Concept
Construction of the real numbers
Related concepts (17)
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
Surreal number
In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than 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 was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.
Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction \tfrac p q of two integers, a numerator p and a non-zero denominator q. For example, \tfrac{-3}{7} is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold \Q. A rational number is a real number.
Least-upper-bound property
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set of all rational numbers with its natural order does not have the least upper bound property.
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers and , there is an integer such that . It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no infinitely large or infinitely small elements.
Irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph 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 B, such that all elements of A are less than all elements of B, and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational.
Constructivism (philosophy of mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it.
Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.
0.999...
In mathematics, 0.999... (also written as 0. or 0.) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence. This number is equal to1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1 - rather, "0.999..." and "1" represent the same number.