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The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...
[9] [6] The method of cylindrical algebraic decomposition, by George E. Collins (1975), improved the time dependence to doubly exponential, [9] [10] of the form () where is the number of bits needed to represent the coefficients in the sentence whose value is to be determined, is the number of polynomials in the sentence, is their total degree ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
The real numbers have various lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property: Every nonempty subset of ...
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.. In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
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Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, , , et cetera, along with all algebraic numbers. Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of ZFC form a field.
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