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rdf:type: owl:Thing
rdfs:label: Φυσικός αριθμός 自然数 Naturliga tal Integer Bilangan asli Número natural Zenbaki oso Entier relatif Heltal 自然数 Zenbaki arrunt Nombre enter عدد صحيح Numero naturale Número natural Numero intero Entjero Celé číslo 整数 Geheel getal Natura nombro Ακέραιος αριθμός 정수 Entier naturel 자연수 Uimhir aiceanta Цілі числа Natuurlijk getal Liczby całkowite Número inteiro عدد طبيعي Número entero Bilangan bulat Slánuimhir 整数 Nombre natural Натуральное число Натуральне число Ganze Zahl Liczby naturalne Přirozené číslo Целое число Natürliche Zahl
rdfs:comment: Bilangan bulat adalah bilangan yang dapat dituliskan tanpa komponen desimal atau pecahan. Sebagai contoh, 21, 4, 0, -3, -67 dan -2048 merupakan bilangan bulat, sedangkan 9,75, 5 12, dan bukan.Himpunan bilangan bulat terdiri dari angka 0, semua bilangan bulat positif (juga disebut dengan bilangan asli), dan invers aditif-nya, semua bilangan bulat negatif . Dalam matematika, himpunan ini sering dilambangkan dengan , atau huruf tebal. Huruf kapital Z yang digunakan berasal dari kata Zahlen, yang berarti bilangan dalam bahasa Jerman. En mathématiques, un entier naturel est un nombre permettant fondamentalement de compter des objets considérés comme des unités équivalentes : un jeton, deux jetons… une carte, deux cartes, trois cartes… Un tel nombre entier peut s'écrire avec une suite finie de chiffres en notation décimale positionnelle (sans signe et sans virgule). Les définitions modernes d’entier naturel sont fondées sur : * 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; … ou * 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; … De gehele getallen zijn alle getallen in de rij …, −3, −2, −1, 0, 1, 2, 3, … die voortgezet wordt door er steeds 1 bij te tellen of er 1 af te trekken. De gehele getallen omvatten 0, de natuurlijke getallen, dus de getallen waarmee wordt geteld, en de tegengestelden daarvan, de negatieve gehele getallen. De wiskundetak die zich met de studie bezighoudt naar de eigenschappen van de gehele getallen, noemt men de getaltheorie. Um número inteiro é um número que pode ser escrito sem um componente fracional. Por exemplo, 21, 4, 0, e −2048 são números inteiros, enquanto 9.75, 52, e √2 não são. O conjunto dos números inteiros é representado pelo símbolo , cuja letra é originada da palavra alemã Zahlen ([ˈtsaːlən], "números"). Os inteiros (juntamente com a operação de adição) formam o menor grupo que contém o monoide aditivo dos números naturais. Como os números naturais, os inteiros formam um conjunto infinito contável. Liczby naturalne – liczby służące podawaniu liczności (trzy osoby, zob. liczebnik główny/kardynalny) i ustalania kolejności (trzecia osoba, zob. liczebnik porządkowy), poddane w matematyce dalszym uogólnieniom (odpowiednio: liczby kardynalne, liczby porządkowe). Badaniem własności liczb naturalnych zajmują się arytmetyka i teoria liczb. W matematyce określenie liczby naturalne oznacza na ogół liczby całkowite dodatnie. Натура́льні чи́сла — числа, що виникають природним чином при лічбі. Це числа: 1, 2, 3, 4, … Множину натуральних чисел прийнято позначати знаком Існують два основних підходи до означення натуральних чисел: * числа, що використовуються при лічбі предметів (перший, другий, третій…) — підхід, загальноприйнятий у більшості країн світу; формалізованим різновидом цього підходу є аксіоматичне описання системи натуральних чисел за допомогою аксіом Пеано. * числа для позначення кількості предметів (один предмет, два предмети…). العدد الصحيح هو الذي يُمكن كتابته بدون استخدام الكسور أو الفواصل العشرية، وتتكون مجموعة الأعداد الصحيحة والتي تعتبر مجموعة جزئية من مجموعة الأعداد الحقيقية- من الأعداد الطبيعية (1، 2، 3.) والصفر والأعداد السالبة المقابلة للأعداد الطبيعية (-1، -2، -3..)، وعليه فمجموعة الأعداد الصحيحة تكون مجموعة غير منتهية شأنها في ذلك شأن مجموعة الأعداد الطبيعية، وعادة ما يرمز لها بالحرف اللاتيني Z. ( 다른 뜻에 대해서는 정수 (동음이의) 문서를 참고하십시오.)( 정수환은 여기로 연결됩니다. 대수적 수체의 부분환에 대해서는 대수적 정수환 문서를 참고하십시오.) ( 정수환은 여기로 연결됩니다. 대한민국의 가수에 대해서는 정수환 (1987년) 문서를 참고하십시오.) 수학에서 정수(整數, 문화어: 옹근수, integer)는 양의 정수(1, 2, 3, 4, 5, 6, 7, 8, ... , n), 음의 정수(-1, -2, -3, -4, -5, -6, -7, -8...) 및 0으로 이루어진 수의 체계이다. 또는 자연수, 자연수의 음수 및 영을 통칭하는 말이다. 수론의 가장 기본적인 연구 대상이다. 정수 전체의 집합의 기호는 이다. An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface Z or blackboard bold . Die natürlichen Zahlen sind die beim Zählen verwendeten Zahlen 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 usw. Je nach Definition kann auch die 0 (Null) zu den natürlichen Zahlen gezählt werden. Die Menge der natürlichen Zahlen bildet mit der Addition und der Multiplikation zusammen eine mathematische Struktur, die als kommutativer Halbring bezeichnet wird. 整数，在電腦應用上也稱為整型，是序列中所有的数的统称，包括负整数、零（0）与正整数。和自然數一樣，整數也是一個可數的無限集合。這個集合在数学上通常表示粗體或，源于德语单词Zahlen（意为“数”）的。在代數數論中，這些屬於有理數的一般整數會被稱為有理整數，用以和高斯整數等的概念加以區分。 Ці́лі чи́сла — в математиці елементи множини, яка утворюється замиканням натуральних чисел відносно віднімання. Таким чином, цілі числа замкнуті відносно додавання, віднімання та множення. Необхідність розгляду цілих чисел викликана неможливістю в загальному випадку відняти від одного натурального числа інше — можна віднімати тільки менше число від більшого. Введення нуля і від’ємних чисел робить віднімання такою ж повноцінною операцією, як додавання. Множина цілих чисел складається з Числа 142857; 0; -273 є цілими.Числа 5½; 9,75 не є цілими. Zenbaki osoen multzoan zenbaki arruntak biltzen dira (0,1,2,...), beren aurkakoekin batera (-0,-1,-2,...). -0 eta 0 berdintzat jotzen dira. Zenbaki osoen multzoa hizkiaz izendatu ohi da ('Zahlen' germanierazko hitzetik). Zenbaki osoak batu, kendu eta biderkatu egin daitezke: emaitza beti izango da zenbaki oso bat. x+a=b motako ekuazioen soluzioa, non a eta b zenbaki osoak diren, zenbaki osoa izango da. Zenbaki arrunten kasuan ez da esaterako gauza bera gertatzen. Zorrotzago, zenbaki osoen multzoak, batuketa eta biderketa eragiketak definitu ondoren, osatzen duela esan behar da. In matematica i numeri naturali sono quei numeri usati per contare e ordinare. Nel linguaggio comune i "numeri cardinali" sono quelli usati per contare e i "numeri ordinali" sono quelli usati per ordinare. I numeri naturali corrispondono all'insieme {0, 1, 2, 3, 4, …}. Essi vengono fatti corrispondere biunivocamente all'insieme dei numeri interi non negativi {0, +1, +2, +3, +4, …}. Talvolta vengono usati anche per indicare l'insieme dei numeri interi positivi {1, 2, 3, 4, …}. Un número entero es un elemento del conjunto numérico que contiene los números naturales; que son o ; dependiendo de cómo se definan, sus opuestos, y en la segunda definición, además el cero. Los enteros negativos, como −1 o −13 (se leen «menos uno», «menos trece», etc.), son menores que cero y también son menores que todos los enteros positivos. Para resaltar la diferencia entre positivos y negativos, se puede escribir un signo «menos» delante de los negativos: -1, -5, etc. Y si no se escribe signo al número se asume que es positivo. En mathématiques, un entier relatif, un entier rationnel ou simplement un nombre entier est un nombre qui se présente comme un entier naturel auquel on a adjoint un signe positif ou négatif indiquant sa position par rapport à 0 sur un axe orienté. Les entiers positifs (supérieurs à zéro) s'identifient aux entiers naturels : 0, 1, 2, 3… tandis que les entiers négatifs sont leurs opposés : 0, −1, −2, −3… L'entier 0 lui-même est donc le seul nombre à la fois positif et négatif. العدد الطبيعي في الرياضيات، هو كل عدد صحيح موجب، مثل 1، 2، 3... 12، 563. ويضيف بعض العلماء الصفر إلى هذه المجموعة من الأعداد. يرمز لمجموعة الأعداد الطبيعية بالحرف اللاتيني N.و هي مجموعة أعداد غير منتهية. يمثل الواحد 1 أصغر الأعداد الطبيعية التي لا تتضمن الصفر ℕ*، بينما يمثل الصفر 0 أصغر الأعداد في مجموعة الأعداد الطبيعية التي تتضمن الصفر ℕ0، ويتم إنشاؤها بواسطة علاقة الترجع:كل عدد طبيعي له موال وهو أيضا عدد صحيح طبيعي، 1 عدد صحيح طبيعي. أي: «1 عدد طبيعي، وإذا كان عدداً طبيعياً، فإن عدد طبيعي أيضاً». ¤ الأعداد الطبيعية تكتب من دون فاصلة /./ ومن دون كسر 1/3 Sa mhatamaitic, is iad na huimhreacha aiceanta ná iad siúd a úsáidtear le haghaidh (mar atá i "tá sé bhonn ar an mbord") agus le h (mar atá i "is í seo an tríú cathair is mó sa tír"). I ngnáththéarmaíocht mhatamaiticiúil, is iad na focail a úsáidtear sa ghnáthchaint chun "" a chomhaireamh, agus is iad na focail a úsáidtear le haghaidh áite nó céime in ord seicheamhach na "h". 自然数（参考ISO 80000-2和中所采用的定义）指非负整数，为免歧义有时也直接以非负整数代替自然数使用。数学中，一般以代表以自然数组成的集合。自然数集是一個可數的，無上界的無窮集合。非零自然数即指正整数。自然数可用于计数（如：桌子上有“三”个苹果）和定序（如：国内“第三”大城市）。 Un nombre natural és qualsevol dels nombres 0, 1, 2, 3…, 19, 20, 21..., que es poden utilitzar per a comptar els elements d'un conjunt finit. Per exemple: 24 pomes, 2 camions o 1123 peixos, són situacions on es compta amb nombres naturals. El conjunt de tots els nombres naturals se simbolitza per la lletra o N. Segons Kronecker, matemàtic alemany (1823–1891): En tot cas, segur que Kronecker es referiria als naturals si a la seva època la nomenclatura fos l'actual. Així, ara hauria dit: "Déu va crear els nombres naturals; tota la resta és obra de l'home". I numeri interi (o numeri interi relativi o, semplicemente, numeri relativi) corrispondono all'insieme ottenuto unendo i numeri naturali (0, 1, 2, ...) e i numeri interi negativi (−1, −2, −3,...), cioè quelli ottenuti ponendo un segno “−” davanti ai naturali. Questo insieme in matematica viene indicato con Z o , perché è la lettera iniziale di “Zahl” che in tedesco significa numero (originariamente "far di conto", infatti l'espressione implica l'utilizzo dei numeri negativi). Це́лые чи́сла — расширение множества натуральных чисел, получаемое добавлением к нему нуля и отрицательных чисел. Необходимость рассмотрения целых чисел продиктована невозможностью в общем случае вычесть из одного натурального числа другое — можно вычитать только меньшее число из большего. Введение нуля и отрицательных чисел делает вычитание такой же полноценной операцией, как сложение. Вещественное число является целым, если его десятичное представление не содержит дробной части (но может содержать знак). Примеры вещественных чисел: 数学における整数（せいすう、英: integer, whole number, 独: Ganze Zahl, 仏: nombre entier, 西: número entero）は、1 とそれに 1 ずつ加えて得られる自然数 (1, 2, 3, 4, …) 、これらに−1を乗じて得られる負数 (−1, −2, −3, −4, …) 、および 0 の総称である。整数の全体からなる集合は、一般に太字のまたは黒板太字ので表す。これはドイツ語 Zahlen（「数」の意・複数形）に由来する。抽象代数学、特に代数的整数論では、しばしば「代数体の整数環」の元という意味で代数的整数あるいは「整数」という言葉を用いる。有理数全体の成す体はそれ自身が代数体の最も簡単な例であり、有理数体の代数体としての整数環すなわち、「有理数の中で整なもの」の全体の成す環は、本項でいう意味での整数全体の成す環である。一般の「整数」との区別のためにここでいう意味の整数を有理整数 (rational integer) と呼ぶことがある。 Een natuurlijk getal is een getal dat het resultaat is van een telling van een eindig aantal dingen, dus een van de getallen De verzameling natuurlijke getallen wordt aangegeven met het symbool . Er is geen overeenstemming of het getal 0 bij de natuurlijke getallen hoort. In de traditionele definitie beginnen de natuurlijke getallen bij 1 – van daaraf begint men immers te tellen. Vanaf de negentiende eeuw ziet men de definitie opduiken die 0 wel tot de natuurlijke getallen rekent (zie geschiedenis). In de wiskunde wordt tegenwoordig vrij algemeen het getal 0 tot de natuurlijke getallen gerekend. Um número natural é um número inteiro não negativo Em alguns contextos, número natural é definido como um número inteiro positivo, sendo também o zero considerado como um número natural (mesmo não sendo positivo ): O conjunto dos números naturais é, comumente, denotado pelo símbolo O símbolo é usado para explicitar que o zero não está sendo incluso, i.e. Uma construção do conjunto dos números naturais que não depende do conjunto dos números inteiros foi desenvolvida por Giuseppe Peano no século XIX e costuma ser chamada de Axiomática de Peano. Dalam matematika, terdapat dua kesepakatan mengenai himpunan bilangan asli. Yang pertama definisi menurut matematikawan tradisional, yaitu himpunan bilangan bulat positif yang bukan nol {1, 2, 3, 4, ...}. Sedangkan yang kedua definisi oleh logikawan dan ilmuwan komputer, adalah himpunan nol dan bilangan bulat positif {0, 1, 2, 3, ...}. Bilangan asli merupakan salah satu konsep matematika yg paling sederhana dan termasuk konsep pertama yang bisa dipelajari dan dimengerti oleh manusia, bahkan beberapa penelitian menunjukkan beberapa jenis kera juga bisa menangkapnya. Die ganzen Zahlen (auch Ganzzahlen, lateinisch numeri integri) sind eine Erweiterung der natürlichen Zahlen. Die ganzen Zahlen umfassen alle Zahlen …, −3, −2, −1, 0, 1, 2, 3, … und enthalten damit alle natürlichen Zahlen sowie deren additive Inverse. Die Menge der ganzen Zahlen wird meist mit dem Buchstaben mit Doppelstrich bezeichnet (das „Z“ steht für das deutsche Wort „Zahlen“). Das alternative Symbol ist mittlerweile weniger verbreitet; ein Nachteil dieses Fettdruck-Symbols ist die schwierige handschriftliche Darstellbarkeit. Der Unicode des Zeichens lautet U+2124 und hat die Gestalt ℤ. En matemáticas, un número natural es cualquiera de los números que se usan para contar los elementos de ciertos conjuntos. En obras más modernas, aparece también como ℕ = {0, 1, 2, 3, 4, …}. De dos números vecinos, el que se encuentra a la derecha se llama siguiente o sucesivo, por lo que el conjunto de los números naturales es ordenado e infinito. El conjunto de todos los números naturales iguales o menores que cierto número natural , es decir, el conjunto , se llama segmento de una sucesión natural y se denota o bien . Натурáльные чи́сла (от лат. naturalis «естественный») — числа, возникающие естественным образом при счёте (1, 2, 3, 4, 5, 6, 7 и так далее). Последовательность всех натуральных чисел, расположенных в порядке возрастания, называется натуральным рядом. Множество натуральных чисел является бесконечным, так как для любого натурального числа найдётся натуральное число, большее чем . Натуральные числа ещё можно называть целыми положительными числами. Поэтому Отрицательные и нецелые (дробные) числа к натуральным не относятся. Is iad na slánuimhir na uimhreacha aiceanta {0, 1, 2, 3, 4, ... } mar aon leis an dhiúl do na huimhreacha aiceanta dheimhneach { -1, -2, -3, -4, ...}. Is fo-thacar iad na slánuimhir {..., -2, -1, 0, 1, 2, ...} do na réaduimhir. Is tacar éigríochta iad na slánuimhir freisin. Úsáidtear an siombail chun an tacar iomlán do uimhreacha aiceanta a léiriú: Úsáidtear an focal Slánuimhir i ríomhchlárú freisin. Is cineál sonraí bunúsach é slánuimhir. Chuirtear é in iúil le 'int' i roinnt ríomhchlár, Java ina measc. San cás seo, ní tacar éigríochta atá i gceist san gcás seo. 自然数（しぜんすう、英: natural number）とは、個数もしくは順番を表す一群の数のことである。集合論においては、自然数は物の個数を数える基数のうちで有限のものであると考えることもできるし、物の並べ方を示す順序数のうちで有限のものであると考えることもできる。自然数を 1, 2, 3, … とする流儀と、0, 1, 2, 3, … とする流儀があり、前者は数論などでよく使われ、後者は集合論、論理学などでよく使われる（詳しくはの節を参照）。日本では高校教育課程においては0を入れないが、大学以降では0を含めることも多い（より正確には、代数学では0を含め、解析学では除外することが多い）。いずれにしても、0 を自然数に含めるかどうかが問題になるときは、その旨を明記する必要がある。自然数の代わりに前者を正整数、後者を非負整数と言い換えることによりこの問題を避けることもある。数学の基礎付けにおいては、自然数の間の加法についての形式的な逆元を考えることによって整数を定義する。正の整数ないしは負でない整数を自然数と同一視し、自然数を整数の一部として取扱うことができる。自然数と同様に整数の全体も可算無限集合である。なお、文脈によっては、その一群に属する個々の数（例えば 3 や 18）を指して自然数ということもある。 Ακέραιοι ονομάζονται όλοι οι φυσικοί αριθμοί μαζί με τους αντίθετους τους και το μηδέν. Το σύνολο των ακεραίων δηλαδή το σύνολο: Συμβολίζεται με το γράμμα , αρχικό της λέξης Zahl που στα γερμανικά σημαίνει αριθμός. Το σύνολο ορίζεται επίσης ως εξής: Όπως και το σύνολο των φυσικών, το σύνολο των ακεραίων είναι άπειρο αριθμήσιμο με πληθάριθμο (άλεφ-μηδέν). Liczby całkowite – liczby naturalne oraz liczby przeciwne do nich a także liczba zero. Są uogólnieniem zbioru liczb naturalnych na zbiór, w którym wykonalne jest odejmowanie. Uogólnieniem liczb całkowitych są liczby wymierne. Zbiór liczb całkowitych oznaczamy w matematyce symbolem (od niem. Zahlen – liczby). W Polsce w większości szkół podstawowych i średnich, w celu ułatwienia skojarzenia z polską nazwą, stosuje się symbol przy czym MEN zaleca używanie oznaczenia . Zenbaki arruntak multzo bateko elementuak zenbatzeko erabiltzen diren zenbakiak dira: 0, 1, 2, 3, 4, 5, 6... Matematikari batzuek (zenbaki-teoriari ekin ziotenak) zero arrunta ez dela deritzote, baina beste batzuk ez dira uste berekoak (multzo-teoria, logika eta informatikari ekin ziotenak). Entziklopedia honetan, zero arrunta dela kontuan hartuko dugu. Zenbaki arruntak zer diren edonork dakien arren, haren definizioa ez da inolaz ere erraza. Peano-ren axiomak zenbaki arrunten multzoa, , adiera bakarreko moduan deskribatzen dute: La entjeroj (aŭ plenaj nombroj) konsistas el la naturaj nombroj (1, 2, 3, …), la respondaj negativaj nombroj (−1, −2, −3, …) kaj 0 (nulo). Matematikistoj kutime signas la aron de la entjeroj per aŭ Z. La naturaj nombroj estas subaro de la entjeroj, kion oni signas per ⊂ . La entjero m estas nomata divizoro de la entjero n, se la kvociento de n per m ankaŭ estas entjero. Ekz-e 3 estas divizoro de 9, kaj 1, 2, 3, 4, 6, 12 estas ĉiuj pozitivaj divizoroj de 12. De naturliga talen är de heltal som är icke-negativa {0, 1, 2, 3, 4, …}, alternativt de heltal som är positiva {1, 2, 3, 4, …}. Den första definitionen är vanlig i Sverige och allmänt i matematisk logik, mängdlära och beräkningsvetenskap, medan den senare kan hittas i bland annat amerikansk litteratur och bland talteoretiker. Mängden av de naturliga talen betecknas ℕ (ett vanligt N i fetstil kan även användas). ℕ är diskret, uppräkneligt oändlig och har kardinalitet Alef-noll (ℵ₀). För att undvika förvirring kan ℤ+ användas för att beteckna de positiva heltalen, och ℕ0 för de icke-negativa. Στα μαθηματικά, οι φυσικοί αριθμοί είναι εκείνοι που χρησιμοποιούνται για τη μέτρηση ("υπάρχουν έξι νομίσματα στο τραπέζι") και για τη σύγκριση ("υπάρχουν περισσότερες καρέκλες από τους πίνακες"). Μια μεταγενέστερη έννοια είναι εκείνη ενός ονομαστικού αριθμού, ο οποίος χρησιμοποιείται μόνο για την ονομασία. ( 자연수(自然水)는 자연에서 나는 물을 가리키기도 합니다.) 수학에서 자연수(自然數, 영어: natural number)는 수를 셀 때나 순서를 매길 때 사용되는 수다. 양의 정수(陽-整數, 영어: positive integer) 1, 2, 3, ...로 정의되거나, 음이 아닌 정수(陰-整數, 영어: non-negative integer) 0, 1, 2, 3, ...로 정의된다. 범자연수(汎自然數, 문화어: 옹근수(-數), 완수(完數), 영어: whole number)라는 용어는 첫째 정의를 택할 경우에 음이 아닌 정수를 가리키는 데 사용되며, 이에 대응하는 문화어와 영어는 둘째 정의를 택할 경우에 정수를 가리키는 데 사용된다. 자연수의 집합은 대문자 N을 써서 표기하며, 보통 칠판 볼드체 ℕ를 사용한다. 자연수의 수를 세는 역할을 일반화하면 기수의 개념을 얻으며, 자연수의 순서를 매기는 기능을 일반화하면 순서수의 개념을 얻는다. 자연수의 집합의 대수적 성질을 일반화하면 반환의 개념을 얻는다. 특히 자연수는 많은 스포츠 점수 같은 경기나 게임에 사용될수 있으며 우리가 가장 흔히 보는 수로도 볼 수 있다. Natura nombro estas unu el la pozitivaj entjeroj (1, 2, 3, 4 ...). Kelkfoje ankaŭ la nombron 0 (nul aŭ nulo) oni kalkulas inter naturaj nombroj. Naturaj nombroj havas du ĉefajn uzojn: Oni uzas ilin por nombri objektojn (ekz-e, "estas tri pomoj sur la tablo") aŭ por ordigi objektojn (ekz-e "ĝi estas la trie plej granda urbo en la lando"). En la dua signifo ili estas nomataj vicmontraj nombroj aŭ numeroj. La simbolo por la aro de ĉiuj naturaj nombroj estas aŭ . Přirozeným číslem (číslem z oboru přirozených čísel) se v matematice rozumí čísla, které je možné použít pro vyjádření počtu („na stole je šest mincí“) nebo pořadí („toto je třetí největší město“). Čísla používaná pro vyjádření počtu se v matematice označují jako kardinální čísla, zatímco čísla určená pro vyjádření pořadí se nazývají ordinální čísla. Přirozená čísla patří mezi základní matematické koncepty, a protože se považují za nejjednodušší na pochopení, začíná výuka matematiky obvykle od přirozených čísel. Množina všech přirozených čísel se obvykle označuje písmenem . Celá čísla se skládají z přirozených čísel (1, 2, 3, …), nuly (0) a záporných celých čísel (−1, −2, −3, …). Množina celých čísel se v matematice většinou označuje Z, nebo , podle Zahlen (německy čísla). Podobně jako přirozená čísla, tvoří celá čísla nekonečnou spočetnou množinu. Studiem celých čísel se zabývá teorie čísel. Els nombres enters són els que designen quantitats no fraccionables en parts més petites que la unitat. Per exemple −3, 80, −4 o 2019 són enters, mentre que ; −1,5; 3,14; o no ho són. Els enters es poden qualificar també amb l'adjectiu "sencer": que no hi manca cap part. Aquesta no és, però, forma correcte d'anomenar-los. Són una extensió dels nombres naturals de forma que a més de comptar coses, permeten comptabilitzar pèrdues o deutes. També són necessaris en magnituds com les altures o la temperatura en què cal considerar valors per sobre o per sota de zero. Heltalen är unionen av mängden naturliga tal {0, 1, 2, ...} och mängden negativa heltal {-1, -2, -3, ...}. Mängden av hela tal betecknas med den dubbelstrukna bokstaven ℤ (ibland fetstilta bokstaven Z), från det tyska ordet Zahlen (tal). Ibland definierar man delmängder av ℤ: ℤ+, ℤ* och ℤ–. * ℤ+ är 1, 2, 3, 4, 5 ... * ℤ* är 0, 1, 2, 3, 4, 5 ... * ℤ– är ... -5, -4, -3, -2, -1 Beroende på definition kan endera ℤ+ eller ℤ* vara detsamma som mängden naturliga tal.
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dbo:thumbnail: n52:Latex_integers.svg?width=300
dbp:id: 403 p/i051290
dbp:title: Integer
dbo:abstract: Zenbaki arruntak multzo bateko elementuak zenbatzeko erabiltzen diren zenbakiak dira: 0, 1, 2, 3, 4, 5, 6... Matematikari batzuek (zenbaki-teoriari ekin ziotenak) zero arrunta ez dela deritzote, baina beste batzuk ez dira uste berekoak (multzo-teoria, logika eta informatikari ekin ziotenak). Entziklopedia honetan, zero arrunta dela kontuan hartuko dugu. Zenbaki arruntak zer diren edonork dakien arren, haren definizioa ez da inolaz ere erraza. Peano-ren axiomak zenbaki arrunten multzoa, , adiera bakarreko moduan deskribatzen dute: * Zero zenbaki arrunta bedi. * a zenbaki arrunt bakoitza, beste a+1 zenbaki arruntak jarraituko du. * Ez dago zenbaki arruntik, zeinen ondorengo zenbakia zeroa den. * Bi zenbaki arrunt desberdinak badira, ondoren datozenak ere desberdinak dira. * Zerorentzat eta edozein zenbaki arrunt harturik honen ondorengoarentzat betetzen den propiatatea, zenbaki arrunt guztientzat beteko da. Azken postulatuak indukzio matematikoaren baliotasuna bermatzen du. Zenbaki natural baten berdinak edo txikiagoak diren zenbaki natural guztien multzoa , hau da, segida natural baten segmentua deitzen da eta edo moduan idatzi ohi da. I numeri interi (o numeri interi relativi o, semplicemente, numeri relativi) corrispondono all'insieme ottenuto unendo i numeri naturali (0, 1, 2, ...) e i numeri interi negativi (−1, −2, −3,...), cioè quelli ottenuti ponendo un segno “−” davanti ai naturali. Questo insieme in matematica viene indicato con Z o , perché è la lettera iniziale di “Zahl” che in tedesco significa numero (originariamente "far di conto", infatti l'espressione implica l'utilizzo dei numeri negativi). Gli interi vengono quindi definiti esattamente come l'insieme dei numeri che sono il risultato tra sottrazioni di numeri naturali. I numeri interi possono essere sommati, sottratti e moltiplicati e il risultato rimane un numero intero. L'inverso di un numero intero non è però un intero in generale, ma un numero razionale; formalmente questo fatto si esprime dicendo che è un anello commutativo, ma non un campo. Die natürlichen Zahlen sind die beim Zählen verwendeten Zahlen 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 usw. Je nach Definition kann auch die 0 (Null) zu den natürlichen Zahlen gezählt werden. Die Menge der natürlichen Zahlen bildet mit der Addition und der Multiplikation zusammen eine mathematische Struktur, die als kommutativer Halbring bezeichnet wird. An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface Z or blackboard bold . The set of natural numbers is a subset of , which in turn is a subset of the set of all rational numbers , itself a subset of the real numbers . Like the natural numbers, is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers. Přirozeným číslem (číslem z oboru přirozených čísel) se v matematice rozumí čísla, které je možné použít pro vyjádření počtu („na stole je šest mincí“) nebo pořadí („toto je třetí největší město“). Čísla používaná pro vyjádření počtu se v matematice označují jako kardinální čísla, zatímco čísla určená pro vyjádření pořadí se nazývají ordinální čísla. Přirozená čísla patří mezi základní matematické koncepty, a protože se považují za nejjednodušší na pochopení, začíná výuka matematiky obvykle od přirozených čísel. Množina všech přirozených čísel se obvykle označuje písmenem . Podle některých z používaných definic (např. standard ) přirozená čísla začínají číslem a označují tak nezáporná čísla (tj. čísla ), zatímco podle jiných definic přirozená čísla začínají číslem a označují tak čísla .. Στα μαθηματικά, οι φυσικοί αριθμοί είναι εκείνοι που χρησιμοποιούνται για τη μέτρηση ("υπάρχουν έξι νομίσματα στο τραπέζι") και για τη σύγκριση ("υπάρχουν περισσότερες καρέκλες από τους πίνακες"). Μια μεταγενέστερη έννοια είναι εκείνη ενός ονομαστικού αριθμού, ο οποίος χρησιμοποιείται μόνο για την ονομασία. Δεν υπάρχει καθολική συμφωνία για το αν θα συμπεριλαμβάνεται το μηδέν στο σύνολο των φυσικών αριθμών: μερικοί ορίζουν τους φυσικούς αριθμούς να είναι οι θετικοί ακέραιοι 1, 2, 3,... ενώ για άλλους ο όρος προσδιορίζει τους μη-αρνητικούς ακέραιους 0, 1, 2, 3, .... Ο πρώτος ορισμός είναι ο παραδοσιακός, με τον τελευταίο ορισμό να εμφανίζεται για πρώτη φορά τον 19ο αιώνα. Μερικοί συγγραφείς χρησιμοποιούν τον όρο φυσικό αριθμό αποκλείοντας το 0 και ακέραιο αριθμό για να το συμπεριλάβουν. Άλλοι χρησιμοποιούν τον όρο ακέραιο αριθμό κατά τρόπο που να περιλαμβάνει τόσο το μηδέν όσο και τους αρνητικούς ακέραιους, δηλαδή ως ισοδύναμο του ακεραίου όρου.Ιδιότητες των φυσικών αριθμών που σχετίζονται με τη διαιρετότητα, όπως η κατανομή των πρώτων αριθμών, μελετούνται στη θεωρία αριθμών. Προβλήματα σχετικά με την καταμέτρηση και την παραγγελία, όπως η , μελετούνται στη Συνδυαστική. Це́лые чи́сла — расширение множества натуральных чисел, получаемое добавлением к нему нуля и отрицательных чисел. Необходимость рассмотрения целых чисел продиктована невозможностью в общем случае вычесть из одного натурального числа другое — можно вычитать только меньшее число из большего. Введение нуля и отрицательных чисел делает вычитание такой же полноценной операцией, как сложение. Вещественное число является целым, если его десятичное представление не содержит дробной части (но может содержать знак). Примеры вещественных чисел: Числа 142857; 0; −273 являются целыми.Числа 5½; 9,75; −12,07 не являются целыми. Множество целых чисел обозначается (от нем. Zahlen — «числа»). Изучением свойств целых чисел занимается раздел математики, называемый теорией чисел. En mathématiques, un entier naturel est un nombre permettant fondamentalement de compter des objets considérés comme des unités équivalentes : un jeton, deux jetons… une carte, deux cartes, trois cartes… Un tel nombre entier peut s'écrire avec une suite finie de chiffres en notation décimale positionnelle (sans signe et sans virgule). L’étude des entiers naturels est l’objet de l’arithmétique, branche des mathématiques, constituée dès l'Antiquité grecque. Chaque nombre entier a un successeur unique, c'est-à-dire un entier qui lui est immédiatement supérieur, et la liste des entiers naturels est infinie. Les définitions modernes d’entier naturel sont fondées sur : * l’axiomatisation de l’arithmétique réalisée par Peano et Dedekind à la fin du XIXe siècle ; * la construction d’ensembles vérifiant les axiomes de l’arithmétique : Ernst Zermelo, quand il a axiomatisé la théorie des ensembles, a montré que les entiers naturels pouvaient être définis en termes ensemblistes (on utilise aujourd'hui le plus souvent une méthode due à von Neumann). La définition originelle, due à Richard Dedekind, de l'ensemble des entiers naturels ne comprend pas le nombre zéro ; plus récemment une autre définition a été proposée qui inclut zéro. Ces deux définitions coexistent encore aujourd'hui. Selon les acceptions, la liste des entiers naturels est donc : * 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; … ou * 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; … Quelle que soit la définition choisie (entiers commençant à zéro ou commençant à un), l'ensemble des entiers naturels est conventionnellement noté « N » ou « ℕ », avec tous les risques induits de mésinterprétation. La notation est due à Dedekind en 1888, qui l'utilise pour l'ensemble des entiers commençant à un. On trouve parfois des notations moins ambiguës exposées dans la section . * Lorsqu'on prend comme définition des entiers naturels les entiers commençant à zéro, l'ensemble des entiers naturels non nuls est couramment noté « » (ou « »). * Lorsqu'on prend pour définition des entiers naturels les entiers commençant à un, l'ensemble des entiers positifs ou nuls, appelés en anglais non-negative integers, est parfois noté « ». Les entiers naturels s'identifient aux entiers relatifs positifs (ou nuls), ainsi qu'aux nombres rationnels positifs (ou nuls) pouvant s'écrire sous la forme d'une fraction de dénominateur 1, et d'une manière plus générale aux réels positifs (ou nuls) de partie fractionnaire nulle. Een natuurlijk getal is een getal dat het resultaat is van een telling van een eindig aantal dingen, dus een van de getallen De verzameling natuurlijke getallen wordt aangegeven met het symbool . Er is geen overeenstemming of het getal 0 bij de natuurlijke getallen hoort. In de traditionele definitie beginnen de natuurlijke getallen bij 1 – van daaraf begint men immers te tellen. Vanaf de negentiende eeuw ziet men de definitie opduiken die 0 wel tot de natuurlijke getallen rekent (zie geschiedenis). In de wiskunde wordt tegenwoordig vrij algemeen het getal 0 tot de natuurlijke getallen gerekend. Als de verzameling van de natuurlijke getallen wordt aangevuld met de negatieve getallen , ontstaat de verzameling van de gehele getallen, aangeduid door het symbool . De natuurlijke getallen vormen dus een strikte deelverzameling van de verzameling van de gehele getallen: De notaties en worden ook gebruikt om de natuurlijke of gehele getallen met uitsluiting van 0, of de (al dan niet strikt) negatieve getallen weer te geven; hun precieze betekenis verschilt echter tussen auteurs. Getallen in de vorm (of ), waarbij behoort tot , noemt men even; dit is de verzameling {0, 2, 4, 6, 8, ...}. De overige getallen in noemt men oneven; dit is de verzameling {1, 3, 5, 7, ...}. Oneven getallen kunnen voor een zeker natuurlijk getal geschreven worden als . Alle verzamelingen waarvoor een bijectie bestaat met , worden aftelbaar oneindige verzamelingen genoemd. Dit is onder meer het geval voor de verzameling van de even getallen, voor de oneven getallen en voor de priemgetallen; alle drie zijn dit deelverzamelingen van . Getallenverzamelingen zijn een belangrijk begrip in de tak van de wiskunde die getaltheorie wordt genoemd. 自然数（しぜんすう、英: natural number）とは、個数もしくは順番を表す一群の数のことである。集合論においては、自然数は物の個数を数える基数のうちで有限のものであると考えることもできるし、物の並べ方を示す順序数のうちで有限のものであると考えることもできる。自然数を 1, 2, 3, … とする流儀と、0, 1, 2, 3, … とする流儀があり、前者は数論などでよく使われ、後者は集合論、論理学などでよく使われる（詳しくはの節を参照）。日本では高校教育課程においては0を入れないが、大学以降では0を含めることも多い（より正確には、代数学では0を含め、解析学では除外することが多い）。いずれにしても、0 を自然数に含めるかどうかが問題になるときは、その旨を明記する必要がある。自然数の代わりに前者を正整数、後者を非負整数と言い換えることによりこの問題を避けることもある。数学の基礎付けにおいては、自然数の間の加法についての形式的な逆元を考えることによって整数を定義する。正の整数ないしは負でない整数を自然数と同一視し、自然数を整数の一部として取扱うことができる。自然数と同様に整数の全体も可算無限集合である。なお、文脈によっては、その一群に属する個々の数（例えば 3 や 18）を指して自然数ということもある。 العدد الطبيعي في الرياضيات، هو كل عدد صحيح موجب، مثل 1، 2، 3... 12، 563. ويضيف بعض العلماء الصفر إلى هذه المجموعة من الأعداد. يرمز لمجموعة الأعداد الطبيعية بالحرف اللاتيني N.و هي مجموعة أعداد غير منتهية. يمثل الواحد 1 أصغر الأعداد الطبيعية التي لا تتضمن الصفر ℕ*، بينما يمثل الصفر 0 أصغر الأعداد في مجموعة الأعداد الطبيعية التي تتضمن الصفر ℕ0، ويتم إنشاؤها بواسطة علاقة الترجع:كل عدد طبيعي له موال وهو أيضا عدد صحيح طبيعي، 1 عدد صحيح طبيعي. أي: «1 عدد طبيعي، وإذا كان عدداً طبيعياً، فإن عدد طبيعي أيضاً». وكل مجموعة مرتبة تخضع لأكسيومات بيانو تسمى مجموعة أعداد طبيعية. ويُرمز إلى هذه المجموعة ب N أو يرمز إليها ب *N إذا حذف منها الصفر. بعض الرياضيين لا يعتبرون الصفر عددا صحيحا طبيعيا. * ومن خصائصها الجبرية: الانغلاق بعمليتي الجمع والضرب * التجميعة، الضرب عملية تجميعية: c × (b × a) = (c × b) × a. * التبادلية، الجمع عملية تبديلية في مجموعة الأعداد الطبيعية: تغيير مكان الطرفين في العملية لا يغير النتيجة:a + b = b + a. الضرب عملية تبديلية في مجموعة الأعداد الطبيعية: تغيير مكان الطرفين في العملية لا يغير النتيجة: a × b = b × a. * وجود العناصر المحايدة، صفر هو العنصر الحيادي لعملية الجمع في مجموعة الأعداد الطبيعية: النتيجة (أو الحاصل) بعد جمع عدد وصفر هو نفس العدد. a + 0 = a. الواحد (1) هو العنصر المحايد لعملية الضرب في مجموعة الأعداد الطبيعية: النتيجة (أو الحاصل) بعد ضرب عدد وواحد هو نفس العدد. a × 1 = a. * توزيعية عملية الضرب على عملية الجمع في مجموعة الأعداد الطبيعية:a × (b + c) = a × b + a × c * لا وجود ، إذا كان a و b عددين طبيعيين حيث 0 = a × b فإن a = 0 أو b = 0.. ¤ الأعداد الطبيعية تكتب من دون فاصلة /./ ومن دون كسر 1/3 ملاحظة: لم يعتبر العديد من علماء الرياضيات الإغريق الواحد عددا. فبالنسبة إليهم، اثنان هو أصغر عدد. Zenbaki osoen multzoan zenbaki arruntak biltzen dira (0,1,2,...), beren aurkakoekin batera (-0,-1,-2,...). -0 eta 0 berdintzat jotzen dira. Zenbaki osoen multzoa hizkiaz izendatu ohi da ('Zahlen' germanierazko hitzetik). Zenbaki osoak batu, kendu eta biderkatu egin daitezke: emaitza beti izango da zenbaki oso bat. x+a=b motako ekuazioen soluzioa, non a eta b zenbaki osoak diren, zenbaki osoa izango da. Zenbaki arrunten kasuan ez da esaterako gauza bera gertatzen. Zorrotzago, zenbaki osoen multzoak, batuketa eta biderketa eragiketak definitu ondoren, osatzen duela esan behar da. Zenbaki osoen kopurua infinitua da, infinito zenbakigarria, zehazki. Un número entero es un elemento del conjunto numérico que contiene los números naturales; que son o ; dependiendo de cómo se definan, sus opuestos, y en la segunda definición, además el cero. Los enteros negativos, como −1 o −13 (se leen «menos uno», «menos trece», etc.), son menores que cero y también son menores que todos los enteros positivos. Para resaltar la diferencia entre positivos y negativos, se puede escribir un signo «menos» delante de los negativos: -1, -5, etc. Y si no se escribe signo al número se asume que es positivo. El conjunto de todos los números enteros se representa por la letra {Z}=\{..., -4, -3, -2, -1, 0, +1, +2, +3,\,...\} letra inicial del vocablo alemán Zahlen («números», pronunciado [ˈtsaːlən]). En la recta numérica los números negativos se encuentran a la izquierda del cero y los positivos a su derecha. Los números enteros pueden sumarse, restarse, multiplicarse y dividirse, siguiendo el modelo de los números naturales añadiendo unas normas para el uso del signo. Los números enteros extienden la utilidad de los números naturales para contar cosas. Pueden utilizarse para contabilizar pérdidas: si en un colegio entran 80 alumnos nuevos de primer curso un cierto año, pero hay 100 alumnos de último curso que pasaron a educación secundaria, en total habrá 100 − 80 = 20 alumnos menos; pero también puede decirse que dicho número ha aumentado en 80 − 100 = −20 alumnos. Ciertas magnitudes, como la temperatura o la altura, usan valores por debajo del cero. La altura del Everest es 8848 metros por encima del nivel del mar, y por el contrario, la orilla del mar Muerto está 423 metros por debajo del nivel del mar; es decir, su altura se puede expresar como −423 m. العدد الصحيح هو الذي يُمكن كتابته بدون استخدام الكسور أو الفواصل العشرية، وتتكون مجموعة الأعداد الصحيحة والتي تعتبر مجموعة جزئية من مجموعة الأعداد الحقيقية- من الأعداد الطبيعية (1، 2، 3.) والصفر والأعداد السالبة المقابلة للأعداد الطبيعية (-1، -2، -3..)، وعليه فمجموعة الأعداد الصحيحة تكون مجموعة غير منتهية شأنها في ذلك شأن مجموعة الأعداد الطبيعية، وعادة ما يرمز لها بالحرف اللاتيني Z. Els nombres enters són els que designen quantitats no fraccionables en parts més petites que la unitat. Per exemple −3, 80, −4 o 2019 són enters, mentre que ; −1,5; 3,14; o no ho són. Els enters es poden qualificar també amb l'adjectiu "sencer": que no hi manca cap part. Aquesta no és, però, forma correcte d'anomenar-los. Són una extensió dels nombres naturals de forma que a més de comptar coses, permeten comptabilitzar pèrdues o deutes. També són necessaris en magnituds com les altures o la temperatura en què cal considerar valors per sobre o per sota de zero. El conjunt dels nombres enters es representa amb el símbol , la lletra inicial de la paraula alemanya Zahlen (nombre). El conjunt dels nombres enters , està format per: * Els nombres naturals anomenats també enters positius: . S'escriuen tant amb el signe més (+) al davant com sense. Si un nombre no porta el signe +, se sobreentén que és positiu. * El nombre 0 (zero) El zero és l'únic nombre enter que no és positiu ni negatiu. * Nombres enters negatius: S'escriuen sempre amb el signe menys (−) al davant, Aquest signe menys forma part de la designació del nombre, no indica l'operació resta. Matemàticament, el conjunt dels nombres enters amb les operacions de suma i multiplicació, , constitueix una estructura algebraica d'anell commutatiu i unitari. és també un conjunt infinit numerable completament ordenat sense fita superior i tampoc fita inferior. La branca de les matemàtiques que estudia les propietats dels nombres enters s'anomena teoria de nombres. Sa mhatamaitic, is iad na huimhreacha aiceanta ná iad siúd a úsáidtear le haghaidh (mar atá i "tá sé bhonn ar an mbord") agus le h (mar atá i "is í seo an tríú cathair is mó sa tír"). I ngnáththéarmaíocht mhatamaiticiúil, is iad na focail a úsáidtear sa ghnáthchaint chun "" a chomhaireamh, agus is iad na focail a úsáidtear le haghaidh áite nó céime in ord seicheamhach na "h". Uaireanta, is féidir na huimhreacha aiceanta a thaispeáint mar shraith áisiúil cód (lipéid nó "ainmneacha"); is é sin, mar a thugann teangeolaithe orthu, ag scaoileadh thairis na hairíonna. go léir nó a lán. a bhaineann le bheith ina n-uimhir sa chiall mhatamaiticiúil. Is minic a chuirtear an tsraith uimhreacha aiceanta in iúl leis an tsiombail {\ displaystyle \ mathbb {N}} \ mathbb {N}. Weisstein, Eric W. https://mathworld.wolfram.com/NaturalNumber.html}Tagraíonn téacsanna, a eisiann an nialas ó na huimhreacha aiceanta, uaireanta do na huimhreacha aiceanta mar aon leis an nialas mar na huimhreacha iomlána, agus i scríbhinní eile, úsáidtear an téarma sin ina ionad sin do na slánuimhreacha (slánuimhreacha diúltacha san áireamh). Die ganzen Zahlen (auch Ganzzahlen, lateinisch numeri integri) sind eine Erweiterung der natürlichen Zahlen. Die ganzen Zahlen umfassen alle Zahlen …, −3, −2, −1, 0, 1, 2, 3, … und enthalten damit alle natürlichen Zahlen sowie deren additive Inverse. Die Menge der ganzen Zahlen wird meist mit dem Buchstaben mit Doppelstrich bezeichnet (das „Z“ steht für das deutsche Wort „Zahlen“). Das alternative Symbol ist mittlerweile weniger verbreitet; ein Nachteil dieses Fettdruck-Symbols ist die schwierige handschriftliche Darstellbarkeit. Der Unicode des Zeichens lautet U+2124 und hat die Gestalt ℤ. Die obige Aufzählung der ganzen Zahlen gibt auch gleichzeitig in aufsteigender Folge deren natürliche Anordnung wieder. Die Zahlentheorie ist der Zweig der Mathematik, der sich mit Eigenschaften der ganzen Zahlen beschäftigt. Die Repräsentation ganzer Zahlen im Computer erfolgt üblicherweise durch den Datentyp Integer. Die ganzen Zahlen werden im Mathematikunterricht üblicherweise in der fünften bis siebten Klasse eingeführt. 整数，在電腦應用上也稱為整型，是序列中所有的数的统称，包括负整数、零（0）与正整数。和自然數一樣，整數也是一個可數的無限集合。這個集合在数学上通常表示粗體或，源于德语单词Zahlen（意为“数”）的。在代數數論中，這些屬於有理數的一般整數會被稱為有理整數，用以和高斯整數等的概念加以區分。自然数（参考ISO 80000-2和中所采用的定义）指非负整数，为免歧义有时也直接以非负整数代替自然数使用。数学中，一般以代表以自然数组成的集合。自然数集是一個可數的，無上界的無窮集合。非零自然数即指正整数。自然数可用于计数（如：桌子上有“三”个苹果）和定序（如：国内“第三”大城市）。 De gehele getallen zijn alle getallen in de rij …, −3, −2, −1, 0, 1, 2, 3, … die voortgezet wordt door er steeds 1 bij te tellen of er 1 af te trekken. De gehele getallen omvatten 0, de natuurlijke getallen, dus de getallen waarmee wordt geteld, en de tegengestelden daarvan, de negatieve gehele getallen. Een geheel getal heet 'geheel' omdat het niet gebroken is en zonder cijfers achter de komma kan worden geschreven. De getallen 21, 4 en −121 zijn bijvoorbeeld gehele getallen, terwijl 9,75, 5½ en geen gehele getallen zijn. De verzameling gehele getallen is een deelverzameling van de reële getallen, en wordt meestal voorgesteld door een vet gedrukte Z of het symbool (Unicode U+2124 ℤ), wat voor Zahlen, het Duits voor getallen, staat. De wiskundetak die zich met de studie bezighoudt naar de eigenschappen van de gehele getallen, noemt men de getaltheorie. Bilangan bulat adalah bilangan yang dapat dituliskan tanpa komponen desimal atau pecahan. Sebagai contoh, 21, 4, 0, -3, -67 dan -2048 merupakan bilangan bulat, sedangkan 9,75, 5 12, dan bukan.Himpunan bilangan bulat terdiri dari angka 0, semua bilangan bulat positif (juga disebut dengan bilangan asli), dan invers aditif-nya, semua bilangan bulat negatif . Dalam matematika, himpunan ini sering dilambangkan dengan , atau huruf tebal. Huruf kapital Z yang digunakan berasal dari kata Zahlen, yang berarti bilangan dalam bahasa Jerman. Subhimpunan yang hanya terdiri dari angka 0 dan bilangan-bilangan bulat positif disebut dengan bilangan cacah. Himpunan sendiri merupakan subhimpunan dari himpunan bilangan rasional, karena nilainya dapat ditulis sebagai pecahan dengan penyebut 1. Bilangan rasional selanjutnya merupakan subhimpunan dari himpunan bilangan real. Um número inteiro é um número que pode ser escrito sem um componente fracional. Por exemplo, 21, 4, 0, e −2048 são números inteiros, enquanto 9.75, 52, e √2 não são. O conjunto dos números inteiros é representado pelo símbolo , cuja letra é originada da palavra alemã Zahlen ([ˈtsaːlən], "números"). Os inteiros (juntamente com a operação de adição) formam o menor grupo que contém o monoide aditivo dos números naturais. Como os números naturais, os inteiros formam um conjunto infinito contável. Os números inteiros podem ser simétricos, quando os números têm sinais opostos, ou pode existir também o valor absoluto de um número inteiro, que é a distância entre a origem e o número. Натура́льні чи́сла — числа, що виникають природним чином при лічбі. Це числа: 1, 2, 3, 4, … Множину натуральних чисел прийнято позначати знаком Існують два основних підходи до означення натуральних чисел: * числа, що використовуються при лічбі предметів (перший, другий, третій…) — підхід, загальноприйнятий у більшості країн світу; формалізованим різновидом цього підходу є аксіоматичне описання системи натуральних чисел за допомогою аксіом Пеано. * числа для позначення кількості предметів (один предмет, два предмети…). Натуральні числа можна записувати за допомогою десяти цифр: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Множина натуральних чисел є нескінченною: для будь-якого натурального числа знайдеться інше натуральне число, більше за нього. ( 다른 뜻에 대해서는 정수 (동음이의) 문서를 참고하십시오.)( 정수환은 여기로 연결됩니다. 대수적 수체의 부분환에 대해서는 대수적 정수환 문서를 참고하십시오.) ( 정수환은 여기로 연결됩니다. 대한민국의 가수에 대해서는 정수환 (1987년) 문서를 참고하십시오.) 수학에서 정수(整數, 문화어: 옹근수, integer)는 양의 정수(1, 2, 3, 4, 5, 6, 7, 8, ... , n), 음의 정수(-1, -2, -3, -4, -5, -6, -7, -8...) 및 0으로 이루어진 수의 체계이다. 또는 자연수, 자연수의 음수 및 영을 통칭하는 말이다. 수론의 가장 기본적인 연구 대상이다. 정수 전체의 집합의 기호는 이다. En matemáticas, un número natural es cualquiera de los números que se usan para contar los elementos de ciertos conjuntos. En obras más modernas, aparece también como ℕ = {0, 1, 2, 3, 4, …}. De dos números vecinos, el que se encuentra a la derecha se llama siguiente o sucesivo, por lo que el conjunto de los números naturales es ordenado e infinito. El conjunto de todos los números naturales iguales o menores que cierto número natural , es decir, el conjunto , se llama segmento de una sucesión natural y se denota o bien . Un nombre natural és qualsevol dels nombres 0, 1, 2, 3…, 19, 20, 21..., que es poden utilitzar per a comptar els elements d'un conjunt finit. Per exemple: 24 pomes, 2 camions o 1123 peixos, són situacions on es compta amb nombres naturals. El conjunt de tots els nombres naturals se simbolitza per la lletra o N. En alguns àmbits matemàtics (especialment en teoria de nombres) és convenient no considerar el zero com un nombre natural, mentre que uns altres, especialment en teoria de conjunts, lògica i informàtica, predomina la postura oposada. En aquest article, el zero és considerat un nombre natural. Segons Kronecker, matemàtic alemany (1823–1891): En tot cas, segur que Kronecker es referiria als naturals si a la seva època la nomenclatura fos l'actual. Així, ara hauria dit: "Déu va crear els nombres naturals; tota la resta és obra de l'home". La entjeroj (aŭ plenaj nombroj) konsistas el la naturaj nombroj (1, 2, 3, …), la respondaj negativaj nombroj (−1, −2, −3, …) kaj 0 (nulo). Matematikistoj kutime signas la aron de la entjeroj per aŭ Z. La naturaj nombroj estas subaro de la entjeroj, kion oni signas per ⊂ . La nocio de negativa nombro aperis pro la bezonoj de evoluo de algebro, kiu donis komunajn principojn solvi aritmetikajn problemojn, sendepende de ilia konkreta enhavo kaj valoroj de originaj nombrosignifoj. Probable, negativa respondo povas esti komprenita kiel grando de inversa direkto, ekz. movo en iu aŭ en ĝia inversa direkto, posedi havaĵon aŭ havi ŝuldon, ktp. Ankoraŭ en 6-11 jarcentoj en Hindio, oni regule uzis negativajn nombrojn ĝuste en tiu senco, kion ili havas en nuntempo. Sed en Eŭropa scienco ĝi eniris difinitive nur de tempoj de Kartezio (17 jc), kiu donis al negativaj nombroj la signifon de direktitaj eltranĉoj. Entjeroj povas esti paraj kaj neparaj. Paraj nombroj oni konsideras tiujn, kiuj dividiĝas je la nombro 2 sen resto, ekz. −4, −2, 0, 2, 4; la aliajn nombrojn el la nefinia vico de entjeroj oni nomas neparaj nombroj, ekz-e −5, −3, −1, 1, 3, 5. Por konstati la parecon de grandaj nombroj, ni rigardas al la fina cifero; se ĝi estas 0, 2, 4, 6, 8 (ekz. 3843924), ĝi estas para nombro, kontraŭokaze (1991) ĝi estas nepara. La entjero m estas nomata divizoro de la entjero n, se la kvociento de n per m ankaŭ estas entjero. Ekz-e 3 estas divizoro de 9, kaj 1, 2, 3, 4, 6, 12 estas ĉiuj pozitivaj divizoroj de 12. Alia grava koncepto estas plej granda komuna divizoro (pgkd) de du nombroj. Ekzemple pgkd(12,16)=4, ĉar 4 estas la plej granda entjero, kiu estas kaj divizoro de 12 kaj divizoro de 16. Heltalen är unionen av mängden naturliga tal {0, 1, 2, ...} och mängden negativa heltal {-1, -2, -3, ...}. Mängden av hela tal betecknas med den dubbelstrukna bokstaven ℤ (ibland fetstilta bokstaven Z), från det tyska ordet Zahlen (tal). Ibland definierar man delmängder av ℤ: ℤ+, ℤ* och ℤ–. * ℤ+ är 1, 2, 3, 4, 5 ... * ℤ* är 0, 1, 2, 3, 4, 5 ... * ℤ– är ... -5, -4, -3, -2, -1 Beroende på definition kan endera ℤ+ eller ℤ* vara detsamma som mängden naturliga tal. Mängden av hela tal är uppräkneligt oändlig och har kardinaltalet Alef-noll. Den är också en delmängd av mängden av rationella tal som i sin tur är en delmängd av mängden av reella tal som är en delmängd av mängden komplexa tal. När det gäller datorsystem används termen heltal (de hela talen) som distinktion till flyttal (de reella talen) eftersom de i datorer hanteras, beräknas och lagras olika. De naturliga talen är de heltal som är icke-negativa {0, 1, 2, 3, 4, …}, alternativt de heltal som är positiva {1, 2, 3, 4, …}. Den första definitionen är vanlig i Sverige och allmänt i matematisk logik, mängdlära och beräkningsvetenskap, medan den senare kan hittas i bland annat amerikansk litteratur och bland talteoretiker. Mängden av de naturliga talen betecknas ℕ (ett vanligt N i fetstil kan även användas). ℕ är diskret, uppräkneligt oändlig och har kardinalitet Alef-noll (ℵ₀). Enligt den definition som görs i Matematikterminologi i skolan, utgiven av Statens skolverk i Sverige, ingår talet 0 bland de naturliga talen. Konventionen att räkna 0 bland de naturliga talen förekom inte alls före 1800-talet och tillämpas inte av alla matematiker. Den infördes i samband med att de naturliga talen gavs en mängdteoretisk definition, enligt vilken de naturliga talen precis motsvarar kardinaltalen för ändliga mängder och 0 måste användas som kardinaltal för den tomma mängden. En fördel med att inkludera 0 är att de naturliga talen då utgör en monoid under både addition och multiplikation. En nackdel är att man inom talteori måste göra undantag för 0 i samband med primtalsfaktorisering, då 0 inte kan primtalsfaktoriseras (1 kan faktoriseras som den tomma produkten). För att undvika förvirring kan ℤ+ användas för att beteckna de positiva heltalen, och ℕ0 för de icke-negativa. Dalam matematika, terdapat dua kesepakatan mengenai himpunan bilangan asli. Yang pertama definisi menurut matematikawan tradisional, yaitu himpunan bilangan bulat positif yang bukan nol {1, 2, 3, 4, ...}. Sedangkan yang kedua definisi oleh logikawan dan ilmuwan komputer, adalah himpunan nol dan bilangan bulat positif {0, 1, 2, 3, ...}. Bilangan asli merupakan salah satu konsep matematika yg paling sederhana dan termasuk konsep pertama yang bisa dipelajari dan dimengerti oleh manusia, bahkan beberapa penelitian menunjukkan beberapa jenis kera juga bisa menangkapnya. Wajar apabila bilangan asli adalah jenis pertama dari bilangan yang digunakan untuk membilang, menghitung, dsb. Sifat yang lebih dalam tentang bilangan asli, termasuk kaitannya dengan bilangan prima, dipelajari dalam teori bilangan. Untuk matematika lanjut, bilangan asli dapat dipakai untuk mengurutkan dan mendefinisikan sifat suatu himpunan. Setiap bilangan, misalnya bilangan 1, adalah konsep abstrak yg tak bisa tertangkap oleh indra manusia, tetapi bersifat universal. Salah satu cara memperkenalkan konsep himpunan semua bilangan asli sebagai sebuah struktur abstrak adalah melalui aksioma Peano (sebagai ilustrasi, lihat aritmetika Peano Diarsipkan 2007-08-19 di Wayback Machine.). Konsep bilangan-bilangan yg lebih umum dan lebih luas memerlukan pembahasan lebih jauh, bahkan kadang-kadang memerlukan kedalaman logika untuk bisa memahami dan mendefinisikannya. Misalnya dalam teori matematika, himpunan semua bilangan rasional bisa dibangun secara bertahap, diawali dari himpunan bilangan-bilangan asli. Um número natural é um número inteiro não negativo Em alguns contextos, número natural é definido como um número inteiro positivo, sendo também o zero considerado como um número natural (mesmo não sendo positivo ): O conjunto dos números naturais é, comumente, denotado pelo símbolo O símbolo é usado para explicitar que o zero não está sendo incluso, i.e. Os usos mais comuns dos números naturais são a contagem, a ordenação e a codificação. Propriedades dos números naturais como, por exemplo, divisibilidade e a distribuição dos números primos, são estudadas na teoria dos números. Propriedades que dizem respeito a contagens e combinações são estudadas pela combinatória. Uma construção do conjunto dos números naturais que não depende do conjunto dos números inteiros foi desenvolvida por Giuseppe Peano no século XIX e costuma ser chamada de Axiomática de Peano. Liczby naturalne – liczby służące podawaniu liczności (trzy osoby, zob. liczebnik główny/kardynalny) i ustalania kolejności (trzecia osoba, zob. liczebnik porządkowy), poddane w matematyce dalszym uogólnieniom (odpowiednio: liczby kardynalne, liczby porządkowe). Badaniem własności liczb naturalnych zajmują się arytmetyka i teoria liczb. W matematyce określenie liczby naturalne oznacza na ogół liczby całkowite dodatnie. To, czy zero jest liczbą naturalną, jest kwestią umowy. W matematyce nie przyjęto ogólnie żadnej konwencji dotyczącej przynależności zera lub jej braku do liczb naturalnych. Interesujące, że z punktu widzenia matematyki obie definicje można uważać w gruncie rzeczy za równoważne. O konkretnym stanowisku decydują często takie sytuacje jak: uproszczenie zapisu pewnych symboli, ograniczenie przypadków szczególnych itp. Is iad na slánuimhir na uimhreacha aiceanta {0, 1, 2, 3, 4, ... } mar aon leis an dhiúl do na huimhreacha aiceanta dheimhneach { -1, -2, -3, -4, ...}. Is fo-thacar iad na slánuimhir {..., -2, -1, 0, 1, 2, ...} do na réaduimhir. Is tacar éigríochta iad na slánuimhir freisin. Úsáidtear an siombail chun an tacar iomlán do uimhreacha aiceanta a léiriú: Úsáidtear an focal Slánuimhir i ríomhchlárú freisin. Is cineál sonraí bunúsach é slánuimhir. Chuirtear é in iúil le 'int' i roinnt ríomhchlár, Java ina measc. San cás seo, ní tacar éigríochta atá i gceist san gcás seo. Ці́лі чи́сла — в математиці елементи множини, яка утворюється замиканням натуральних чисел відносно віднімання. Таким чином, цілі числа замкнуті відносно додавання, віднімання та множення. Необхідність розгляду цілих чисел викликана неможливістю в загальному випадку відняти від одного натурального числа інше — можна віднімати тільки менше число від більшого. Введення нуля і від’ємних чисел робить віднімання такою ж повноцінною операцією, як додавання. Множина цілих чисел складається з * множини натуральних чисел , * нуля — розв'язку рівняння , * множини від'ємних чисел — множини розв'язків усіх рівнянь виду . Для позначення множини цілих чисел використовується символ ℤ, який може в різних авторів використовуватися для позначення групи множин: ℤ+, ℤ+ або ℤ> для позначення додатних цілих чисел, ℤ≥ для не від'ємних цілих чисел, ℤ≠ для всіх цілих чисел крім нуля. Деякі автори використовують позначення ℤ* для всіх цілих чисел крім нуля, інші для позначення не від'ємних цілих чисел, або для {–1, 1}. Дійсне число є цілим, якщо його десяткове подання не містить дробової частини (але може містити знак). Приклади дійсних чисел: Числа 142857; 0; -273 є цілими.Числа 5½; 9,75 не є цілими. Natura nombro estas unu el la pozitivaj entjeroj (1, 2, 3, 4 ...). Kelkfoje ankaŭ la nombron 0 (nul aŭ nulo) oni kalkulas inter naturaj nombroj. Naturaj nombroj havas du ĉefajn uzojn: Oni uzas ilin por nombri objektojn (ekz-e, "estas tri pomoj sur la tablo") aŭ por ordigi objektojn (ekz-e "ĝi estas la trie plej granda urbo en la lando"). En la dua signifo ili estas nomataj vicmontraj nombroj aŭ numeroj. La simbolo por la aro de ĉiuj naturaj nombroj estas aŭ . In matematica i numeri naturali sono quei numeri usati per contare e ordinare. Nel linguaggio comune i "numeri cardinali" sono quelli usati per contare e i "numeri ordinali" sono quelli usati per ordinare. I numeri naturali corrispondono all'insieme {0, 1, 2, 3, 4, …}. Essi vengono fatti corrispondere biunivocamente all'insieme dei numeri interi non negativi {0, +1, +2, +3, +4, …}. Talvolta vengono usati anche per indicare l'insieme dei numeri interi positivi {1, 2, 3, 4, …}. ( 자연수(自然水)는 자연에서 나는 물을 가리키기도 합니다.) 수학에서 자연수(自然數, 영어: natural number)는 수를 셀 때나 순서를 매길 때 사용되는 수다. 양의 정수(陽-整數, 영어: positive integer) 1, 2, 3, ...로 정의되거나, 음이 아닌 정수(陰-整數, 영어: non-negative integer) 0, 1, 2, 3, ...로 정의된다. 범자연수(汎自然數, 문화어: 옹근수(-數), 완수(完數), 영어: whole number)라는 용어는 첫째 정의를 택할 경우에 음이 아닌 정수를 가리키는 데 사용되며, 이에 대응하는 문화어와 영어는 둘째 정의를 택할 경우에 정수를 가리키는 데 사용된다. 자연수의 집합은 대문자 N을 써서 표기하며, 보통 칠판 볼드체 ℕ를 사용한다. 약수 관계나 소수 분포를 비롯한 자연수의 성질들은 수론의 연구 대상이며, 분할이나 를 비롯한 자연수의 문제들은 조합론의 연구 대상이다. 자연수는 많은 연산에 대하여 닫혀있지 않다. 정수는 자연수를 뺄셈에 대하여 닫혀있도록 확장하여 얻는 수 체계이며, 유리수는 자연수를 추가로 나눗셈에 대하여 닫혀있도록 확장한 수 체계이다. 실수는 추가로 코시 수열의 극한에 대하여 닫혀있도록 확장한 것이며, 복소수는 추가로 다항식의 근에 대하여 닫혀있도록 확장한 것이다. 하나하나가 유한하지만, 무한 집합을 이룬다. 자연수의 집합은 "가장 작은 크기"의 무한 집합이며, 자연수와 크기가 같은 집합을 가산 무한 집합이라고 한다. 자연수가 만족시켜야 하는 일련의 공리들을 제시하여 자연수를 일종의 무정의 개념으로 간주할 수 있으며, 이러한 자연수의 공리들이 이루는 체계 가운데 가장 자주 사용되는 하나는 페아노 공리계이다. 수리논리학에서 이는 자연수의 이론에 해당된다. 자연수를 특별한 집합으로서 간주하여 다룰 수도 있는데, 이 경우 보통 자연수의 집합은 최소 재귀 집합으로 정의된다. 수리논리학에서 이는 자연수의 모형에 해당된다. 자연수의 수를 세는 역할을 일반화하면 기수의 개념을 얻으며, 자연수의 순서를 매기는 기능을 일반화하면 순서수의 개념을 얻는다. 자연수의 집합의 대수적 성질을 일반화하면 반환의 개념을 얻는다. 특히 자연수는 많은 스포츠 점수 같은 경기나 게임에 사용될수 있으며 우리가 가장 흔히 보는 수로도 볼 수 있다. Celá čísla se skládají z přirozených čísel (1, 2, 3, …), nuly (0) a záporných celých čísel (−1, −2, −3, …). Množina celých čísel se v matematice většinou označuje Z, nebo , podle Zahlen (německy čísla). Podobně jako přirozená čísla, tvoří celá čísla nekonečnou spočetnou množinu. Studiem celých čísel se zabývá teorie čísel. Liczby całkowite – liczby naturalne oraz liczby przeciwne do nich a także liczba zero. Są uogólnieniem zbioru liczb naturalnych na zbiór, w którym wykonalne jest odejmowanie. Uogólnieniem liczb całkowitych są liczby wymierne. Zbiór liczb całkowitych oznaczamy w matematyce symbolem (od niem. Zahlen – liczby). W Polsce w większości szkół podstawowych i średnich, w celu ułatwienia skojarzenia z polską nazwą, stosuje się symbol przy czym MEN zaleca używanie oznaczenia . En mathématiques, un entier relatif, un entier rationnel ou simplement un nombre entier est un nombre qui se présente comme un entier naturel auquel on a adjoint un signe positif ou négatif indiquant sa position par rapport à 0 sur un axe orienté. Les entiers positifs (supérieurs à zéro) s'identifient aux entiers naturels : 0, 1, 2, 3… tandis que les entiers négatifs sont leurs opposés : 0, −1, −2, −3… L'entier 0 lui-même est donc le seul nombre à la fois positif et négatif. Un nombre réel est entier s'il est sans partie fractionnaire, c'est-à-dire si son écriture décimale ne comprend pas de chiffre (autre que zéro) « après la virgule ». Les entiers relatifs permettent d'exprimer la différence de deux entiers naturels quelconques. Entre autres significations de la différence, on peut citer la position sur un axe orienté par rapport à un point de référence (un axe à positions discrètes, c'est-à-dire discontinues) ; le déplacement depuis une position d'origine, dans un sens ou dans l'autre ; ou encore la variation d'une valeur entière, donc comptée en unités (variation positive pour un gain, négative pour une perte). L'ensemble des entiers relatifs est noté « Z », lettre capitale grasse dans les textes dactylographiés, peu à peu supplantée par la graphie manuscrite avec une barre oblique ajourée : « ℤ ».La présence d'un astérisque en exposant (« Z* ») désigne en général l'ensemble des entiers relatifs non nuls, même si cette notation est utilisée parfois pour l'ensemble des éléments inversibles de Z[réf. souhaitée], c'est-à-dire la paire d'entiers {−1, 1}.La notation « Z− » désigne l'ensemble des entiers négatifs. Il est plus rare de trouver la notation « Z+ », remplacée par la notation « N » des entiers naturels par identification. Cet ensemble est (totalement) ordonné pour la relation de comparaison usuelle héritée des entiers naturels. Il est aussi muni des opérations d'addition et de multiplication qui fondent la notion d'anneau en algèbre. Les entiers relatifs sont aussi appelés entiers rationnels, appellation qui ne doit pas entraîner de confusion avec les nombres rationnels ou fractions. Cette dénomination vient de l'anglais rational integer, et désigne un cas particulier d'entiers algébriques, construit sur le corps de nombres des rationnels. On trouve cette appellation chez Nicolas Bourbaki et certains mathématiciens s'inscrivant dans le mouvement des mathématiques modernes, parmi lesquels Georges Papy. Натурáльные чи́сла (от лат. naturalis «естественный») — числа, возникающие естественным образом при счёте (1, 2, 3, 4, 5, 6, 7 и так далее). Последовательность всех натуральных чисел, расположенных в порядке возрастания, называется натуральным рядом. Множество натуральных чисел является бесконечным, так как для любого натурального числа найдётся натуральное число, большее чем . Натуральные числа ещё можно называть целыми положительными числами. Поэтому Отрицательные и нецелые (дробные) числа к натуральным не относятся. Свойства натуральных чисел и операций с ними изучают арифметика и (более углублённо) теория чисел. Ακέραιοι ονομάζονται όλοι οι φυσικοί αριθμοί μαζί με τους αντίθετους τους και το μηδέν. Το σύνολο των ακεραίων δηλαδή το σύνολο: Συμβολίζεται με το γράμμα , αρχικό της λέξης Zahl που στα γερμανικά σημαίνει αριθμός. Το σύνολο ορίζεται επίσης ως εξής: Όπως και το σύνολο των φυσικών, το σύνολο των ακεραίων είναι άπειρο αριθμήσιμο με πληθάριθμο (άλεφ-μηδέν). 数学における整数（せいすう、英: integer, whole number, 独: Ganze Zahl, 仏: nombre entier, 西: número entero）は、1 とそれに 1 ずつ加えて得られる自然数 (1, 2, 3, 4, …) 、これらに−1を乗じて得られる負数 (−1, −2, −3, −4, …) 、および 0 の総称である。整数の全体からなる集合は、一般に太字のまたは黒板太字ので表す。これはドイツ語 Zahlen（「数」の意・複数形）に由来する。抽象代数学、特に代数的整数論では、しばしば「代数体の整数環」の元という意味で代数的整数あるいは「整数」という言葉を用いる。有理数全体の成す体はそれ自身が代数体の最も簡単な例であり、有理数体の代数体としての整数環すなわち、「有理数の中で整なもの」の全体の成す環は、本項でいう意味での整数全体の成す環である。一般の「整数」との区別のためにここでいう意味の整数を有理整数 (rational integer) と呼ぶことがある。
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Subject Item: dbr:Siteswap
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Subject Item: dbr:Range_coding
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Subject Item: dbr:Rational_reconstruction_(mathematics)
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Subject Item: dbr:Scanf_format_string
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Subject Item: dbr:XSLT_elements
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Subject Item: dbr:Squaring_the_square
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Subject Item: dbr:T1_space
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Subject Item: dbr:Tarski's_high_school_algebra_problem
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Subject Item: dbr:Zero-product_property
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Subject Item: dbr:Subgroup_series
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Subject Item: dbr:Wave_height
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Subject Item: dbr:Zeisel_number
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Subject Item: dbr:Riemann_form
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Subject Item: dbr:Ring_of_mixed_characteristic
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Subject Item: dbr:Risk_register
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Subject Item: dbr:TI-57
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Subject Item: dbr:Tacnode
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Subject Item: dbr:Stirling_polynomials
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Subject Item: dbr:Transcendental_equation
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Subject Item: dbr:Steenrod_problem
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Subject Item: dbr:Intger
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Subject Item: dbr:Spline_wavelet
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Subject Item: dbr:Standard_Commands_for_Programmable_Instruments
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Subject Item: dbr:Three-way_comparison
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Subject Item: dbr:Ring_of_rational_integers
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Subject Item: dbr:Set_of_integers
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Subject Item: dbr:Signed_Numbers
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Subject Item: dbr:Construction_of_the_integers
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Subject Item: dbr:Integar
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Subject Item: dbr:Integer-valued
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Subject Item: dbr:Integer_number
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Subject Item: dbr:Integer_numbers
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Subject Item: dbr:Integer_value
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Subject Item: dbr:Intergar
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Subject Item: dbr:Negative_integer
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Subject Item: dbr:Z_(set)
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Subject Item: wikipedia-en:Integer
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