real numbers - ορισμός. Τι είναι το real numbers
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Τι (ποιος) είναι real numbers - ορισμός

REAL NUMBER THAT CAN BE COMPUTED TO WITHIN ANY DESIRED PRECISION BY A FINITE, TERMINATING ALGORITHM
Computable numbers; Recursive number; Recursive numbers; Uncomputable number; Non-computable numbers; Noncomputable number; Non-computable number; Computable real; Computable real number; Computable reals; Uncomputable numbers; Uncomputable real number
  • π]] can be computed to arbitrary precision, while [[almost every]] real number is not computable.

real number         
  • A symbol for the set of real numbers
  • Real numbers <math>(\mathbb{R})</math> include the rational numbers <math>(\mathbb{Q})</math>, which include the integers <math>(\mathbb{Z})</math>, which in turn include the natural numbers <math>(\mathbb{N})</math>
  • Real numbers can be thought of as all points on a number line
QUANTITY ALONG A CONTINUOUS LINE
Real numbers; Real Numbers; Bounded real-valued data; Real number field; Real (numbers); ℝ; Field of reals; Axiomatic real number; Complete ordered field; The complete ordered field; Reall numbers; Real number system; Real (number); Real Number System; Set of real numbers; R (math); R (maths)
<mathematics> One of the infinitely divisible range of values between positive and negative infinity, used to represent continuous physical quantities such as distance, time and temperature. Between any two real numbers there are infinitely many more real numbers. The integers ("counting numbers") are real numbers with no fractional part and real numbers ("measuring numbers") are complex numbers with no imaginary part. Real numbers can be divided into rational numbers and {irrational numbers}. Real numbers are usually represented (approximately) by computers as floating point numbers. Strictly, real numbers are the equivalence classes of the Cauchy sequences of rationals under the {equivalence relation} "real number", where a real number b if and only if a-b is Cauchy with limit 0. The real numbers are the minimal topologically closed field containing the rational field. A sequence, r, of rationals (i.e. a function, r, from the natural numbers to the rationals) is said to be Cauchy precisely if, for any tolerance delta there is a size, N, beyond which: for any n, m exceeding N, | r[n] - r[m] | < delta A Cauchy sequence, r, has limit x precisely if, for any tolerance delta there is a size, N, beyond which: for any n exceeding N, | r[n] - x | < delta (i.e. r would remain Cauchy if any of its elements, no matter how late, were replaced by x). It is possible to perform addition on the reals, because the equivalence class of a sum of two sequences can be shown to be the equivalence class of the sum of any two sequences equivalent to the given originals: ie, areal numberb and creal numberd implies a+creal numberb+d; likewise a.creal numberb.d so we can perform multiplication. Indeed, there is a natural embedding of the rationals in the reals (via, for any rational, the sequence which takes no other value than that rational) which suffices, when extended via continuity, to import most of the algebraic properties of the rationals to the reals. (1997-03-12)
Real number         
  • A symbol for the set of real numbers
  • Real numbers <math>(\mathbb{R})</math> include the rational numbers <math>(\mathbb{Q})</math>, which include the integers <math>(\mathbb{Z})</math>, which in turn include the natural numbers <math>(\mathbb{N})</math>
  • Real numbers can be thought of as all points on a number line
QUANTITY ALONG A CONTINUOUS LINE
Real numbers; Real Numbers; Bounded real-valued data; Real number field; Real (numbers); ℝ; Field of reals; Axiomatic real number; Complete ordered field; The complete ordered field; Reall numbers; Real number system; Real (number); Real Number System; Set of real numbers; R (math); R (maths)
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials.
Construction of the real numbers         
  • irrational]], [[real number]]s.
AXIOMATIC DEFINITIONS OF THE REAL NUMBERS
Constructions of the real numbers; Construction of reals; Construction of real numbers; Construction of the reals; Axiomatic theory of real numbers; Constructions of real numbers; Hyperrational numbers; Hyperrational number
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 any smaller complete ordered field.

Βικιπαίδεια

Computable number

In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals. The concept of a computable real number was introduced by Emile Borel in 1912, using the intuitive notion of computability available at the time.

Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

Παραδείγματα από το σώμα κειμένου για real numbers
1. According to Schmid, though, the real numbers could be higher.
2. Local authorities are protesting that existing population figures do not reflect real numbers.
3. Officials in affected areas say the death toll given by the ministry is far below the real numbers.
4. Meanwhile, China constitutes a security threat, too: It‘s spending billions on an arms buildup, and lying about the real numbers.
5. After months when it was all about expectations and momentum, not to mention confusion, real numbers finally became important.