The real numbers form a field. This is a set of numbers with two [binary] operations defined on it: addition (usually denoted by +) and multiplication (usually denoted by *) such that:
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No, they are not the same. Axioms cannot be proved, most properties can.
There are too many to enumerate.
Real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced.
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which mixed number or improper fraction is closest to the decimal 5.27?