AlaaElhilo/Wikipedia_ComputerScience · Datasets at Hugging Face

Number int64 1 7.61k	Text stringlengths 2 3.11k
4,401	Bill-NOM
4,402	3-come-PL-FUT-INFER
4,403	John-NOM Bill-NOM 3-come-PL-FUT-INFER
4,404	'John or Bill will come.'
4,405	An operand of a conjunction is a conjunct.
4,406	Beyond logic, the term "conjunction" also refers to similar concepts in other fields:
4,407	Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.
4,408	The conjunctive identity is true, which is to say that AND-ing an expression with true will never change the value of the expression. In keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction is often defined as having the result true.
4,409	In systems where logical conjunction is not a primitive, it may be defined as
4,410	or
4,411	or in logical operator notation:
4,412	Here is an example of an argument that fits the form conjunction introduction:
4,413	Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.
4,414	...or alternatively,
4,415	In logical operator notation:
4,416	...or alternatively,
4,417	This formula can be seen as a special case of
4,418	In other words, a conjunction can actually be proven false just by knowing about the relation of its conjuncts, and not necessary about their truth values.
4,419	This formula can be seen as a special case of
4,420	Either of the above are constructively valid proofs by contradiction.
4,421	commutativity: yes
4,422	associativity: yes
4,423	distributivity: with various operations, especially with or
4,424	with material nonimplication:
4,425	with itself:
4,426	idempotency: yes
4,427	monotonicity: yes
4,428	truth-preserving: yes When all inputs are true, the output is true.
4,429	falsehood-preserving: yes When all inputs are false, the output is false.
4,430	Walsh spectrum:
4,431	Nonlinearity: 1
4,432	If using binary values for true and false , then logical conjunction works exactly like normal arithmetic multiplication.
4,433	In high-level computer programming and digital electronics, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "AND", an algebraic multiplication, or the ampersand symbol & . Many languages also provide short-circuit control structures corresponding to logical conjunction.
4,434	Logical conjunction is often used for bitwise operations, where 0 corresponds to false and 1 to true:
4,435	The operation can also be applied to two binary words viewed as bitstrings of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. For example:
4,436	This can be used to select part of a bitstring using a bit mask. For example, 10011101 AND 00001000 = 00001000 extracts the fourth bit of an 8-bit bitstring.
4,437	In computer networking, bit masks are used to derive the network address of a subnet within an existing network from a given IP address, by ANDing the IP address and the subnet mask.
4,438	Logical conjunction "AND" is also used in SQL operations to form database queries.
4,439	The Curry–Howard correspondence relates logical conjunction to product types.
4,440	As with other notions formalized in mathematical logic, the logical conjunction and is related to, but not the same as, the grammatical conjunction and in natural languages.
4,441	English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order having the sense of "then". For example, "They got married and had a child" in common discourse means that the marriage came before the child.
4,442	The word "and" can also imply a partition of a thing into parts, as "The American flag is red, white, and blue." Here, it is not meant that the flag is at once red, white, and blue, but rather that it has a part of each color.
4,443	In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content , other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false".
4,444	In classical logic, with its intended semantics, the truth values are true , and untrue or false ; that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws:
4,445	Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.
4,446	In intuitionistic logic, and more generally, constructive mathematics, statements are assigned a truth value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if one can build a proof of the statement from those axioms. A statement is false if one can deduce a contradiction from it. This leaves open the possibility of statements that have not yet been assigned a truth value. Unproven statements in intuitionistic logic are not given an intermediate truth value . Indeed, one can prove that they have no third truth value, a result dating back to Glivenko in 1928.
4,447	Instead, statements simply remain of unknown truth value, until they are either proven or disproven.
4,448	There are various ways of interpreting intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation. See also Intuitionistic logic § Semantics.
4,449	Multi-valued logics allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval such structure is a total order; this may be expressed as the existence of various degrees of truth.
4,450	Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the necessary truth of formulae.
4,451	But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Boolean algebra semantics of classical propositional calculus.
4,452	Intuitionistic type theory uses types in the place of truth values.
4,453	Topos theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.
4,454	The real numbers are fundamental in calculus , in particular by their role in the classical definitions of limits, continuity and derivatives.
4,455	The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers. Some irrational numbers are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers. There are also real numbers which are not, such as π = 3.1415...; these are called transcendental numbers.
4,456	Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers are equally spaced.
4,457	Conversely, analytic geometry is the association of points on lines to real numbers such that geometric displacements are proportional to differences between corresponding numbers.
4,458	The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current axiomatic definition is that real numbers form the unique Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences , Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.
4,459	Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence.
4,460	The real numbers form an ordered field. Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations, addition and multiplication, and a total order that have the following properties.
4,461	Many other properties can be deduced from the above ones. In particular:
4,462	Several other operations are commonly used, which can be deduced from the above ones.
4,463	The real numbers 0 and 1 are commonly identified with the natural numbers 0 and 1. This allows identifying any natural number n with the sum of n real numbers equal to 1.
4,464	The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties . So, the identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with 1 taken as the successor function.
4,465	These identifications are formally abuses of notation, and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming. In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by the compiler.
4,466	Dedekind completeness implies other sorts of completeness , but also has some important consequences.
4,467	The last two properties are summarized by saying that the real numbers form a real closed field. This implies the real version of the fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.
4,468	A key property of real numbers is their decimal representation. A decimal representation consists of a nonnegative integer k and an infinite sequence of decimal digits
4,469	that is written
4,470	The real number defined by the sequence is the least upper bound of the D n , {\displaystyle D_{n},} which exists by Dedekind completeness.
4,471	In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9.
4,472	A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete :
4,473	A sequence of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N such that the distance \|xn − xm\| is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the xn eventually come and remain arbitrarily close to each other.
4,474	A sequence converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N such that the distance \|xn − x\| is less than ε for n greater than N.
4,475	Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.
4,476	The set of rational numbers is not complete. For example, the sequence , where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number .
4,477	The completeness property of the reals is the basis on which calculus, and more generally mathematical analysis, are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.
4,478	For example, the standard series of the exponential function
4,479	converges to a real number for every x, because the sums
4,480	The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
4,481	First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element .
4,482	Additionally, an order can be Dedekind-complete, see § Axiomatic approach. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field and then forms the Dedekind-completion of it in a standard way.
4,483	As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
4,484	The real numbers form a metric space: the distance between x and y is defined as the absolute value \|x − y\|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible , separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.
4,485	The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.
4,486	A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.
4,487	Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava , who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.
4,488	The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects . Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation . In Europe, such numbers, not commensurable with the numerical unit, were called irrational or surd .
4,489	In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
4,490	In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.
4,491	In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert gave a flawed proof that π cannot be rational; Legendre completed the proof and showed that π is not the square root of a rational number. Liouville showed that neither e nor e2 can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor extended and greatly simplified this proof. Hermite proved that e is transcendental, and Lindemann , showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass , Hilbert , Hurwitz, and Gordan.
4,492	The developers of calculus used real numbers without having defined them rigorously. The first rigorous definition was published by Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891.
4,493	These properties imply the Archimedean property , which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N.
4,494	The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like converges to a unique real number—in this case π. For details and other constructions of real numbers, see construction of the real numbers.
4,495	In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.
4,496	Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.
4,497	The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.
4,498	The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.
4,499	Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are elements of the set of the real numbers .
4,500	Electronic calculators and computers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary definable real numbers, which are inconvenient to manipulate.

4,401

Bill-NOM

4,402

3-come-PL-FUT-INFER

4,403

John-NOM Bill-NOM 3-come-PL-FUT-INFER

4,404

'John or Bill will come.'

4,405

An operand of a conjunction is a conjunct.

4,406

Beyond logic, the term "conjunction" also refers to similar concepts in other fields:

4,407

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

4,408

The conjunctive identity is true, which is to say that AND-ing an expression with true will never change the value of the expression. In keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction is often defined as having the result true.

4,409

In systems where logical conjunction is not a primitive, it may be defined as

4,410

or

4,411

or in logical operator notation:

4,412

Here is an example of an argument that fits the form conjunction introduction:

4,413

Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.

4,414

...or alternatively,

4,415

In logical operator notation:

4,416

...or alternatively,

4,417

This formula can be seen as a special case of

4,418

In other words, a conjunction can actually be proven false just by knowing about the relation of its conjuncts, and not necessary about their truth values.

4,419

This formula can be seen as a special case of

4,420

Either of the above are constructively valid proofs by contradiction.

4,421

commutativity: yes

4,422

associativity: yes

4,423

distributivity: with various operations, especially with or

4,424

with material nonimplication:

4,425

with itself:

4,426

idempotency: yes

4,427

monotonicity: yes

4,428

truth-preserving: yes When all inputs are true, the output is true.

4,429

falsehood-preserving: yes When all inputs are false, the output is false.

4,430

Walsh spectrum:

4,431

Nonlinearity: 1

4,432

If using binary values for true and false , then logical conjunction works exactly like normal arithmetic multiplication.

4,433

In high-level computer programming and digital electronics, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "AND", an algebraic multiplication, or the ampersand symbol & . Many languages also provide short-circuit control structures corresponding to logical conjunction.

4,434

Logical conjunction is often used for bitwise operations, where 0 corresponds to false and 1 to true:

4,435

The operation can also be applied to two binary words viewed as bitstrings of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. For example:

4,436

This can be used to select part of a bitstring using a bit mask. For example, 10011101 AND 00001000 = 00001000 extracts the fourth bit of an 8-bit bitstring.

4,437

In computer networking, bit masks are used to derive the network address of a subnet within an existing network from a given IP address, by ANDing the IP address and the subnet mask.

4,438

Logical conjunction "AND" is also used in SQL operations to form database queries.

4,439

The Curry–Howard correspondence relates logical conjunction to product types.

4,440

As with other notions formalized in mathematical logic, the logical conjunction and is related to, but not the same as, the grammatical conjunction and in natural languages.

4,441

English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order having the sense of "then". For example, "They got married and had a child" in common discourse means that the marriage came before the child.

4,442

The word "and" can also imply a partition of a thing into parts, as "The American flag is red, white, and blue." Here, it is not meant that the flag is at once red, white, and blue, but rather that it has a part of each color.

4,443

In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content , other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false".

4,444

In classical logic, with its intended semantics, the truth values are true , and untrue or false ; that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws:

4,445

Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.

4,446

In intuitionistic logic, and more generally, constructive mathematics, statements are assigned a truth value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if one can build a proof of the statement from those axioms. A statement is false if one can deduce a contradiction from it. This leaves open the possibility of statements that have not yet been assigned a truth value. Unproven statements in intuitionistic logic are not given an intermediate truth value . Indeed, one can prove that they have no third truth value, a result dating back to Glivenko in 1928.

4,447

Instead, statements simply remain of unknown truth value, until they are either proven or disproven.

4,448

There are various ways of interpreting intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation. See also Intuitionistic logic § Semantics.

4,449

Multi-valued logics allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval such structure is a total order; this may be expressed as the existence of various degrees of truth.

4,450

Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the necessary truth of formulae.

4,451

But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Boolean algebra semantics of classical propositional calculus.

4,452

Intuitionistic type theory uses types in the place of truth values.

4,453

Topos theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.

4,454

The real numbers are fundamental in calculus , in particular by their role in the classical definitions of limits, continuity and derivatives.

4,455

The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers. Some irrational numbers are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers. There are also real numbers which are not, such as π = 3.1415...; these are called transcendental numbers.

4,456

Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers are equally spaced.

4,457

Conversely, analytic geometry is the association of points on lines to real numbers such that geometric displacements are proportional to differences between corresponding numbers.

4,458

The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current axiomatic definition is that real numbers form the unique Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences , Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

4,459

Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence.

4,460

The real numbers form an ordered field. Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations, addition and multiplication, and a total order that have the following properties.

4,461

Many other properties can be deduced from the above ones. In particular:

4,462

Several other operations are commonly used, which can be deduced from the above ones.

4,463

The real numbers 0 and 1 are commonly identified with the natural numbers 0 and 1. This allows identifying any natural number n with the sum of n real numbers equal to 1.

4,464

The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties . So, the identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with 1 taken as the successor function.

4,465

These identifications are formally abuses of notation, and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming. In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by the compiler.

4,466

Dedekind completeness implies other sorts of completeness , but also has some important consequences.

4,467

The last two properties are summarized by saying that the real numbers form a real closed field. This implies the real version of the fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.

4,468

A key property of real numbers is their decimal representation. A decimal representation consists of a nonnegative integer k and an infinite sequence of decimal digits

4,469

that is written

4,470

The real number defined by the sequence is the least upper bound of the D n , {\displaystyle D_{n},} which exists by Dedekind completeness.

4,471

In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9.

4,472

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete :

4,473

A sequence of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N such that the distance |xn − xm| is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the xn eventually come and remain arbitrarily close to each other.

4,474

A sequence converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N such that the distance |xn − x| is less than ε for n greater than N.

4,475

Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.

4,476

The set of rational numbers is not complete. For example, the sequence , where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number .

4,477

The completeness property of the reals is the basis on which calculus, and more generally mathematical analysis, are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.

4,478

For example, the standard series of the exponential function

4,479

converges to a real number for every x, because the sums

4,480

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

4,481

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element .

4,482

Additionally, an order can be Dedekind-complete, see § Axiomatic approach. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field and then forms the Dedekind-completion of it in a standard way.

4,483

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

4,484

The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible , separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

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The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.

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A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.

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Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava , who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.

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The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects . Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation . In Europe, such numbers, not commensurable with the numerical unit, were called irrational or surd .

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In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.

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In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.

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In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert gave a flawed proof that π cannot be rational; Legendre completed the proof and showed that π is not the square root of a rational number. Liouville showed that neither e nor e2 can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor extended and greatly simplified this proof. Hermite proved that e is transcendental, and Lindemann , showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass , Hilbert , Hurwitz, and Gordan.

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The developers of calculus used real numbers without having defined them rigorously. The first rigorous definition was published by Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891.

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These properties imply the Archimedean property , which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N.

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The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like converges to a unique real number—in this case π. For details and other constructions of real numbers, see construction of the real numbers.

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In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.

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Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.

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The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.

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The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.

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Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are elements of the set of the real numbers .

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Electronic calculators and computers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary definable real numbers, which are inconvenient to manipulate.