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# Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch"). 

111 relations: Abelian group, Absolute value, Abstract algebra, Additive identity, Additive inverse, Aleph number, Algebraic expression, Algebraic integer, Algebraic number field, Algebraic number theory, Algebraic structure, ALGOL 68, Arbitrary-precision arithmetic, Associative property, Automated theorem proving, −1, Bijection, Binary operation, Blackboard bold, C (programming language), Cardinality, Category of rings, Closure (mathematics), Commutative property, Commutative ring, Computer language, Congruence relation, Countable set, Cyclic group, Data type, Discrete valuation ring, Distributive property, Division (mathematics), Embedding, Emphasis (typography), Equality (mathematics), Equivalence class, Equivalence relation, Eric Temple Bell, Euclidean algorithm, Euclidean division, Euclidean domain, Exponentiation, Field (mathematics), Field of fractions, Floor and ceiling functions, Fraction (mathematics), French language, Fundamental theorem of arithmetic, Garrett Birkhoff, ... Expand index (61 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

## Aleph number

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

## Algebraic expression

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).

## Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).

## Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

## Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

## Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

## ALGOL 68

ALGOL 68 (short for Algorithmic Language 1968) is an imperative computer programming language that was conceived as a successor to the ALGOL 60 programming language, designed with the goal of a much wider scope of application and more rigorously defined syntax and semantics.

## Arbitrary-precision arithmetic

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Automated theorem proving

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs.

## −1

In mathematics, −1 is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0.

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## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Binary operation

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

## Blackboard bold

Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.

## C (programming language)

C (as in the letter ''c'') is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).

## Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

## Computer language

A computer language is a method of communication with a computer.

## Congruence relation

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

## Data type

In computer science and computer programming, a data type or simply type is a classification of data which tells the compiler or interpreter how the programmer intends to use the data.

## Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

## Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

## Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

## Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

## Emphasis (typography)

In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them.

## Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

## Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Eric Temple Bell

Eric Temple Bell (February 7, 1883 &ndash; December 21, 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life.

## Euclidean algorithm

. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.

## Euclidean division

In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.

## Euclidean domain

In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers.

## Exponentiation

Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Field of fractions

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.

## Floor and ceiling functions

In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).

## Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

## French language

French (le français or la langue française) is a Romance language of the Indo-European family.

## Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

## Garrett Birkhoff

Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician.

## German language

German (Deutsch) is a West Germanic language that is mainly spoken in Central Europe.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

## Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

## Hyperinteger

In non-standard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part.

## Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Integer lattice

In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted Zn, is the lattice in the Euclidean space Rn whose lattice points are ''n''-tuples of integers.

## Integer sequence

In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.

## Integer-valued function

In mathematics, an integer-valued function is a function whose values are integers.

## Integral domain

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

## Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

## Isabelle (proof assistant)

The Isabelle theorem prover is an interactive theorem prover, a Higher Order Logic (HOL) theorem prover.

## Java (programming language)

Java is a general-purpose computer-programming language that is concurrent, class-based, object-oriented, and specifically designed to have as few implementation dependencies as possible.

## Latin

Latin (Latin: lingua latīna) is a classical language belonging to the Italic branch of the Indo-European languages.

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## List of mathematical symbols

This is a list of symbols used in all branches of mathematics to express a formula or to represent a constant.

## Men of Mathematics

Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré is a book on the history of mathematics published in 1937 by Scottish-born American mathematician and science fiction writer E. T. Bell (1883–1960).

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

## Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

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## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.

## Number

A number is a mathematical object used to count, measure and also label.

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## Object Pascal

Object Pascal refers to a branch of object-oriented derivatives of Pascal, mostly known as the primary programming language of Delphi.

## On-Line Encyclopedia of Integer Sequences

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences.

## Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

## Ordered ring

In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R.

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.

## Profinite integer

In mathematics, a profinite integer is an element of the ring where p runs over all prime numbers, \mathbb_p is the ring of ''p''-adic integers and \widehat.

## Proof assistant

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Remainder

In mathematics, the remainder is the amount "left over" after performing some computation.

## Rewriting

In mathematics, computer science, and logic, rewriting covers a wide range of (potentially non-deterministic) methods of replacing subterms of a formula with other terms.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

## Ring of integers

In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.

## Saunders Mac Lane

Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

## Subring

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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## Subtraction

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Term algebra

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature.

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Two's complement

Two's complement is a mathematical operation on binary numbers, best known for its role in computing as a method of signed number representation.

## Unicode

Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems.

## Universal property

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.

## Universal quantification

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".

## Valuation ring

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x &minus;1 belongs to D. Given a field F, if D is a subring of F such that either x or x &minus;1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.

## Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

## Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

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## 0.999...

In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).

## 1

1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.

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## References

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