Question

In real numbers what they will include

Answer 1

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In mathematics, a real numberis a value of a continuous quantity that can represent a distance along a line. ... The real numbers include all therational numbers, such as theinteger −5 and the fraction 4/3, and all the irrationalnumbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number).


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Answer 2

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In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century byRené 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 √2(1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such astime, mass, energy, velocity, and many more.

Real numbers can be thought of as points on an infinitely long line called the number line orreal line, where the points corresponding tointegers are equally spaced. Any real number can be determined by a possibly infinitedecimal 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 thecomplex 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 current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; <), up to an isomorphism,[a] whereas popular constructive definitions of real numbers include declaring them asequivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinitedecimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.

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 {\displaystyle {\mathfrak {c}}and calledcardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted {\displaystyle \aleph _{0} 'aleph-naught'). The statement that there is no subset of the reals with cardinality strictly greater than {\displaystyle \aleph _{0}}and strictly smaller than {\displaystyle {\mathfrak {c} is known as thecontinuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theoryincluding the axiom of choice (ZFC), the standard foundation of modern mathematics, in the sense that some models of ZFC satisfy CH, while others violate it.

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