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  2. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    The real number x defined by the sequence is the least upper bound of the , which exists by Dedekind completeness. Conversely, given a nonnegative real number x, one can define a decimal representation of x by induction, as follows.

  3. Construction of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Construction_of_the_real...

    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 ...

  4. Extended real number line - Wikipedia

    en.wikipedia.org/wiki/Extended_real_number_line

    Extended real numbers (top) vs projectively extended real numbers (bottom). In mathematics, the extended real number system [a] is obtained from the real number system by adding two elements denoted + and [b] that are respectively greater and lower than every real number.

  5. Completeness of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Completeness_of_the_real...

    The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom.Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields ...

  6. Real analysis - Wikipedia

    en.wikipedia.org/wiki/Real_analysis

    The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique complete ordered field, in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers.

  7. List of numeral systems - Wikipedia

    en.wikipedia.org/wiki/List_of_numeral_systems

    "A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." [1]: 38 The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. [1]

  8. Real coordinate space - Wikipedia

    en.wikipedia.org/wiki/Real_coordinate_space

    Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers. In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted R n or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors.

  9. List of types of numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_types_of_numbers

    Positive numbers: Real numbers that are greater than zero. Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal ...