Confused about incompleteness of rational numbers The following two facts about rational numbers confuse me a bit:


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*Rational numbers are ordered but unlike real numbers are not complete. The incompleteness is due to the existence of gaps in the rational numbers. When we say gaps for this ordered objects, doesn't it intuitively mean that there should exist a pair of rational numbers with a gap in between them. 

*There exist infinitely many rational numbers in between any two rational numbers. Doesn't this intuitively mean there is no gap between any two rational numbers?

A gap in the set of rational numbers is a pair of non-empty sets $A,B$ of rational numbers such that


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*every rational number is a member of $A$ or of $B$, and

*every member of $A$ is strictly less than every member of $B$ (and consequently no number can be a member of both), and

*$A$ has no largest member and $B$ has no smallest member. For every number $a_1\in A$ there is some number $a_2\in A$ such that $a_2>a_1$ and for every number $b_1\in B$ there is sum number $b_2\in B$ such that $b_2< b_1.$


If $A$ is the set of all rational numbers less than $\sqrt 2$ and $B$ is the set of all rational numbers greater than $\sqrt 2,$ that is a gap.
The set of all real numbers, by contrast, has no gaps.
