Real Number

A Real Number is a number that when squared results in a positive number.

• Context:
  • It can range from being a Rational Number to being an Irrational Number.
  • It can range from being a Non-Negative Real Number to being a Negative Real Number to being a Non-Positive Real Number to being a Positive Real Number.
  • It can be a member of a Real Number Set, Real Number Multiset, Real Number List, ...
• Example(s):
  • an Integer Number, such as 0.
  • a Real-Valued Prediction.
  • an Irrational Number, such as: [math]\displaystyle{ \pi }[/math].
  • a Random Variable Value.
  • …
• Counter-Example(s):
  • an Imaginary Number.
• See: Complex Number, Continuous Function, Scalar Output Function, Real Matrix, Non-Terminating Decimal Number.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/real_number Retrieved:2015-6-22.
  • In mathematics, a real number is a value that represents a quantity along a continuous line. The adjective real in this context was introduced in the 17th century by Descartes, who distinguished between real and imaginary roots of polynomials.

    The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as (1.41421356…, the square root of two, an irrational algebraic number) and pi (3.14159265…, a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.

    These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of classical mathematics.

    The reals are uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted [math]\displaystyle{ \mathfrak c }[/math] and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted [math]\displaystyle{ \aleph_0 }[/math]). The statement that there is no subset of the reals with cardinality strictly greater than [math]\displaystyle{ \aleph_0 }[/math] and strictly smaller than [math]\displaystyle{ \mathfrak c }[/math] is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory (ZF), the standard foundation of modern mathematics, in the sense that some models of ZF satisfy CH while others violate it.

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/real_number#Definition Retrieved:2015-6-22.
  • The real number system [math]\displaystyle{ (\mathbb R ; + ; \cdot ; \lt ) }[/math] can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.

2009

• (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=real%20number
  • S: (n) real number, real (any rational or irrational number)

2000

• (Witten & Frank, 2000) ⇒ Ian H. Witten, and Eibe Frank. (2000). “Data Mining: Practical Machine Learning Tools and Techniques with Java implementations." Morgan Kaufmann.

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