Real number

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number".

Definition

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.

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Axiomatic approach

Let R denote the set of all real numbers. Then:

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• The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
• if x ≥ y then x + z ≥ y + z;
• if x ≥ 0 and y ≥ 0 then xy ≥ 0.
• The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.

The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.

Related Topics:
Rationals - Square root

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The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object.

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