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Concept
Construction of the real numbers
Summary
In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition.
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. 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 numbers and denoted respectively with + and ×; the binary relation is inequality, denoted Moreover, the following properties called axioms must be satisfied.
The existence of such a structure is a theorem, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
is a field under addition and multiplication. In other words,
For all x, y, and z in , x + (y + z) = (x + y) + z and x × (y × z) = (x × y) × z. (associativity of addition and multiplication)
For all x and y in , x + y = y + x and x × y = y × x. (commutativity of addition and multiplication)
For all x, y, and z in , x × (y + z) = (x × y) + (x × z). (distributivity of multiplication over addition)
For all x in , x + 0 = x.
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