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Dedekind completeness

Previous properties do not distinguish real numbers from rational numbers. This distinction is provided by Dedekind completeness, which states that every set of real numbers with an upper bound admits a least upper bound. This means the following. A set of real numbers S {\displaystyle S} is bounded above if there is a real number u {\displaystyle u} such that s ≤ u {\displaystyle s\leq u} for all s ∈ S {\displaystyle s\in S}; such a u {\displaystyle u} is called an upper bound of S. {\displaystyle S.} So, Dedekind completeness means that, if S is bounded above, it has an upper bound that is less than any other upper bound.

Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences.

The last two properties are summarized by saying that the real numbers form a real closed field. This implies the real version of the fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.