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The real numbers can be generalized and extended in several different directions: The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field. The affinely extended real number system adds two elements +∞ and −∞.
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 ...
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.
In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number. Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom ), or may be a theorem proven from the ...
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.
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.
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.
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation =; every complex number can be expressed in the form +, where a and b are real numbers.