Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
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An integer is the number zero (0), a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .).^[1] The negations or additive inverses of the positive natural numbers are referred to as negative integers.^[2] The set of all integers is often denoted by the boldface Z or blackboard bold ${\displaystyle \mathbb {Z} }$.^[3][4]

The set of natural numbers ${\displaystyle \mathbb {N} }$ is a subset of ${\displaystyle \mathbb {Z} }$, which in turn is a subset of the set of all rational numbers ${\displaystyle \mathbb {Q} }$, itself a subset of the real numbers ${\displaystyle \mathbb {R} }$.^[a] Like the set of natural numbers, the set of integers ${\displaystyle \mathbb {Z} }$ 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, 5/4 and √2 are not.^[8]

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.

History

The word integer comes from the Latin integer meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via the French word entier, which means both entire and integer.^[9] Historically the term was used for a number that was a multiple of 1,^[10][11] or to the whole part of a mixed number.^[12][13] Only positive integers were considered, making the term synonymous with the natural numbers. The definition of integer expanded over time to include negative numbers as their usefulness was recognized.^[14] For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers.^[15]

The phrase the set of the integers was not used before the end of the 19th century, when Georg Cantor introduced the concept of infinite sets and set theory. The use of the letter Z to denote the set of integers comes from the German word Zahlen ("numbers")^[3][4] and has been attributed to David Hilbert.^[16] The earliest known use of the notation in a textbook occurs in Algèbre written by the collective Nicolas Bourbaki, dating to 1947.^[3][17] The notation was not adopted immediately, for example another textbook used the letter J^[18] and a 1960 paper used Z to denote the non-negative integers.^[19] But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.^[20]

The symbol ${\displaystyle \mathbb {Z} }$ is often annotated to denote various sets, with varying usage amongst different authors: ${\displaystyle \mathbb {Z} ^{+}}$,${\displaystyle \mathbb {Z} _{+}}$ or ${\displaystyle \mathbb {Z} ^{>}}$ for the positive integers, ${\displaystyle \mathbb {Z} ^{0+}}$ or ${\displaystyle \mathbb {Z} ^{\geq }}$ for non-negative integers, and ${\displaystyle \mathbb {Z} ^{\neq }}$ for non-zero integers. Some authors use ${\displaystyle \mathbb {Z} ^{*}}$ for non-zero integers, while others use it for non-negative integers, or for {–1, 1} (the group of units of ${\displaystyle \mathbb {Z} }$). Additionally, ${\displaystyle \mathbb {Z} _{p}}$ is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p-adic integers.^[21][22]

The whole numbers were synonymous with the integers up until the early 1950s.^[23][24][25] In the late 1950s, as part of the New Math movement,^[26] American elementary school teachers began teaching that whole numbers referred to the natural numbers, excluding negative numbers, while integer included the negative numbers.^[27][28] The whole numbers remain ambiguous to the present day.^[29]

Algebraic properties

Algebraic structure → Ring theory Ring theory
Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring ${\displaystyle \mathbb {Z} }$ • Terminal ring ${\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring ${\displaystyle \mathbb {Z} /1\mathbb {Z} }$ • Integers modulo pn ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ • Prüfer p-ring ${\displaystyle \mathbb {Z} (p^{\infty })}$ • Base-p circle ring ${\displaystyle \mathbb {T} }$ • Base-p integers ${\displaystyle \mathbb {Z} }$ • p-adic rationals ${\displaystyle \mathbb {Z} }$ • Base-p real numbers ${\displaystyle \mathbb {R} }$ • p-adic integers ${\displaystyle \mathbb {Z} _{p}}$ • p-adic numbers ${\displaystyle \mathbb {Q} _{p}}$ • p-adic solenoid ${\displaystyle \mathbb {T} _{p}}$ Algebraic geometry • Affine variety
Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
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Like the natural numbers, ${\displaystyle \mathbb {Z} }$ is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, 0), ${\displaystyle \mathbb {Z} }$, unlike the natural numbers, is also closed under subtraction.^[30]

The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring ${\displaystyle \mathbb {Z} }$.

${\displaystyle \mathbb {Z} }$ is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).

The following table lists some of the basic properties of addition and multiplication for any integers a, b and c:

Zdroj:https://en.wikipedia.org?pojem=Integers
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Properties of addition and multiplication on integers