aritmetika definována na nějaké konečné množině ℤn. Tato množina vznikne ze ℤ tak, že jsou všechna čísla se stejným zbytkem po dělení číslem (zbytková třída) brána jako kongruentní a ztotožněna s jediným reprezentantem. Taková množina se pak nazývá množina zbytkových tříd. Modulair rekenen, of rekenen modulo een getal, is een vorm van geheeltallig rekenen met een getal dat als bovengrens fungeert, de modulus. Een typisch voorbeeld is de klok waarop modulo 12, of modulo 24, gerekend wordt. Als het 6 uur is, dan staat de klok 8 uur later niet op 14, maar op 14 − 12 = 2 uur. De verzameling getallen waarmee modulo gerekend wordt, wordt aangeduid als , naar het symbool dat de verzameling gehele getallen aanduidt. En matemàtiques, i més concretament en teoria de nombres algebraics, l'aritmètica modular és un conjunt de mètodes que permeten la resolució de problemes sobre els nombres enters. Aquests mètodes sorgeixen de l'estudi del residu obtingut per una divisió. La idea de base de l'aritmètica modular és de treballar no sobre els nombres mateixos, sinó sobre els residus de la seva divisió per alguna cosa. Quan es fa, per exemple, la prova del nou, s'efectua una operació d'aritmètica modular sense saber-ho: el divisor és el valor 9. En mathématiques et plus précisément en théorie algébrique des nombres, l’arithmétique modulaire est un ensemble de méthodes permettant la résolution de problèmes sur les nombres entiers. Ces méthodes dérivent de l’étude du reste obtenu par une division euclidienne. L'idée de base de l'arithmétique modulaire est de travailler non sur les nombres eux-mêmes, mais sur les restes de leur division par quelque chose. Quand on fait par exemple une preuve par neuf à l'école primaire, on effectue un peu d'arithmétique modulaire sans le savoir : le diviseur est alors le nombre 9. Dalam matematika dan khususnya pada teori bilangan aljabar, aritmetika modular adalah metode aritmetika untuk menyelesaikan permasalahan mengenai bilangan bulat. Ide dasar dari aritmetika modular adalah bekerja dengan sisa hasil pembagian bilangan, bukan dengan bilangan itu sendiri. Salah satu contoh dari aritmetika modular ada pada sistem 12-jam, di mana hari dibagi menjadi dua periode 12-jam. Jika sekarang jarum jam menunjukan pukul 7:00, maka 8 jam kemudian akan menunjukan pukul 3:00. Penambahan sederhana akan menghasilkan 7 + 8 = 15. Namun karena jam "berulang" setiap 12 jam, angka 15 "sama dengan" angka 3; ini adalah contoh aritmetika modulo 12. Modula aritmetiko estas sistemo de aritmetiko por entjeroj, kie nombroj "turniĝas reen" post kiam ili atingas certan valoron — la modulon. Modulan aritmetikon prezentis Carl Friedrich Gauss en lia libro Disquisitiones Arithmeticae (publikigita en 1801). Ekzemplo por modula aritmetiko estas kutima horloĝo: la aritmetiko de horoj sur la horloĝo. Se la tempo estas 7 horoj do post 8 horoj estas 15 horoj (kiel en kutima aldono). Se la tempo estas 7 horoj do post 19 horoj estas (7+19)=26 horoj (laŭ en kutima aldono), sed horloĝo uzas modulon 24, do estas 26 mod 24=2 horoj (de la sekva diurno). Aritmetika modularra, matematikaren esparruan, zenbaki osoko kongruentzia klaseetarako sistema aritmetikoa da. 1801ean Carl Friendrich Gaussek garatu zuen eta Disquisitiones Arithmeticae izeneko liburuan argitara eman zuen. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. 数学、特に初等代数的整数論における合同算術(ごうどうさんじゅつ、英: modular arithmetic; モジュラ計算)は、(剰余を持つ除法の意味で))自然数あるいは整数をある特定の自然数で割ったときの剰余に注目して、自然数あるいは整数に関する問題を解決する一連の方法の総称である。合同算術の起源は、一般にはガウスが著作『Disquisitiones Arithmeticae』を出版する1801年にまで遡れるものとされる。ガウスによる合同を用いたこの新しい手法は、有名な平方剰余の相互法則を明らかにし、より抽象的な観点からウィルソンの定理などの定理の記述の簡素化に一役を買った。ガウスの研究は自然数を扱う整数論のみならず、代数学や幾何学といった数学のほかの主要な分野にまで影響を与えるものであった。 この手法の基本は、「数それ自体」ではなくそれを別な数で割った(商がいくらになるかということは無視して)「剰余だけ」を考えるということにある。こういった考え方は何か特殊で高尚なものというようなものではなく、実際に日常生活においても時刻や角度といったものの計算や単位の換算などで、ちょっとした合同算術が特別な知識無くあるいは無意識に行われているのである。
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Aritmetika modularra, matematikaren esparruan, zenbaki osoko kongruentzia klaseetarako sistema aritmetikoa da. 1801ean Carl Friendrich Gaussek garatu zuen eta Disquisitiones Arithmeticae izeneko liburuan argitara eman zuen. Erlojuaren aritmetika moduan ere ezagutzen da, orduak 12:00-etara iristean berriro 00:00-tik kontatzen hasten baitira; 12 orduko erlojua aritmetika modularraren adibide argia da. Erloju analogikoetan, eguna 12 orduko bi zatitan banatuta dago; goizeko 7:00-ak badira eta 8 ordu pasa badira, 15:00-ak izan beharko lirateke, baina 0tik 12ra kontatzen duenez, 3:00-ak direla esaten dugu. Modu berean, 12:00-ak badira eta 21 ordu pasa badira, 9:00-ak direla esaten dugu 33:00-ak direla esan beharrean, erlojuaren orratza 12:00-ra iristean berriro ere 0tik kontatzen hasten delako. Hori horrela izanik, erloju analogikoetan 12 moduluko aritmetika egiten dela esango dugu. Modulär aritmetik, moduloräkning eller kongruensräkning är ett område inom aritmetiken, där man räknar med ett begränsat antal tal. Andra tal räknas som jämlika ("kongruenta") med ett av dessa, nämligen med det av talen som blir rest vid division med antalet tal man räknar med. Den modulära aritmetiken används bland annat inom kryptologin. I den modulära matematiken analyseras och används kongruensrelationen. Två tal a och b sägs vara kongruenta modulo n om n delar differensen mellan a och b, vilket för alla nollskilda n är ekvivalent med att de har samma principala rest vid division med n.Detta betecknas , och ibland även . Talen a och b är kongruenta modulo 0 om och endast om a = b. Detta triviala slags kongruens bortser man ofta från, och förutsätter då i stället att n är nollskilt, alltså inte är lika med noll. Under det extraantagandet kan man formellt beskriva definitionen och dess grundläggande egenskaper så här: har samma rest vid division med n . Модульна арифметика — це система арифметики цілих чисел,в якій числа «обертаються навколо» деякого значення — модуля. Найбільш відомий приклад модульної арифметики — це запис часу в 12-годинному форматі, в якому день ділиться на два 12-годинних періоди. Якщо зараз 9:00, то через 4 години на годиннику буде 1:00. Якщо просто додати, то 9 + 4 = 13, але це неправильна відповідь, тому що на годиннику по досягненні стрілки 12-ї години, замість 12:00 ми отримуємо 00:00. Тому правильна відповідь, що на годиннику буде 1:00. Аналогічним чином, якщо годинник починає відлік о 12:00 (опівдні) і пройде 21 година, то час буде 9:00 наступного дня, а не 33:00. Оскільки годинник починає новий відлік часу після досягнення 12, то це буде арифметика за модулем 12. 12 відповідає не тільки значенню 12, але також і 0, так що час, який називається «12:00», також може бути названий «0:00», оскільки 0 ≡ 12 mod 12. Ще один підхід до модульної арифметики пов'язаний з остачами від ділення цілих чисел на певне задане натуральне число. Фактично в ній розглядаються класи еквівалентності певного натурального числа. У сучасному вигляді модульна арифметика була розвинута Гаусом в (1801). En matemática, la aritmética modular es un sistema aritmético para clases de equivalencia de números enteros llamadas clases de congruencia. La aritmética modular fue introducida en 1801 por Carl Friedrich Gauss en su libro Disquisitiones Arithmeticae.​ Algunas veces se le llama, sugerentemente, aritmética del reloj, ya que los números «dan la vuelta» tras alcanzar cierto valor llamado módulo.​ In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Modulair rekenen, of rekenen modulo een getal, is een vorm van geheeltallig rekenen met een getal dat als bovengrens fungeert, de modulus. Een typisch voorbeeld is de klok waarop modulo 12, of modulo 24, gerekend wordt. Als het 6 uur is, dan staat de klok 8 uur later niet op 14, maar op 14 − 12 = 2 uur. Bij modulair rekenen met modulus of rekenen modulo wordt gerekend met de getallen , waarna niet volgt maar weer opnieuw met 0 begonnen wordt. De getallen 0 tot en met staan als het ware in een kring. Het resultaat van een berekening modulo is de rest van het resultaat na geheeltallige deling door de modulus . De normale definities van optelling en vermenigvuldiging worden gebruikt, maar als het resultaat groter is dan of gelijk aan wordt net zo vaak afgetrokken tot het resultaat weer kleiner is dan . Getallen die modulo gelijk zijn, die dus een veelvoud van van elkaar verschillen, noemt men congruent modulo , genoteerd met het symbool . Bijvoorbeeld De verzameling getallen waarmee modulo gerekend wordt, wordt aangeduid als , naar het symbool dat de verzameling gehele getallen aanduidt. Em matemática, aritmética modular (chamada também de aritmética do relógio) é um sistema de aritmética para inteiros, onde os números "retrocedem" quando atingem um certo valor, o módulo. O matemático suíço Euler foi o pioneiro na abordagem de congruência por volta de 1750, quando ele explicitamente introduziu a ideia de congruência módulo um número natural N. A abordagem moderna da aritmética modular foi desenvolvida por Carl Friedrich Gauss em seu livro Disquisitiones Arithmeticae, publicado em 1801. Um uso familiar da aritmética modular é no relógio de ponteiro, no qual o dia é divido em dois períodos de 12 horas cada. Se são 7:00 agora, então 8 horas depois serão 3:00. A adição usual sugere que o tempo futuro deveria ser , mas o relógio "retrocede" a cada 12 horas. Da mesma forma, se o relógio começa em 12:00(meio-dia) e 21 horas passam, então a hora será 9:00 do dia seguinte, em vez de 33:00. Como o número de horas começa de novo depois que atinge 12, esta aritmética é chamada aritmética módulo 12. Em termos da definição abaixo, é congruente com módulo , então "15:00" em um é exibido "3:00 "em um relógio de 12 horas. في الرياضيات وبالتحديد في مجال النظرية الجبرية للأعداد، الحسابيات النمطية (بالإنجليزية: modular arithmetics)‏ هي مجموعة من الطرق التي تتيح حل بعض المسائل الخاصة بالأعداد الصحيحة و من ضمنها الطبيعية. وهي ترتكز على دراسة الباقي الحاصل من القسمة الإقليدية. ترتكز الحسابيات النمطية أساسا على النظر إلى باقي قسمة الأعداد الطبيعية على عدد طبيعي معين ثابت ما، بدلا من النظر إلى هذه الأعداد ذاتها. يظهر هذا جليا في مثال حسابيات المنبه، الذي يوافق حالة n=12 : العقرب الصغير يوجد في نفس الموضع في لحظتين تفصل بينهما اثنتا عشرة ساعة، وبهذا تصير الساعة 1 كالساعة 13. En mathématiques et plus précisément en théorie algébrique des nombres, l’arithmétique modulaire est un ensemble de méthodes permettant la résolution de problèmes sur les nombres entiers. Ces méthodes dérivent de l’étude du reste obtenu par une division euclidienne. L'idée de base de l'arithmétique modulaire est de travailler non sur les nombres eux-mêmes, mais sur les restes de leur division par quelque chose. Quand on fait par exemple une preuve par neuf à l'école primaire, on effectue un peu d'arithmétique modulaire sans le savoir : le diviseur est alors le nombre 9. Si ses origines remontent à l’Antiquité, les historiens associent généralement sa naissance à l’année 1801, date de la publication du livre Disquisitiones arithmeticae de Carl Friedrich Gauss. Sa nouvelle approche permet d’élucider de célèbres conjectures et simplifie les démonstrations d’importants résultats par une plus grande abstraction. Si le domaine naturel de ces méthodes est la théorie des nombres, les conséquences des idées de Gauss se retrouvent dans d’autres champs des mathématiques, comme l’algèbre ou la géométrie. Le XXe siècle modifie le statut de l’arithmétique modulaire. L'arithmétique de base des ordinateurs, celle qui travaille sur des mots mémoire de taille fixe, est nécessairement une arithmétique modulaire. Le développement de nombreuses applications industrielles impose la mise au point d’algorithmes pour l'arithmétique modulaire. Ils résolvent essentiellement des questions soulevées par le développement de l'informatique. L’article « Congruence sur les entiers » propose une introduction plus mathématique ; « Anneau ℤ/nℤ » traite le même sujet de manière moins didactique et plus exhaustive. 模算數(英語:Modular arithmetic)是一個整数的算术系統,其中數字超過一定值後(稱為模)後會「捲回」到較小的數值,模算數最早是出現在卡爾·弗里德里希·高斯在1801年出版的《算术研究》一書中。 模算數常見的應用是在十二小時制,將一天分為二個以十二小時計算的單位。假設現在七點,八小時後會是三點。用一般的算術加法,會得到7 + 8 = 15,但在十二小時制中,超過十二小時會歸零,不存在「十五點」。類似的情形,若時鐘目前是十二時,二十一小時後會是九點,而不是三十三點。小時數超過十二後會再回到一,為模12的模算數系統。依照上述的定義,12和12本身同餘,也和0同餘,因此12:00的時間也可以稱為是0:00,因為模12時,12和0同餘。 数学、特に初等代数的整数論における合同算術(ごうどうさんじゅつ、英: modular arithmetic; モジュラ計算)は、(剰余を持つ除法の意味で))自然数あるいは整数をある特定の自然数で割ったときの剰余に注目して、自然数あるいは整数に関する問題を解決する一連の方法の総称である。合同算術の起源は、一般にはガウスが著作『Disquisitiones Arithmeticae』を出版する1801年にまで遡れるものとされる。ガウスによる合同を用いたこの新しい手法は、有名な平方剰余の相互法則を明らかにし、より抽象的な観点からウィルソンの定理などの定理の記述の簡素化に一役を買った。ガウスの研究は自然数を扱う整数論のみならず、代数学や幾何学といった数学のほかの主要な分野にまで影響を与えるものであった。 この手法の基本は、「数それ自体」ではなくそれを別な数で割った(商がいくらになるかということは無視して)「剰余だけ」を考えるということにある。こういった考え方は何か特殊で高尚なものというようなものではなく、実際に日常生活においても時刻や角度といったものの計算や単位の換算などで、ちょっとした合同算術が特別な知識無くあるいは無意識に行われているのである。 20世紀には、合同算術にまつわる状況は大きく様変わりをしている。計算機やウェブの普及に伴って情報セキュリティの観点からの暗号化アルゴリズムの開発や取り扱いといったような場面で古典的な合同算術に関する理論の工業的・商業的応用が頻繁に見られるようになった。 Στα μαθηματικά, η αριθμητική υπολοίπων είναι ένα σύστημα αριθμητικής για ακέραιους αριθμούς, όπου οι αριθμοί «αναδιπλώνονται» έως την επίτευξη μιας ορισμένης τιμής — συντελεστή αναδίπλωσης (αναδιπλωτής: modulo πληθυντικός moduli). Η σύγχρονη προσέγγιση για την αριθμητική υπολοίπων αναπτύχθηκε από τον Καρλ Φρίντριχ Γκάους, στο βιβλίο του , που δημοσιεύθηκε το 1801. Μία γνωστή χρήση της αριθμητικής υπολοίπων είναι το , κατά το οποίο η μέρα χωρίζεται σε δύο 12-ωρες περιόδους. Αν τώρα η ώρα είναι 7:00, τότε σε 8 ώρες αργότερα θα είναι 3:00. Επιπλέον, το συνηθισμένο θα ήταν ο μεταγενέστερος χρόνος να πρέπει να είναι 7 + 8 = 15, αλλά αυτό δεν είναι ορθό, διότι η ώρα του ρολογιού «αναδιπλώνεται» κάθε 12 ώρες: σε ένα 12-ωρο, δεν υπάρχει «η ώρα 15:00». Ομοίως, αν το ρολόι ξεκινά στις 12:00 (το μεσημέρι) και έχουν παρέλθει 21 ώρες, τότε ο χρόνος θα είναι 9:00 την επόμενη μέρα, αντί για 33:00. Επειδή η ώρα ξεκινά ξανά από την αρχή αφού φτάσει στις 12:00, αυτό είναι αριθμητική modulo 12. Σύμφωνα με τον ορισμό παρακάτω, το 12 είναι όχι μόνο με το 12, αλλά και με το 0, έτσι ώστε ο χρόνος που ονομάζεται "12:00" θα μπορούσε επίσης να ονομάζεται "0:00", αφού το 12 είναι ισότιμο με το 0 modulo 12. Arytmetyka modularna, arytmetyka reszt – system liczb całkowitych, w którym liczby „zawijają się” po osiągnięciu pewnej wartości nazywanej modułem, często określanej terminem modulo (skracane mod). Pierwszy pełny wykład arytmetyki reszt przedstawił Carl Friedrich Gauss w Disquisitiones Arithmeticae („Badania arytmetyczne”, 1801). Arytmetyka modularna pojawia się wszędzie tam, gdzie występuje powtarzalność i cykliczność; dotyczy ona samego mierzenia czasu i jako taka jest podstawą działania kalendarza (zob. ). Ponadto korzysta się z niej w teorii liczb, teorii grup, kryptografii, informatyce, przy tworzeniu sum kontrolnych, a nawet przy tworzeniu wzorów. Zasada działania szyfru RSA oraz Test Millera-Rabina opierają się na własnościach mnożenia w arytmetyce modularnej liczb całkowitych. L'aritmetica modulare (a volte detta aritmetica dell'orologio poiché su questo principio si basa il calcolo delle ore a cicli di 12 o 24) rappresenta un importante ramo della matematica. Trova applicazioni nella crittografia, nella teoria dei numeri (in particolare nella ricerca dei numeri primi) ed è alla base di molte delle più comuni operazioni aritmetiche e algebriche. Si tratta di un sistema di aritmetica degli interi, in cui i numeri "si avvolgono su loro stessi" ogni volta che raggiungono i multipli di un determinato numero , detto modulo. Per capire, si pensi al funzionamento di un orologio in formato da 12 ore: trascorse quest'ultime "si ricomincia" dal numero 1 a contare le ore. Dire "sono le 3 del pomeriggio" (formato 12 ore) equivale a dire "sono le 15" (formato 24 ore). Tradotto in termini matematici, significa che . Si legge, è congruente a , modulo . L'aritmetica modulare e la notazione usuale delle congruenze vennero formalmente introdotte da Carl Friedrich Gauss nel suo trattato Disquisitiones Arithmeticae, pubblicato nel 1801. Dalam matematika dan khususnya pada teori bilangan aljabar, aritmetika modular adalah metode aritmetika untuk menyelesaikan permasalahan mengenai bilangan bulat. Ide dasar dari aritmetika modular adalah bekerja dengan sisa hasil pembagian bilangan, bukan dengan bilangan itu sendiri. Salah satu contoh dari aritmetika modular ada pada sistem 12-jam, di mana hari dibagi menjadi dua periode 12-jam. Jika sekarang jarum jam menunjukan pukul 7:00, maka 8 jam kemudian akan menunjukan pukul 3:00. Penambahan sederhana akan menghasilkan 7 + 8 = 15. Namun karena jam "berulang" setiap 12 jam, angka 15 "sama dengan" angka 3; ini adalah contoh aritmetika modulo 12. Walau sudah dipelajari sejak kuno, sejarawan umumnya mengasosiasikan kemunculan aritmetika modular dengan tahun 1801, tahun publikasi buku Disquisitiones Arithmeticae karya Carl Friedrich Gauss. Pendekatan yang ia gunakan mempermudah penjelasan konjektur-konjektur terkenal, dan menyederhanakan bukti-bukti penting. Implikasi karya Gauss juga ditemukan pada bidang selain teori bilangan, seperti aljabar dan geometri. Kegunaan aritmetika modular meningkat pada abad ke-20. Operasi aritmetika dasar pada komputer yang bekerja pada jumlah bit yang tetap, pada dasarnya adalah operasi aritmetika modular. Penerapan pada bidang industri memicu perkembangan algoritma aritmetika modular. Modula aritmetiko estas sistemo de aritmetiko por entjeroj, kie nombroj "turniĝas reen" post kiam ili atingas certan valoron — la modulon. Modulan aritmetikon prezentis Carl Friedrich Gauss en lia libro Disquisitiones Arithmeticae (publikigita en 1801). Ekzemplo por modula aritmetiko estas kutima horloĝo: la aritmetiko de horoj sur la horloĝo. Se la tempo estas 7 horoj do post 8 horoj estas 15 horoj (kiel en kutima aldono). Se la tempo estas 7 horoj do post 19 horoj estas (7+19)=26 horoj (laŭ en kutima aldono), sed horloĝo uzas modulon 24, do estas 26 mod 24=2 horoj (de la sekva diurno). 수론에서 모듈러 산술(영어: modular arithmetic) 또는 합동 산술(合同算術)은 정수의 합과 곱을 어떤 주어진 수의 나머지에 대하여 정의하는 방법이다. 정수환의 몫환 의 환 구조로 생각할 수 있다. En matemàtiques, i més concretament en teoria de nombres algebraics, l'aritmètica modular és un conjunt de mètodes que permeten la resolució de problemes sobre els nombres enters. Aquests mètodes sorgeixen de l'estudi del residu obtingut per una divisió. La idea de base de l'aritmètica modular és de treballar no sobre els nombres mateixos, sinó sobre els residus de la seva divisió per alguna cosa. Quan es fa, per exemple, la prova del nou, s'efectua una operació d'aritmètica modular sense saber-ho: el divisor és el valor 9. Tot i que els seus orígens es remunten a l'antiguitat, generalment, els historiadors associen el seu naixement l'any 1801, data de la publicació del llibre Disquisitiones arithmeticae de Carl Friedrich Gauss (1777 - 1855). El seu nou enfocament permet elucidar cèlebres conjectures i simplifica les demostracions d'importants resultats gràcies a una major abstracció. Si bé l'àmbit natural d'aquests mètodes és la teoria dels nombres, les conseqüències de les idees de Gauss es troben també en altres camps de les matemàtiques, com l'àlgebra o la geometria. El segle xx modifica l'estatut de l'aritmètica modular. D'una banda, es necessiten altres mètodes per progressar en la teoria dels nombres. D'altra banda, el desenvolupament de nombroses aplicacions industrials imposa la posada a punt d'algorismes procedents de les tècniques modulars. Resolen essencialment qüestions sorgides en la teoria de la informació, una branca considerada actualment, sobretot, com matemàtiques aplicades. Na rozdíl od běžné aritmetiky je modulární aritmetika definována na nějaké konečné množině ℤn. Tato množina vznikne ze ℤ tak, že jsou všechna čísla se stejným zbytkem po dělení číslem (zbytková třída) brána jako kongruentní a ztotožněna s jediným reprezentantem. Taková množina se pak nazývá množina zbytkových tříd.
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