Completeness of the real numbers

Common card · No. 380653

Completeness is a property of the real numbers that, intuitively, implies that there are no "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 a 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 forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space). Depending on which other properties are assumed, Dedekind completeness and Cauchy completeness may be equivalent properties, or Dedekind completeness may be a stronger property than Cauchy completeness.

Read on Wikipedia