In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain... 29 KB (4,049 words) - 02:17, 26 April 2024 |
construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction. There... 11 KB (1,521 words) - 21:13, 9 December 2023 |
Rational number (redirect from Set of rational numbers) decimals (see Construction of the real numbers). The term rational in reference to the set Q {\displaystyle \mathbb {Q} } refers to the fact that a rational... 24 KB (3,494 words) - 12:24, 28 April 2024 |
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular... 49 KB (7,673 words) - 19:10, 25 December 2023 |
Hyperreal number (redirect from Nonstandard real numbers) mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number... 33 KB (4,892 words) - 07:43, 21 April 2024 |
Complex number (redirect from Non real numbers) an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i... 89 KB (11,600 words) - 17:44, 26 April 2024 |
Dedekind cut (category Real numbers) Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two sets A and... 13 KB (2,069 words) - 18:57, 26 April 2024 |
Surreal number (redirect from SurrealNumbers) any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction... 75 KB (11,406 words) - 10:29, 11 April 2024 |
Least-upper-bound property (category Real analysis) and the Heine–Borel theorem. It is usually taken as an axiom in synthetic constructions of the real numbers, and it is also intimately related to the construction... 12 KB (1,470 words) - 20:02, 7 December 2023 |
Cauchy sequence (section In real numbers) complete. The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves... 20 KB (3,218 words) - 06:27, 23 January 2024 |
Multiplication (redirect from Product of two negative numbers) arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers; see construction of the real numbers. There are many... 48 KB (6,245 words) - 13:49, 27 April 2024 |
Complete metric space (redirect from Completion of a metric space) Cantor's construction of the real numbers is similar to the above construction; the real numbers are the completion of the rational numbers using the ordinary... 16 KB (2,519 words) - 07:00, 19 April 2024 |
Transcendental number (redirect from Real transcendental numbers) set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers... 53 KB (6,898 words) - 04:32, 8 April 2024 |
Number (redirect from History of numbers) included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes... 62 KB (7,755 words) - 02:08, 25 January 2024 |
mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and... 4 KB (496 words) - 23:15, 25 April 2024 |
0.999... (redirect from Infinite series of nines) such as series, limits, formal construction of real numbers, etc. The proof given below is a direct formalization of the intuitive fact that, if one draws... 96 KB (13,418 words) - 14:22, 18 April 2024 |
Computable number (redirect from Computable numbers) the recursive numbers, effective numbers or the computable reals or recursive reals.[citation needed] The concept of a computable real number was introduced... 23 KB (3,168 words) - 07:27, 23 September 2023 |
real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of... 11 KB (1,502 words) - 02:55, 9 April 2024 |
the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers... 18 KB (2,226 words) - 10:26, 6 March 2024 |
Nested intervals (redirect from Nested sequence of closed intervals) a construction method for the real numbers (in order to complete the field of rational numbers). As stated in the introduction, historic users of mathematics... 22 KB (4,101 words) - 17:17, 19 December 2023 |
+\infty \right\}.} It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol + ∞ {\displaystyle... 14 KB (2,129 words) - 13:10, 7 December 2023 |
Integer (redirect from Construction of the integers) 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... 34 KB (3,924 words) - 08:26, 22 April 2024 |
ring F of finite hyperrationals (i.e. ratio of a pair of hyperintegers), see construction of the real numbers. The quotients R[X] / (X), R[X] / (X + 1), and... 16 KB (2,064 words) - 20:12, 15 March 2024 |
ISO 4157 (category Construction standards) parts of buildings" ISO 4157-2:1998 "Construction drawings -- Designation systems -- Part 2: Room names and numbers" ISO 4157-3:1998 "Construction drawings... 1 KB (144 words) - 10:11, 17 January 2024 |
Natural number (redirect from Set of natural numbers) measurement—a hallmark characteristic of real numbers. The natural numbers can be used for counting (as in "there are six coins on the table"), in which case they... 48 KB (5,485 words) - 11:10, 9 April 2024 |
Cantor's first set theory article (redirect from On a Property of the Collection of All Real Algebraic Numbers) from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber... 101 KB (7,536 words) - 20:48, 11 February 2024 |
Archimedean property (redirect from Axiom of Archimedes) the usual absolute value (from the order) is the field of real numbers. By this construction the field of real numbers is Archimedean both as an ordered... 16 KB (2,386 words) - 11:34, 3 September 2023 |
show that it is the crowning good. Euclid Euclid's Elements Eudoxus reals (a fairly recently discovered construction of the real numbers, named in his honor)... 22 KB (2,685 words) - 03:03, 7 April 2024 |
in the closed and bounded interval [ a , b ] {\displaystyle [a,b]} , then it must attain a maximum and a minimum Construction of the real numbers Natural... 14 KB (1,595 words) - 07:42, 1 January 2024 |