# Completeness of The Real Numbers

Completeness Of The Real Numbers

Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line. This contrasts with the rational numbers, whose corresponding number line has a “gap” at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually the 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 construction. There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space).

Read more about Completeness Of The Real Numbers:  Forms of Completeness

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Completeness Of The Real Numbers - Forms of Completeness - Bolzano–Weierstrass Theorem
... The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence ... Again, this theorem is equivalent to the other forms of completeness given above ...

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