Search results
Results from the WOW.Com Content Network
The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers . Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414... ; these are called algebraic numbers .
Considering the rational fractions with real coefficients, radical expressions representing numbers, such as / , are also rational fractions, as are a transcendental numbers such as /, since all of ,, and are real numbers, and thus considered as coefficients.
The rationals are a dense subset of the real numbers; every real number has rational numbers arbitrarily close to it. [6] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. [18] In the usual topology of the real numbers, the rationals are neither an open set nor a closed set. [19]
Computable number: A real number whose digits can be computed by some algorithm. Period: A number which can be computed as the integral of some algebraic function over an algebraic domain. Definable number: A real number that can be defined uniquely using a first-order formula with one free variable in the language of set theory.
For a positive number written in a conventional positional numeral system (such as binary or decimal), its fractional part hence corresponds to the digits appearing after the radix point, such as the decimal point in English. The result is a real number in the half-open interval [0, 1).
The term "real number" is a retronym coined in response to "imaginary number". Together with the p-adic numbers , the reals are a limit set of the rational numbers . Real numbers may be represented in a variety of ways: as binary numbers , as continued fractions , and in other interesting ways, such as rational zeta series .
In reverse mathematics, one way of constructing the real numbers is to represent them as functions from unary numbers to dyadic rationals, where the value of one of these functions for the argument is a dyadic rational with denominator that approximates the given real number. Defining real numbers in this way allows many of the basic results of ...
The real numbers are more numerous than the ... and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, ...