Integer

The integers consist of the positive natural numbers (1 (number), 2 (number), 3 (number), …), their negative and non-negative numberss (−1, −2, −3, ...) and the number 0 (number). The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold,

\mathbb{Z}

), which stands for ''Zahlen'' (German language for "numbers"). They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set. == Algebraic properties == Like the natural numbers, Z is closure (mathematics) under the binary operation 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 (number), Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division (mathematics), since the quotient of two integers (''e.g.'', 1 divided by 2), need not be an integer. The following table lists some of the basic properties of addition and multiplication for any integers ''a'', ''b'' and ''c''. {| | || addition || multiplication |- | Closure (mathematics): || ''a'' + ''b'' is an integer || ''a'' × ''b'' is an integer |- | associativity: || ''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c'' || ''a'' × (''b'' × ''c'') = (''a'' × ''b'') × ''c'' |- | commutativity: || ''a'' + ''b'' = ''b'' + ''a'' || ''a'' × ''b'' = ''b'' × ''a'' |- | existence of an identity element: || ''a'' + 0 = ''a'' || ''a'' × 1 = ''a'' |- | existence of inverse elements: || ''a'' + (−''a'') = 0 || |- | distributivity: || colspan=2 align=center| ''a'' × (''b'' + ''c'') = (''a'' × ''b'') + (''a'' × ''c'') |} In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the ''only'' infinite cyclic group, in the sense that any infinite cyclic group is group isomorphism to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, note that not every integer has a multiplicative inverse; e.g. there is no integer ''x'' such that 2''x'' = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the properties from the above table taken together say that Z together with addition and multiplication is a commutative ring (mathematics) with unity. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field (mathematics). The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ''ab'' = 0, either ''a'' = 0 or ''b'' = 0. Although ordinary division is not defined on Z, it does possess an important property called the division algorithm: that is, 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'' is called the ''quotient'' and ''r'' is called the ''remainder'', resulting from division of ''a'' by ''b''. This is the basis for the Euclidean algorithm for computing greatest common divisors. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of prime number in an essentially unique way. This is the fundamental theorem of arithmetic. ==Order-theoretic properties == Z is a total order without upper or lower bound. The ordering of Z is given by : ... < −2 < −1 < 0 < 1 < 2 < ... An integer is ''positive'' if it 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: # if ''a'' < ''b'' and ''c'' < ''d'', then ''a'' + ''c'' < ''b'' + ''d'' # if ''a'' < ''b'' and 0 < ''c'', then ''ac'' < ''bc''. (From this fact, one can show that if ''c'' < 0, then ''ac'' > ''bc''.) ==Integers in computing== An integer is often one of the primitive datatypes in computer languages. However, these "integers" can only represent a subset of all mathematical integers, since "real-world" computers are of finite capacity. Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers. On the other hand, theoretical models of digital computers, e.g., Turing machines, usually do have infinite (but only countable) capacity. For more information, see Integer (computer science). ==Quotations== ''God invented the integers, all else is the work of man.'' Leopold Kronecker ==External links== * [http://www.positiveintegers.org The Positive Integers - divisor tables and numeral representation tools] Elementary mathematics Group theory Integers Number theory Set theory fa:اعداد صحیح scn:Nummuru rilativu su:Integer th:จำนวนเต็ม

Integer

What was described here before is entirely inaccurate - the :whole numbers are the nonnegative integers, not the other way around, and are often not distinguished from the natural numbers. ---- I know this complicates things but is the "unique" in the first sentence not supposed to be "unique up to isomorphism"? -- User:Jan Hidders ---- What is here is not wrong, per se. it just is a bit mathematical if you are discussing the way that the word integer is used in the context of computers. In that context it is slightly different because it has to do with the type of hardware used for math, and the storage of the numbers in computer memory. integer is commonly used for either the numbers that can be stored in one word, or it is the number range for the 'natural' address space of the computer.

I thought that Z was commonly used for complex variables, and x was most commonly used for reals. If I'm missing something, just delete this please (I doubt that I'll remember to check back). :''The letter Z is commonly used for the set of all integers, the letter C is commonly used for the set of all complex numbers, and the letter R is commonly used for the set of all real numbers. ''n'' or ''k'' are commonly used for integer variables, ''z'' is commonly used for complex variables, and ''x'' is commonly used for real variables. --AxelBoldt'' == Is zero positive? == I think this is a matter of convention. Some mean ≥ 0, others > 0 when they use the word ''positive''. I think that it is important to mention this somewhere, lest readers be confused when reading other things. User:Lupin 22:09, 7 Sep 2004 (UTC) No. Zero cannot be positive, even partially. This is not merely a matter of convention. --User:OmegaMan Strictly, yes, but Lupin is quiet correct in pointing out that some readers and writers are imprecise on that, and many (wrongly) interpret ''positive'' as meaning ''not negative''. So it never hurts to spell it out, rather then allow them to continue being muddled... User:Quota :It is a matter of convention: see my comment below. User:MFH 14:39, 7 Apr 2005 (UTC) Inconsistencies in mathematical terminology (which unfortunately exist) should not be confused with inconsistencies in mathematical definitions (which do not exist). The approach "Lupin" advocated was to spread those inconsistencies to mathematical definitions as well. --User:OmegaMan :I'd have to take issue with this. Reading what I actually wrote, I don't think I have advocated inconsistent definitions. The fact is that some people's definitions of positivity and negativity say that zero is both positive and negative, and some people's definitions say that zero is neither positive or negative. This is a matter of terminology. User:Lupin 14:26, 5 Apr 2005 (UTC) ::Proposed wording: In an informal context, the phrase \"positive numbers\" may occasionally be intended to include zero. More correctly, this is called \"non-negative numbers\". ::However, while a couple of sentences like these may be appropriate somewhere in the wikipedia, I'm not sure this article is the place. The same issue exists with the real number line. Perhaps it really belongs to one of the articles Positive, Negative and non-negative numbers, or 0 (number)? ::Can anyone substantiate the claim made above that some consider zero to be both positive and negative? I haven't come across it myself (unless, of course, "some" means uninformed people, which really isn't the point).--User:Noe 20:13, Apr 5, 2005 (UTC) :::I agree that this is not the article to be making this distinction clear. :::Incidentally, the distinction some people make is between "positive", meaning "greater than or equal to zero" and "strictly positive", meaning "strictly greater than zero". I have heard some attribute this difference in terminology to which side of the Atlantic you come from. User:Lupin 01:49, 7 Apr 2005 (UTC) :The meaning of "positive" is a matter of convention (= definition), which is proved by the fact that this word does mean "≥ 0" in many other languages, e.g. in French language, where one adds "strictly" for "> 0"; see Lupine's comment just above. :In some sense, this is even more natural, in view of the definition of an order relation as opposed to a strict order. What this concerns, all mathematicians agree. Of course, we also all agree that there should be no doubt that "positive" does mean "> 0" in English, by definition (= convention); I do not intend at all to advocate any misunderstanding of this. User:MFH 14:39, 7 Apr 2005 (UTC) == Why is it "Z"? == Why is exactly the letter "Z" chosen? Possible explanation: "Z" looks like "N"-tilted, which kind of shows a relationship between Z and N. But then "M" would be an even better letter, since it is almost N mirrored, which is exactly (the interpretation of) how Z is usually defined. Perhaps M was already taken, but I haven't seen any indication of that. I was told it was because of the German word Zahl. --User:Georg Muntingh 10:23, 19 Sep 2004 (UTC) == How is the set of integers constructed? == I understand how the set of natural numbers is constructed by the Peano postulates, but I can't find any explanation as to how the set of integers is constructed. If 1 = what is -1 supposed to be? --User:Toper As far as I know (but I could be wrong), the construction of Z is just a hack, to make sure all natural numbers have an inverse. So you take the natural numbers N={0, 1, 2, ....}, append to it the set {1n, 2n, 3n, ...}. It does not matter what 1n, 2n, 3n, are, as long as they are distinct from 0, 1, 2, 3... Then you define 1n+1=0 5n+2=3n 7n+10=3 2n+9n=11n etc I think you get the idea. You define the addition on the union of {0, 1, 2, ..} with the set {1n, 2n, 3n, ...} in such a way that what you get looks like integer addition (whether the integers are positive, or negative). Then you declare this thing to be the integers Z. This kind of hack is used in algebra all the time, and nothing is wrong with it, but it looks unnatural. Did this answer your question? User:Oleg Alexandrov | User_talk:Oleg Alexandrov 19:02, 24 Jan 2005 (UTC) You're explanation makes sense; thank you. I guess my problem stemmed from viewing set members as having structural equivalence. I believe you're example shows, instead, that set members actually have name equivalence. I was incorrectly? trying to view additive inverse as a unitary function that took a set structured according to the Peano postulates (an ordinal?) and returned something representative of that sets opposite. If I understand correctly, you're saying that the integers are not so much the set that contains ~~the~~ members, but more so the collection of relations defined for a set, a set with members which happend to be named {..., -3, -2, -1, 0, 1, 2, 3, ...}. Are there ways to recursively define the integers along with their essential relations in a similar fashion to the natural numbers in Primitive recursive function#Examples? --User:Toper 18:56, 25 Jan 2005 (UTC) You are right. For integers, just as for naturals, it does not matter what the nature of the numbers is. It is the properties and the relationships that matter. I think you could generalize the stuff in Primitive recursive function#Examples to integers. Give it a hard thought, it should be an interesting exercise. User:Oleg Alexandrov | User_talk:Oleg Alexandrov 19:07, 25 Jan 2005 (UTC) :Well, I'm sure this is way over the top for most people, but from category theory, Z arises naturally (no pun intended) from the non-negative integers N = {0, 1, 2, ...} by taking the left adjoint of the forgetful functor from the category of groups to the category of monoids. Does that clear it up? ==== trying to be useful ==== IMHO the very first answer phrase is almost the best one among the above. (After elimination of the last one, being the "true" winner, but completely useless to 99.9% of all visitors, who don't know what "abstract nonsense" really means.) I like the definition :: Z = N×N / { ((a,b),(c,d)) | a+d=b+c } . The idea is called ''symmetrization of a semigroup'', which is a simplified version of a quotient field. Just like fractions ''a/b'', ''c/d'' are couples of integers ''(a,b),(c,d)'' that are identified ("equal") iff ''ad=bc'', here we identify couples of natural numbers if ''a+d=b+c'', which makes ''(a,b)'' represent the integer ''a-b''. An element ''a'' of N is seen as element of Z by taking the class of (a,0). Its additive inverse does then always exist and is the class of (0,a), denoted by ''– a''. (This ''injection'' is compatible with componentwise addition of couples, its what mathematicians call a morphism.) So far, the additive structure was concerned. But the above identification is also compatible with multiplication defined by ''(a,b)×(c,d)=(ac+bd,ad+bc)'' (just separate positive and negative terms of ''(a-b)(c-d)''). Mathematicians would call this a semiring morphism. I hope this helped. (Maybe I should move/copy this (or an improved version) to the main page and/or another of the cited pages... User:MFH 20:23, 6 Apr 2005 (UTC) ==Circular== Integers are currently defined in terms of natural numbers, and vice-versa. User:24.91.43.225 17:17, 14 Jun 2005 (UTC) :Well not really. This article defines the integers using the natural numbers. The article natural numbers does use the term "integer" to ''describe'' the natural numbers in the intro, but it ''defines'' the natural numbers Natural number#Formal definitions, without reference to the term "integer". Perhaps the first sentence of Natural number could be rewritten more clearly as: "Natural number can mean either an element of the set {1, 2, 3, ... } (i.e the negative and non-negative numbers integers) or the set {0, 1, 2, 3, ... } (i.e. the non-negative integers)." User:Paul August User_talk:Paul August 20:10, Jun 15, 2005 (UTC)

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Words begining with Integer:

Integer
Integer
Integer-valued_polynomial
IntegerNumbers
Integers
Integers
Integers
Integers_mod_n
Integer_(computer_science)
Integer_(computer_science)
Integer_BASIC
Integer_BASIC_programming_language
Integer_factorisation
Integer_factorization
Integer_factorization
Integer_factorization_algorithms
Integer_factorization_problem
Integer_linear_programming
Integer_notation
Integer_number
Integer_part
Integer_partition
Integer_partition
Integer_polynomial
Integer_program
Integer_programming
Integer_quaternion
Integer_Research
Integer_research
Integer_sequence
Integer_sequence
Integer_sequences
Integer_sequences
Integer_sequences
Integer_square_root
Integer_square_root