forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space). The real numbers can be...
11 KB (1,521 words) - 21:13, 9 December 2023
Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences. Archimedean property: for every real number...
61 KB (8,208 words) - 21:57, 25 August 2024
completeness, complete, completed, or incompleteness in Wiktionary, the free dictionary. Complete may refer to: Completeness (logic) Completeness of a...
4 KB (509 words) - 14:24, 14 May 2024
use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment. Cantor's construction of the real...
16 KB (2,525 words) - 07:00, 19 April 2024
(poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices...
13 KB (1,924 words) - 01:21, 18 August 2023
equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field...
31 KB (4,202 words) - 00:15, 12 August 2024
Cauchy sequence (section In real numbers)
constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. One of the standard illustrations...
20 KB (3,219 words) - 14:10, 2 July 2024
Infimum and supremum (category Pages displaying short descriptions of redirect targets via Module:Annotated link)
number). The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset S {\displaystyle S} of the real numbers has an...
24 KB (4,346 words) - 03:49, 4 August 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 (11,605 words) - 23:07, 1 September 2024
Least-upper-bound property (redirect from Dedekind completeness)
the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers. More...
11 KB (1,329 words) - 22:00, 9 September 2024
a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the...
21 KB (2,982 words) - 05:08, 28 August 2024
Rational number Irrational number Completeness of the real numbers Least-upper-bound property Real line Extended real number line Dedekind cut 0 1 0.999...
14 KB (1,603 words) - 09:24, 23 July 2024
Computable number (redirect from Computable numbers)
known as the recursive numbers, effective numbers or the computable reals or recursive reals. The concept of a computable real number was introduced by...
24 KB (3,263 words) - 06:14, 11 August 2024
(infinite) sets of elements. The basic property of the completeness of the real numbers that is required for defining and using real numbers involves a quantification...
52 KB (6,887 words) - 08:33, 23 August 2024
critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least...
49 KB (7,673 words) - 10:05, 6 August 2024
Transcendental number (redirect from Real transcendental numbers)
all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex...
51 KB (6,761 words) - 14:52, 25 August 2024
theory T but true in the "standard" model of the natural numbers: φu is false in some other, "non-standard" models of T.) The completeness theorem makes a...
17 KB (2,329 words) - 02:29, 10 September 2024
Rational number (redirect from Set of rational numbers)
number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example:...
24 KB (3,494 words) - 14:12, 30 July 2024
decode other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all...
29 KB (3,163 words) - 23:27, 21 May 2024
Sequence (redirect from Function of an integer variable)
sequence of rational numbers (e.g. via its decimal expansion, also see completeness of the real numbers). As another example, π is the limit of the sequence...
39 KB (6,156 words) - 17:44, 7 September 2024
Intermediate value theorem (category Theorems in real analysis)
and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers Q because gaps exist...
26 KB (4,331 words) - 16:08, 10 July 2024
Number (redirect from History of numbers)
real numbers such as the square root of 2 ( 2 ) {\displaystyle \left({\sqrt {2}}\right)} and π, and complex numbers which extend the real numbers with...
62 KB (7,747 words) - 11:27, 10 September 2024
proof must use the completeness of the real numbers, which is not an algebraic property). This article describes the history of the theory of equations, called...
120 KB (16,878 words) - 07:06, 26 July 2024
The following assertions are equivalent to ACA0 over RCA0: The sequential completeness of the real numbers (every bounded increasing sequence of real...
37 KB (4,665 words) - 19:08, 5 June 2024
+\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) - 07:06, 23 May 2024
This is a list of notable numbers and articles about notable numbers. The list does not contain all numbers in existence as most of the number sets are...
58 KB (3,872 words) - 14:53, 5 September 2024
Nested intervals (redirect from Nested sequence of closed intervals)
Real Analysis, Springer, p. 45, ISBN 9780817642112. Königsberger, Konrad (2003), "2.3 Die Vollständigkeit von R (the completeness of the real numbers)"...
22 KB (4,101 words) - 19:08, 25 July 2024
Irrational number (redirect from History of irrational numbers)
mathematics, the irrational numbers (in- + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed...
39 KB (5,254 words) - 23:16, 8 September 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
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