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56-561: The word integrity evolved from the Latin adjective integer , meaning whole or complete . In this context, integrity is the inner sense of "wholeness" deriving from qualities such as honesty and consistency of character . In ethics , a person is said to possess the virtue of integrity if the person's actions are based upon an internally consistent framework of principles. These principles should uniformly adhere to sound logical axioms or postulates. A person has ethical integrity to

112-445: A . To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule: precisely when Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using [( a , b )] to denote the equivalence class having ( a , b ) as a member, one has: The negation (or additive inverse) of an integer

168-520: A fractional component . For example, 21, 4, 0, and −2048 are integers, while 9.75, ⁠5 + 1 / 2 ⁠ , 5/4, 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 . The word integer comes from

224-415: A moral responsibility as well as a virtue. A person's value system provides a framework within which the person acts in ways that are consistent and expected. Integrity can be seen as the state of having such a framework and acting congruently within it. One essential aspect of a consistent framework is its avoidance of any unwarranted (arbitrary) exceptions for a particular person or group—especially

280-407: A UK Government agency, stated that they upheld a line of government policy in advance of the outcome of a study that they had commissioned. The concept of integrity may also feature in business contexts that go beyond the issues of employee/employer honesty and ethical behavior, notably in marketing or branding contexts. Brand "integrity" gives a company's brand a consistent, unambiguous position in

336-480: A consistent life in light of the truth that value judgments, including the command "Thou shalt not kill," are merely subjective assertions. Politicians are given power to make, execute, or control policy, which can have important consequences. They typically promise to exercise this power in a way that serves society, but may not do so, which opposes the notion of integrity. Aristotle said that because rulers have power they will be tempted to use it for personal gain. In

392-564: A crucial role in detecting people who have low integrity. Naive respondents really believe this pretense and behave accordingly, reporting some of their past deviance and their thoughts about the deviance of others, fearing that if they do not answer truthfully their untrue answers will reveal their "low integrity". These respondents believe that the more candid they are in their answers, the higher their "integrity score" will be. Disciplines and fields with an interest in integrity include philosophy of action , philosophy of medicine , mathematics ,

448-452: A disk drive to a computer display. Such integrity is a fundamental principle of information assurance . Corrupted information is untrustworthy ; uncorrupted information is of value. integer#Latin 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, . . .). The negations or additive inverses of

504-473: A few basic operations (e.g., zero , succ , pred ) and using natural numbers , which are assumed to be already constructed (using the Peano approach ). There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2), and the types of arguments accepted by these operations;

560-458: A finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1) . In fact, Z {\displaystyle \mathbb {Z} } under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to Z {\displaystyle \mathbb {Z} } . The first four properties listed above for multiplication say that Z {\displaystyle \mathbb {Z} } under multiplication

616-400: A set P − {\displaystyle P^{-}} which is disjoint from P {\displaystyle P} and in one-to-one correspondence with P {\displaystyle P} via a function ψ {\displaystyle \psi } . For example, take P − {\displaystyle P^{-}} to be

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672-461: A world in which everyone was a thief. The philosopher Immanuel Kant formally described the principle of universality of application for one's motives in his categorical imperative . The concept of integrity implies a wholeness—a comprehensive corpus of beliefs often referred to as a worldview . This concept of wholeness emphasizes honesty and authenticity , requiring that one act at all times in accordance with one's worldview. Ethical integrity

728-433: Is a commutative monoid . However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that Z {\displaystyle \mathbb {Z} } under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that Z {\displaystyle \mathbb {Z} } together with addition and multiplication

784-422: Is a commutative ring with unity . It is the prototype of all objects of such algebraic structure . Only those equalities of expressions are true in  Z {\displaystyle \mathbb {Z} } for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings. The lack of zero divisors in the integers (last property in

840-422: Is a subset of the set of all rational numbers Q {\displaystyle \mathbb {Q} } , itself a subset of the real numbers R {\displaystyle \mathbb {R} } . Like the set of natural numbers, the set of integers Z {\displaystyle \mathbb {Z} } is countably infinite . An integer may be regarded as a real number that can be written without

896-535: Is an area in philosophy concerned with theories about the processes causing willful human bodily movements of a more or less complex kind. This area of thought involves epistemology , ethics , metaphysics , jurisprudence , and philosophy of mind , and has attracted the strong interest of philosophers ever since Aristotle 's Nicomachean Ethics (Third Book). With the advent of psychology and later neuroscience , many theories of action are now subject to empirical testing . Philosophical action theory, or

952-469: Is called the quotient and r is called the remainder of the division of a by b . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. The above says that Z {\displaystyle \mathbb {Z} } is a Euclidean domain . This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain , and any positive integer can be written as

1008-475: Is equivalent to the statement that any Noetherian valuation ring is either a field —or a discrete valuation ring . In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, zero , and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the Peano axioms , call this P {\displaystyle P} . Then construct

1064-416: Is generally accepted as moral, what others think, but primarily with what is ethical, what politicians should do based on reasonable arguments. Important virtues of politicians are faithfulness, humility, and accountability. Furthermore, they should be authentic and a role model. Aristotle identified dignity ( megalopsychia , variously translated as proper pride, greatness of soul, and magnanimity) as

1120-468: Is greater than zero , and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: Thus it follows that Z {\displaystyle \mathbb {Z} } together with the above ordering is an ordered ring . The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered . This

1176-401: Is identified with the class [( n ,0)] (i.e., the natural numbers are embedded into the integers by map sending n to [( n ,0)] ), and the class [(0, n )] is denoted − n (this covers all remaining classes, and gives the class [(0,0)] a second time since –0 = 0. Thus, [( a , b )] is denoted by If the natural numbers are identified with the corresponding integers (using

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1232-401: Is not a one-dimensional concept. In his book he presents a multifaceted perspective of integrity. Integrity relates, for example, to compliance to the rules as well as to social expectations, to morality as well as to ethics, and to actions as well as to attitude. Electronic signals are said to have integrity when there is no corruption of information between one domain and another, such as from

1288-437: Is not defined on Z {\displaystyle \mathbb {Z} } , the division "with remainder" is defined on them. It is called Euclidean division , and possesses the following important property: given two integers a and b with b ≠ 0 , there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b | , where | b | denotes the absolute value of b . The integer q

1344-425: Is not free since the integer 0 can be written pair (0,0), or pair (1,1), or pair (2,2), etc.. This technique of construction is used by the proof assistant Isabelle ; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers. Philosophy of action Action theory or theory of action

1400-424: Is not synonymous with the good, as Zuckert and Zuckert show about Ted Bundy : When caught, he defended his actions in terms of the fact-value distinction . He scoffed at those, like the professors from whom he learned the fact-value distinction, who still lived their lives as if there were truth-value to value claims. He thought they were fools and that he was one of the few who had the courage and integrity to live

1456-431: Is obtained by reversing the order of the pair: Hence subtraction can be defined as the addition of the additive inverse: The standard ordering on the integers is given by: It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form ( n ,0) or (0, n ) (or both at once). The natural number n

1512-508: 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 . The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that whole numbers referred to the natural numbers , excluding negative numbers, while integer included

1568-523: 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 . Historically the term was used for a number that was a multiple of 1, or to the whole part of a mixed number . Only positive integers were considered, making the term synonymous with

1624-512: The efficient cause , the agent, and the final cause , the intention. In some theories a desire plus a belief about the means of satisfying that desire are always what is behind an action. Agents aim, in acting, to maximize the satisfaction of their desires. Such a theory of prospective rationality underlies much of economics and other social sciences within the more sophisticated framework of rational choice . However, many theories of action argue that rationality extends far beyond calculating

1680-441: The mind , cognition , consciousness , materials science , structural engineering , and politics . Popular psychology identifies personal integrity, professional integrity, artistic integrity, and intellectual integrity. For example, to behave with scientific integrity, a scientific investigation shouldn't determine the outcome in advance of the actual results. As an example of a breach of this principle, Public Health England ,

1736-443: The natural numbers . The definition of integer expanded over time to include negative numbers as their usefulness was recognized. For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers. 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

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1792-426: The ordered pairs ( 1 , n ) {\displaystyle (1,n)} with the mapping ψ = n ↦ ( 1 , n ) {\displaystyle \psi =n\mapsto (1,n)} . Finally let 0 be some object not in P {\displaystyle P} or P − {\displaystyle P^{-}} , for example the ordered pair (0,0). Then

1848-433: The philosophy of action , should not be confused with sociological theories of social action , such as the action theory established by Talcott Parsons . Nor should it be confused with activity theory . Basic action theory typically describes action as intentional behavior caused by an agent in a particular situation . The agent's desires and beliefs (e.g. a person wanting a glass of water and believing that

1904-605: The Normative Phenomenon of Morality, Ethics, and Legality". The authors model integrity as the state of being whole and complete, unbroken, unimpaired, sound, and in perfect condition. They posit a model of integrity that provides access to increased performance for individuals, groups, organizations, and societies. Their model "reveals the causal link between integrity and increased performance, quality of life, and value-creation for all entities, and provides access to that causal link." According to Muel Kaptein, integrity

1960-417: The best means to achieve one's ends. For instance, a belief that I ought to do X, in some theories, can directly cause me to do X without my having to want to do X (i.e. have a desire to do X). Rationality, in such theories, also involves responding correctly to the reasons an agent perceives, not just acting on wants. While action theorists generally employ the language of causality in their theories of what

2016-460: The book The Servant of the People , Muel Kaptein says integrity should start with politicians knowing what their position entails, because the consistency required by integrity applies also to the consequences of one's position. Integrity also demands knowledge and compliance with both the letter and the spirit of the written and unwritten rules. Integrity is also acting consistently not only with what

2072-448: The clear liquid in the cup in front of them is water) lead to bodily behavior (e.g. reaching across for the glass). In the simple theory (see Donald Davidson ), the desire and belief jointly cause the action. Michael Bratman has raised problems for such a view and argued that we should take the concept of intention as basic and not analyzable into beliefs and desires. Aristotle held that a thorough explanation must give an account of both

2128-544: The crown of the virtues, distinguishing it from vanity, temperance, and humility. "Integrity tests" or (more confrontationally) "honesty tests" aim to identify prospective employees who may hide perceived negative or derogatory aspects of their past, such as a criminal conviction or drug abuse. Identifying unsuitable candidates can save the employer from problems that might otherwise arise during their term of employment. Integrity tests make certain assumptions, specifically: The claim that such tests can detect "fake" answers plays

2184-407: The embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {..., −2, −1, 0, 1, 2, ...} . Some examples are: In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines . Integers are represented as algebraic terms built using

2240-407: The extent that the person's actions, beliefs, methods, measures, and principles align with a well-integrated core group of values . A person must, therefore, be flexible and willing to adjust these values to maintain consistency when these values are challenged—such as when observed results are incongruous with expected outcomes. Because such flexibility is a form of accountability , it is regarded as

2296-1179: The integers are defined to be the union P ∪ P − ∪ { 0 } {\displaystyle P\cup P^{-}\cup \{0\}} . The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows: − x = { ψ ( x ) , if  x ∈ P ψ − 1 ( x ) , if  x ∈ P − 0 , if  x = 0 {\displaystyle -x={\begin{cases}\psi (x),&{\text{if }}x\in P\\\psi ^{-1}(x),&{\text{if }}x\in P^{-}\\0,&{\text{if }}x=0\end{cases}}} The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey

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2352-458: 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 : The first five properties listed above for addition say that Z {\displaystyle \mathbb {Z} } , under addition, is an abelian group . It is also a cyclic group , since every non-zero integer can be written as

2408-448: The integers as a subring is the field of rational numbers . The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes Z {\displaystyle \mathbb {Z} } as its subring . Although ordinary division

2464-447: The integers into this ring. This universal property , namely to be an initial object in the category of rings , characterizes the ring  Z {\displaystyle \mathbb {Z} } . 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 ,

2520-510: The letter Z to denote the set of integers comes from the German word Zahlen ("numbers") and has been attributed to David Hilbert . The earliest known use of the notation in a textbook occurs in Algèbre written by the collective Nicolas Bourbaki , dating to 1947. The notation was not adopted immediately. For example, another textbook used the letter J, and a 1960 paper used Z to denote

2576-510: The mind of their audience. This is established for example via consistent messaging and a set of graphics standards to maintain visual integrity in marketing communications . Kaptein and Wempe developed a theory of corporate integrity that includes criteria for businesses dealing with moral dilemmas. Another use of the term "integrity" appears in Michael Jensen 's and Werner Erhard 's paper, "Integrity: A Positive Model that Incorporates

2632-437: The negative numbers. The whole numbers remain ambiguous to the present day. Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra Like the natural numbers , Z {\displaystyle \mathbb {Z} } is closed under the operations of addition and multiplication , that is,

2688-544: The non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers. The symbol Z {\displaystyle \mathbb {Z} } is often annotated to denote various sets, with varying usage amongst different authors: Z + {\displaystyle \mathbb {Z} ^{+}} , Z + {\displaystyle \mathbb {Z} _{+}} , or Z > {\displaystyle \mathbb {Z} ^{>}} for

2744-430: The person or group that holds the framework. In law, this principle of universal application requires that even those in positions of official power can be subjected to the same laws as pertain to their fellow citizens. In personal ethics, this principle requires that one should not act according to any rule that one would not wish to see universally followed. For example, one should not steal unless one would want to live in

2800-702: The positive integers, Z 0 + {\displaystyle \mathbb {Z} ^{0+}} or Z ≥ {\displaystyle \mathbb {Z} ^{\geq }} for non-negative integers, and Z ≠ {\displaystyle \mathbb {Z} ^{\neq }} for non-zero integers. Some authors use Z ∗ {\displaystyle \mathbb {Z} ^{*}} for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the group of units of Z {\displaystyle \mathbb {Z} } ). Additionally, Z p {\displaystyle \mathbb {Z} _{p}}

2856-405: The positive natural numbers are referred to as negative integers . The set of all integers is often denoted by the boldface Z or blackboard bold Z {\displaystyle \mathbb {Z} } . The set of natural numbers N {\displaystyle \mathbb {N} } is a subset of Z {\displaystyle \mathbb {Z} } , which in turn

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2912-727: The presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms. The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair ( x , y ) {\displaystyle (x,y)} that takes as arguments two natural numbers x {\displaystyle x} and y {\displaystyle y} , and returns an integer (equal to x − y {\displaystyle x-y} ). This operation

2968-412: The products of primes in an essentially unique way. This is the fundamental theorem of arithmetic . Z {\displaystyle \mathbb {Z} } is a totally ordered set without upper or lower bound . The ordering of Z {\displaystyle \mathbb {Z} } is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... . An integer is positive if it

3024-404: The sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, 0 ), Z {\displaystyle \mathbb {Z} } , unlike the natural numbers, is also closed under subtraction . The integers form a ring which is the most basic one, in the following sense: for any ring, there is a unique ring homomorphism from

3080-422: The table) means that the commutative ring  Z {\displaystyle \mathbb {Z} } is an integral domain . The lack of multiplicative inverses, which is equivalent to the fact that Z {\displaystyle \mathbb {Z} } is not closed under division, means that Z {\displaystyle \mathbb {Z} } is not a field . The smallest field containing

3136-416: The various laws of arithmetic. In modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers ( a , b ) . The intuition is that ( a , b ) stands for the result of subtracting b from

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