Contents

1. Overview
2. Etymology
3. Cultural Impact

Number in {…, –2, –1, 0, 1, 2, …}

This article delves into the mathematical concept of integers , a foundational set of numbers . For a discussion of integers as a specific data type within computing contexts, please refer to the article on Integer (computer science) .

The integers, when meticulously arranged upon a number line , reveal a perfectly ordered, infinitely extending sequence.

An integer, at its core, is a number that possesses no fractional component . It can be the solitary zero , any positive natural number (such as 1, 2, 3, and so forth, extending indefinitely), or the precise negation of a positive natural number (−1 , −2, −3, and so on). The negations, or more formally, the additive inverses , of these positive natural numbers are distinctly recognized as negative integers. The complete set encompassing all integers is routinely symbolized in boldface as Z or, more commonly and elegantly in mathematical texts, as blackboard bold

{\displaystyle \mathbb {Z} }

The set of natural numbers , typically denoted as

{\displaystyle \mathbb {N} }

, is, rather predictably, a subset of this broader collection,

{\displaystyle \mathbb {Z} }

. This set of integers, in turn, forms a subset of the rational numbers , denoted as

{\displaystyle \mathbb {Q} }

, which itself is a subset of the real numbers ⁠

{\displaystyle \mathbb {R} }

⁠. To be more precise, each of these number systems is embedded into the subsequent one, meaning there’s an isomorphic mapping of the initial set to a subset of the next. For those who prefer a more streamlined, if less constructive, view, one might define the real numbers first and then consider the other sets as mere subsets within that grand continuum.

Much like its progenitor, the set of natural numbers , the collection of integers,

{\displaystyle \mathbb {Z} }

, possesses the property of being countably infinite . This means, despite its boundless extent, its elements can be put into a one-to-one correspondence with the natural numbers . Fundamentally, an integer can be conceived as a real number that conveniently presents itself without any cumbersome fractional component . To illustrate, numbers like 21, 4, 0 , and −2048 are undeniably integers. However, 9.75, ⁠5+1/2⁠, 5/4, and the enigmatic square root of 2 decidedly are not. These are the numbers that, for better or worse, maintain their ‘wholeness’.

The integers hold a rather significant position in algebraic structure . They constitute the smallest possible group and the smallest possible ring that contain the natural numbers . This is not a trivial observation; it underscores their fundamental role. Within the sophisticated realm of algebraic number theory , these straightforward integers are sometimes given the slightly more specific designation of rational integers. This is done to prevent confusion, distinguishing them from the broader, more abstract category of algebraic integers . Indeed, (rational) integers are precisely those algebraic integers that also happen to be rational numbers —a neat little intersection, for those who appreciate such things.

History

The very word “integer” has its roots in the ancient Latin term integer, which translates to “whole” or, quite literally, “untouched.” It stems from the prefix in- (“not”) combined with tangere (“to touch”). The English word “entire” also traces its lineage back to this same origin, having passed through the French language word entier, which, rather conveniently, means both “entire” and “integer.”

Historically, the term “integer” was initially applied to any number that was a direct multiple of 1, or, alternatively, to the whole portion of a mixed number . In those early days, the mathematical community primarily concerned itself with positive integers, rendering the term effectively synonymous with the natural numbers . It took a considerable amount of time, and perhaps a fair bit of philosophical hand-wringing, for the definition of “integer” to expand and embrace negative numbers , as their practical utility and conceptual necessity gradually became undeniable. For instance, the formidable Leonhard Euler , in his seminal 1765 work, Elements of Algebra , explicitly defined integers to include both their positive and negative counterparts, marking a significant shift in mathematical thought. One might wonder what took them so long to grasp the concept of ’less than nothing.'

The precise phrase “the set of the integers” only began to see widespread use towards the close of the 19th century. This timing is no mere coincidence, as it aligns directly with the period when Georg Cantor pioneered the revolutionary concepts of infinite sets and set theory , forever altering the landscape of mathematics. The adoption of the letter Z to symbolically represent the set of integers is derived from the German language word Zahlen, meaning “numbers.” This particular notation is widely credited to the influential mathematician David Hilbert . The earliest documented appearance of this notation in a published textbook can be found in Algèbre , a work produced by the collective known as Nicolas Bourbaki , dating back to 1947.

However, the notation wasn’t universally adopted overnight. For a period, some textbooks opted for alternative symbols, such as the letter J. As late as 1960, a notable academic paper utilized Z to specifically denote the non-negative integers, demonstrating the lack of a unified standard. Yet, by 1961, the symbol Z had largely become the standard and generally accepted notation within modern algebra texts for representing the complete set of positive and negative integers. It seems even mathematicians, eventually, can agree on some things.

The symbol

{\displaystyle \mathbb {Z} }

is frequently adorned with various annotations to specify particular subsets of integers, though usage can, frustratingly, vary among different authors. For instance:

{\displaystyle \mathbb {Z} ^{+}}

, ⁠

{\displaystyle \mathbb {Z} _{+}}

, or ⁠

{\displaystyle \mathbb {Z} ^{>}}

are commonly employed to denote the set of positive integers.

0 +

{\displaystyle \mathbb {Z} ^{0+}}

or ⁠

≥

{\displaystyle \mathbb {Z} ^{\geq }}

typically represent the set of non-negative integers (i.e., 0 and all positive integers).

≠

{\displaystyle \mathbb {Z} ^{\neq }}

is used by some to signify the set of non-zero integers.

The notation ⁠

∗

{\displaystyle \mathbb {Z} ^{*}}

can be particularly ambiguous: some authors use it for non-zero integers, others for non-negative integers, and still others for the rather specific set {−1, 1} (which represents the group of units of ⁠

{\displaystyle \mathbb {Z} }

⁠). One would think clarity would be paramount, but apparently not.

Furthermore, ⁠

{\displaystyle \mathbb {Z} _{p}}

is employed to denote either the set of integers modulo p (that is, the set of congruence classes of integers) or the set of p-adic integers. Such varied usage, of course, adds a certain… spice to mathematical communication.

For a period up until the early 1950s, the term “whole numbers” was considered synonymous with “integers.” However, in the late 1950s, as part of the somewhat controversial “New Math ” educational movement, American elementary school teachers began to distinguish these terms. Under this new pedagogical approach, “whole numbers” came to refer exclusively to the natural numbers , explicitly excluding negative numbers, while “integer” was broadened to encompass the negative numbers as well. This semantic shift, as is often the case with educational reforms, has left “whole numbers” a term somewhat ambiguous even to the present day.

Algebraic properties

Integers can be vividly imagined as distinct, uniformly spaced points stretching infinitely in both directions along a number line . As depicted, the non-negative integers are often visually represented in blue, while their negative counterparts are shown in red. A rather neat, if simplistic, visual.

Algebraic structure → Group theory Group theory

Basic notions

Group homomorphisms

Finite groups

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} }

)

Topological and Lie groups

Infinite dimensional Lie group

O(∞)
SU(∞)
Sp(∞)

Algebraic groups

Linear algebraic group
Reductive group
Abelian variety
Elliptic curve
v
t
e
Algebraic structure → Ring theory Ring theory

Basic concepts

Rings

Ring homomorphisms

Algebraic structures

{\displaystyle \mathbb {Z} }

Terminal ring

0

{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }

Related structures

Commutative algebra

Commutative rings

Algebraic number theory

{\displaystyle \mathbb {Z} _{p}}

p -adic numbers

{\displaystyle \mathbb {Q} _{p}}

Prüfer p -ring

(

∞

)

{\displaystyle \mathbb {Z} (p^{\infty })}

Noncommutative algebra

Noncommutative rings

Noncommutative algebraic geometry

Free algebra

Clifford algebra

Much like their simpler cousins, the natural numbers , the set ⁠

{\displaystyle \mathbb {Z} }

⁠ exhibits closure under the fundamental operations of addition and multiplication . This simply means that if you take any two integers and add them together, or multiply them, the result will, predictably, also be an integer. Anything less would be chaos, wouldn’t it? However, the inclusion of the negative natural numbers (and, critically, 0 ) bestows upon ⁠

{\displaystyle \mathbb {Z} }

⁠ a significant advantage over the natural numbers : it is also closed under subtraction . This seemingly minor addition introduces a whole new level of algebraic utility.

The integers, as a collective, form a ring , which can be considered the most fundamental of its kind. This isn’t just a casual observation; it’s a profound statement about its structure. In essence, for any given ring in the mathematical universe, there exists a single, unique ring homomorphism that maps from the integers into that specific ring . This universal property , the distinction of being an initial object within the entire category of rings , uniquely defines the ring ⁠

{\displaystyle \mathbb {Z} }

⁠. This singular homomorphism is injective if and only if the characteristic of the target ring is zero. A direct consequence of this is that any ring possessing a characteristic of zero must, by definition, contain a subring that is isomorphic to ⁠

{\displaystyle \mathbb {Z} }

⁠, and this subring will always be its smallest. It’s the ultimate foundation, in a sense.

However, ⁠

{\displaystyle \mathbb {Z} }

⁠ is conspicuously not closed under division . The quotient of two integers (for example, 1 divided by 2) is, more often than not, not an integer itself. Expecting a tidy result every time is simply naive. Furthermore, while the natural numbers are closed under exponentiation (positive integer powers of natural numbers yield natural numbers), the integers are not. This is because negative exponents would inevitably lead to fractions, pulling the result outside the integer set.

The following table, if you can bear to look at it, outlines some of the fundamental properties governing addition and multiplication for any arbitrary integers a, b, and c:

Properties of addition and multiplication on integers	Addition	Multiplication
Closure :	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	The only invertible integers (called units ) are −1 and 1.
Distributivity :		a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No zero divisors :		If a × b = 0, then a = 0 or b = 0 (or both)

The first five properties articulated above, specifically concerning addition, collectively demonstrate that ⁠

{\displaystyle \mathbb {Z} }

⁠, when considered under the operation of addition, forms an abelian group . Furthermore, it is also a cyclic group , a rather elegant characteristic, as every non-zero integer can be expressed as a finite sum of either 1 + 1 + … + 1 or (−1) + (−1) + … + (−1). In fact, ⁠

{\displaystyle \mathbb {Z} }

⁠ under addition holds the unique distinction of being the only infinite cyclic group —meaning any other infinite cyclic group is structurally isomorphic to ⁠

{\displaystyle \mathbb {Z} }

⁠. A certain kind of fundamental elegance, wouldn’t you say?

Conversely, the initial four properties enumerated above for multiplication indicate that ⁠

{\displaystyle \mathbb {Z} }

⁠ under multiplication forms a commutative monoid . However, it’s crucial to note that not every integer possesses a multiplicative inverse (for instance, the number 2 has no integer inverse that, when multiplied, yields 1). This absence of universal multiplicative inverses means that ⁠

{\displaystyle \mathbb {Z} }

⁠ under multiplication, despite its other virtues, does not constitute a group .

When all the rules from the preceding property table (with the exception of the very last one) are considered in their entirety, they collectively declare that ⁠

{\displaystyle \mathbb {Z} }

⁠, endowed with both addition and multiplication, forms a commutative ring with unity . It stands as the quintessential prototype for all mathematical objects possessing such an algebraic structure . Only those equalities of expressions that hold true in ⁠

{\displaystyle \mathbb {Z} }

⁠ for all possible values of variables are also true in any unital commutative ring . It’s worth remembering that certain non-zero integers may, under specific mappings, correspond to the additive identity (zero) in certain other rings .

The absence of zero divisors within the integers (as noted in the final property in the table) signifies that the commutative ring ⁠

{\displaystyle \mathbb {Z} }

⁠ is, in fact, an integral domain . This is a desirable quality, implying a certain ‘integrity’ in its multiplicative structure.

The aforementioned lack of multiplicative inverses, which is simply another way of stating that ⁠

{\displaystyle \mathbb {Z} }

⁠ is not closed under division, explicitly means that ⁠

{\displaystyle \mathbb {Z} }

⁠ does not qualify as a field . The smallest field that comprehensively contains the integers as a subring is the field of rational numbers . The elegant process by which the rational numbers are constructed from the integers can be analogously applied to form the field of fractions for any given integral domain . Conversely, one can embark from an algebraic number field (which is essentially an extension of rational numbers ) and systematically extract its specific ring of integers , a structure that, predictably, includes ⁠

{\displaystyle \mathbb {Z} }

⁠ as its inherent subring .

While standard division, as a binary operation yielding a single result, is not universally defined over ⁠

{\displaystyle \mathbb {Z} }

⁠, a more nuanced form of division—“division with remainder”—is indeed well-defined. This is formally known as Euclidean division , and it possesses a remarkably important property: given any two integers, a and b, where b is emphatically not zero , there exist a unique pair of integers, q (the quotient) and r (the remainder ), such that the equation a = q × b + r holds true, and the remainder r satisfies the condition 0 ≤ r < |b|, where |b| signifies the absolute value of b. This elegant principle forms the bedrock of the Euclidean algorithm , a venerable method for efficiently computing greatest common divisors through a precise sequence of Euclidean divisions .

These properties reveal that ⁠

{\displaystyle \mathbb {Z} }

⁠ is, in mathematical parlance, a Euclidean domain . This designation carries significant implications, as it inherently implies that ⁠

{\displaystyle \mathbb {Z} }

⁠ is also a principal ideal domain . Consequently, this means that any positive integer can be uniquely expressed as a product of primes in an essentially unique manner, disregarding the order of the prime factors and their signs. This profound insight is formally enshrined as the fundamental theorem of arithmetic , a cornerstone of number theory.

Order-theoretic properties

The set ⁠

{\displaystyle \mathbb {Z} }

⁠ is characterized as a totally ordered set , meaning that for any two distinct integers, one is always demonstrably greater or smaller than the other. Furthermore, this set exists without an upper or lower bound ; it extends infinitely in both positive and negative directions. The inherent ordering of ⁠

{\displaystyle \mathbb {Z} }

⁠ is universally understood and given by the familiar sequence: :… −3 < −2 < −1 < 0 < 1 < 2 < 3 < …. An integer is unambiguously defined as positive if its value is greater than zero , and negative if its value is less than zero . The number zero itself holds a neutral position, being explicitly defined as neither negative nor positive. It simply is.

This inherent ordering of integers is, quite conveniently, perfectly compatible with the standard algebraic operations, in the following manner:

If a < b and c < d, then it logically follows that a + c < b + d. The sums maintain the inequality.
If a < b and 0 < c (meaning c is a positive integer), then ac < bc. Multiplication by a positive integer preserves the inequality.

Thus, it is definitively established that ⁠

{\displaystyle \mathbb {Z} }

⁠, when considered in conjunction with its inherent ordering, forms an ordered ring .

The integers hold a singular distinction as the only nontrivial totally ordered abelian group whose positive elements exhibit the property of being well-ordered . This rather technical statement is, intriguingly, mathematically equivalent to the assertion that any Noetherian valuation ring is either a field (a structure where every non-zero element has a multiplicative inverse) or, more specifically, a discrete valuation ring . A subtle but crucial point for those who appreciate such nuances.

Construction

Traditional development

In the realm of elementary school pedagogy, integers are frequently introduced and intuitively defined as the direct union of the (positive) natural numbers , the singular entity known as zero , and the precise negations of those natural numbers . This seemingly straightforward approach can be formalized with a bit more rigor. One first constructs the set of natural numbers according to the foundational Peano axioms , let’s call this set ⁠

{\displaystyle P}

⁠. Subsequently, one constructs a distinct set ⁠

−

{\displaystyle P^{-}}

⁠ that is entirely disjoint from ⁠

{\displaystyle P}

⁠ and stands in a perfect one-to-one correspondence with ⁠

{\displaystyle P}

⁠ via some function, let’s say ⁠

{\displaystyle \psi }

⁠. For instance, one could define ⁠

−

{\displaystyle P^{-}}

⁠ as the ordered pairs ⁠

( 1 , n )

{\displaystyle (1,n)}

⁠ with the mapping ⁠

ψ

n ↦ ( 1 , n )

{\displaystyle \psi =n\mapsto (1,n)}

⁠. Finally, one designates 0 as some unique object not belonging to either ⁠

{\displaystyle P}

⁠ or ⁠

−

{\displaystyle P^{-}}

⁠—perhaps the ordered pair (0,0) for neatness. The integers are then formally defined as the union of these three distinct components: ⁠

P ∪

−

∪ { 0 }

{\displaystyle P\cup P^{-}\cup {0}}

⁠.

The traditional arithmetic operations (addition, subtraction, multiplication) are then defined on these integers in a piecewise manner, painstakingly addressing each combination of positive numbers, negative numbers, and zero . For example, the operation of negation is defined with a certain case-by-case precision:

⁠

− x

{

ψ ( x ) ,

x ∈ P

− 1

( x ) ,

x ∈

−

0 ,

x

{\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}}}

This traditional style of definition, while intuitive for beginners, unfortunately leads to a proliferation of distinct cases. Each arithmetic operation necessitates a separate definition for every conceivable combination of integer types (positive, negative, zero), making the task of rigorously proving that integers adhere to the various fundamental laws of arithmetic—such as associativity or commutativity—an undeniably tedious and laborious undertaking. One might even call it inefficient.

Equivalence classes of ordered pairs

Red points represent ordered pairs of natural numbers . Linked red points are equivalence classes representing the blue integers at the end of the line.

In the more sophisticated landscape of modern set-theoretic mathematics , a significantly more abstract and, frankly, elegant construction is frequently employed. This approach allows for the definition of arithmetical operations without the cumbersome need for numerous case distinctions, a welcome relief from the traditional method. The integers can, under this framework, be formally constructed as the equivalence classes of ordered pairs of natural numbers (a, b).

The underlying intuition behind this construction is that an ordered pair (a, b) conceptually represents the outcome of subtracting b from a. To align with our inherent understanding that, for example, 1 − 2 and 4 − 5 should denote the identical numerical value, we introduce an equivalence relation ~ on these pairs. This relation is governed by the following precise rule: ⁠

( a , b ) ∼ ( c , d )

{\displaystyle (a,b)\sim (c,d)}

⁠ holds true if and only if ⁠

a + d

b + c

{\displaystyle a+d=b+c}

⁠.

Addition and multiplication of these newly defined integers can then be elegantly articulated in terms of the corresponding operations on the underlying natural numbers . By utilizing the notation [( a, b)] to signify the equivalence class that includes (a, b) as one of its members, the operations are defined as follows:

For addition: ⁠

[ ( a , b ) ] + [ ( c , d ) ] := [ ( a + c , b + d ) ]

{\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)]}

⁠.

For multiplication: ⁠

[ ( a , b ) ] ⋅ [ ( c , d ) ] := [ ( a c + b d , a d + b c ) ]

{\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]}

⁠.

The negation (or, more accurately, the additive inverse ) of an integer is obtained by the straightforward act of reversing the order of the elements within its representative pair: ⁠

− [ ( a , b ) ] := [ ( b , a ) ]

{\displaystyle -[(a,b)]:=[(b,a)]}

⁠.

Consequently, the operation of subtraction can be elegantly defined as the addition of the additive inverse : ⁠

[ ( a , b ) ] − [ ( c , d ) ] := [ ( a + d , b + c ) ]

{\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)]}

⁠.

The standard ordering relationship among integers is established by the following condition: ⁠

[ ( a , b ) ] < [ ( c , d ) ]

{\displaystyle [(a,b)]<[(c,d)]}

⁠ if and only if ⁠

a + d < b + c

{\displaystyle a+d<b+c}

⁠.

It is a relatively straightforward matter to verify that these definitions for operations and ordering are robust and entirely independent of the specific choice of representatives from within the equivalence classes .

Every equivalence class resulting from this construction possesses a unique member that conforms to either the form (n, 0) or (0, n) (or, in the singular case of zero , both simultaneously). A natural number n is conceptually identified with the equivalence class [( n, 0)]—meaning the natural numbers are embedded into the integers via a mapping that sends n to [( n, 0)]. The equivalence class [(0, n)] is then designated as −n. This elegant convention covers all remaining classes and, as a side effect, provides the equivalence class [(0,0)] a second representation, as −0 is, quite logically, equal to 0.

Thus, any equivalence class [( a, b)] is ultimately denoted by:

⁠

{

a − b ,

a ≥ b

− ( b − a ) ,

a < b

{\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\-(b-a),&{\mbox{if }}a<b\end{cases}}}

If the natural numbers are identified with their corresponding integers (through the embedding process mentioned earlier), this established convention introduces no ambiguity whatsoever. This notation, ultimately, recovers the familiar and universally recognized representation of the integers as the infinite sequence {…, −2, −1, 0, 1, 2, …}.

A few illustrative examples, for those who appreciate concrete instances:

⁠

= [ ( 0 , 0 ) ]

= [ ( 1 , 1 ) ]

= ⋯

= [ ( k , k ) ]

= [ ( 1 , 0 ) ]

= [ ( 2 , 1 ) ]

= ⋯

= [ ( k + 1 , k ) ]

− 1

= [ ( 0 , 1 ) ]

= [ ( 1 , 2 ) ]

= ⋯

= [ ( k , k + 1 ) ]

= [ ( 2 , 0 ) ]

= [ ( 3 , 1 ) ]

= ⋯

= [ ( k + 2 , k ) ]

— 2

= [ ( 0 , 2 ) ]

= [ ( 1 , 3 ) ]

= ⋯

= [ ( k , k + 2 ) ]

{\displaystyle {\begin{aligned}0&=[(0,0)]&=[(1,1)]&=\cdots &&=[(k,k)]\1&=[(1,0)]&=[(2,1)]&=\cdots &&=[(k+1,k)]\-1&=[(0,1)]&=[(1,2)]&=\cdots &&=[(k,k+1)]\2&=[(2,0)]&=[(3,1)]&=\cdots &&=[(k+2,k)]\-2&=[(0,2)]&=[(1,3)]&=\cdots &&=[(k,k+2)]\end{aligned}}}

Other approaches

Within the specialized field of theoretical computer science , alternative methodologies for constructing integers are frequently employed, particularly by automated theorem provers and term rewrite engines . In these computational contexts, integers are typically represented as algebraic terms that are meticulously built using a select few fundamental operations—such as zero, succ (successor), and pred (predecessor). This approach assumes that the natural numbers have already been successfully constructed, often utilizing the venerable Peano approach .

There exist, quite remarkably, at least ten distinct constructions for signed integers, each with its own set of characteristics. These various constructions differ in several key aspects: the precise number of basic operations required for the construction; the number of arguments (typically between 0 and 2) and their types accepted by these operations; whether natural numbers are included as arguments for some of these operations; and, crucially, whether these operations function as free constructors or not. A free constructor implies that a given integer can be represented by only one unique algebraic term , whereas non-free constructors allow for multiple equivalent representations.

The elegant technique for constructing integers presented in the preceding section (using equivalence classes of ordered pairs ) corresponds to a specific case where there is a single basic operation, pair ⁠

( x , y )

{\displaystyle (x,y)}

⁠, which accepts two natural numbers ⁠

{\displaystyle x}

⁠ and ⁠

{\displaystyle y}

⁠ as arguments and yields an integer (conceptually equivalent to ⁠

x − y

{\displaystyle x-y}

⁠). This particular operation is not free, given that the integer 0 can be represented by pair(0,0), or pair(1,1), or pair(2,2), and so on, ad infinitum. This construction technique, despite its non-free nature, is notably employed by the proof assistant Isabelle . However, many other computational tools opt for alternative construction techniques, particularly those grounded in free constructors, which tend to be simpler to implement and can often be executed with greater efficiency within computer systems. Practicality, it seems, sometimes trumps theoretical elegance.

Computer science

Main article: Integer (computer science)

An integer is very frequently designated as a primitive data type across a multitude of computer languages . However, it’s a critical distinction that integer data types within computing environments can only ever represent a finite subset of all mathematical integers. This limitation arises directly from the finite capacity inherent in all practical computer hardware. Furthermore, in the ubiquitous two’s complement representation scheme, the fundamental definition of sign inherently differentiates between “negative” and “non-negative” values, rather than the more nuanced mathematical distinction of “negative, positive, and 0 ”. (It is, of course, entirely feasible for a computer system to ascertain whether an integer value is definitively positive.) Fixed-length integer approximation data types (or subsets thereof) are conventionally denoted by keywords such as int or Integer in various prominent programming languages, including Algol68 , C , Java , and Delphi , among others.

For situations demanding the representation of arbitrarily large integers that exceed the bounds of fixed-size types, variable-length representations, often referred to as bignums , are employed. These sophisticated data structures can store any integer, limited only by the total available memory within the computer system. Other integer data types are typically implemented with a predetermined, fixed size, usually a number of bits that is a power of 2 (e.g., 4, 8, 16, 32, 64 bits) or a human-memorable number of decimal digits (e.g., 9 or 10 digits). Humans, ever trying to fit the infinite into their finite boxes. A valiant, if ultimately futile, effort to tame the untamable.

Cardinality

The set of integers is classified as countably infinite . This designation signifies that, despite its boundless extent, it is possible to establish a one-to-one correspondence, or a bijection , between each integer and a unique natural number . It’s not as impressive as it sounds, simply a matter of clever mapping. An example of such a pairing, which demonstrates this bijection , is as follows:

(0, 1), (1, 2), (−1, 3), (2, 4), (−2, 5), (3, 6), . . . ,(1 − k , 2k − 1), (k, 2k ), . . .

More technically precise, the cardinality of ⁠

{\displaystyle \mathbb {Z} }

⁠ is formally stated to be equal to ℵ₀ (aleph-null ), which is the cardinality of the natural numbers . The existence of this pairing between the elements of ⁠

{\displaystyle \mathbb {Z} }

⁠ and ⁠

{\displaystyle \mathbb {N} }

⁠ is precisely what is defined as a bijection . Meaning you can line them up, one by one, into eternity. Thrilling, I’m sure.

Footnotes

^a More precisely, each system is embedded in the next, isomorphically mapped to a subset. The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the reals.

History

Algebraic properties

0

Order-theoretic properties

Construction

Traditional development

ψ

− x

x

Equivalence classes of ordered pairs

a + d

Other approaches

Computer science

Cardinality

See also

Footnotes