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...

distinguishes real numbers from imaginary numbers such as the square roots of −1. The real numbers include the rational numbers, such as the integer −5 and the fraction...

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...

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...

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular...

mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

the recursive numbers, effective numbers or the computable reals or recursive reals.[citation needed] The concept of a computable real number was introduced...

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...

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...

+\infty \right\}.} It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol + ∞ {\displaystyle...

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...

the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers...

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...

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...

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...

parts of buildings" ISO 4157-2:1998 "Construction drawings -- Designation systems -- Part 2: Room names and numbers" ISO 4157-3:1998 "Construction drawings...

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...

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...

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)...