Completeness of the real numbers
Concept in mathematics / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Completeness of the real numbers?
Summarize this article for a 10 year old
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 equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space).