"Gaps" or "holes" in rational number system In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$. Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $p^2<2$, and $B$ consists of all positive rationals $p$ such that $p^2>2$. He shows that $A$ contains no largest number and $B$ contains no smallest.
And then in 1.2, Rudin remarks that what he has done above is to show that the rational number system has certain gaps. His remarks confused me.
My questions are:


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*If he had shown that no rational number $p$ with $p^2=2$, this already gave the conclusion that rational number system has "gaps" or "holes". Why did he need to set up the second argument about the two sets $A$ and $B$?

*How does the second argument that "$A$ contains no largest number and $B$ contains no smallest" showed gaps in rational number system? My intuition does not work here. Or it is nothing to do with intuition?

There is a difference between a thing not existing in some set, and the existence of "gap" corresponding to that thing.  For example, there is no rational number $p$ such that $p > q$ for all rational numbers $q$.  Does this mean that there is a "gap" in the rationals corresponding to some "largest" rational number?  I think that most people would argue that, no, there is no "gap" there.
Or, perhaps more interestingly, there is no rational number $p$ such that $p^2 = -1$.  In order to solve the equation $p^2 + 1 = 0$, it is necessary to introduce the imaginary unit $i$ and the complex number system (or, perhaps, the Gaussian rationals; we don't need a continuum, really).  Is the lack of existence of a rational $p$ such that $p^2 = -1$ a "gap"?  Again, I think that most people would argue that it is not.
Similarly, it is not a priori obvious that the non-existence of a (positive) rational number $p$ such that $p^2 = 2$ represents any kind of gap in the rational number system.  By showing that no such $p$ exists, all Rudin has done is show that no such $p$ exists.  This seems tautological (because it is), but the situation is analogous to the non-existence of a largest rational number or the imaginary unit.
What Rudin then does is demonstrate that there is a "rational number like object", $s$, which can meaningfully be said to have the following properties:


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*$s^2 = 2$,

*there is a set of positive rational numbers $A$ such that $a \in A$ implies that $a < s$, and

*there is a set of positive rational numbers $B$ such that $b \in B$ implies that $b > s$.
Thus, in a very meaningful sense, this object $s$ fits into the rational number system in a natural way.  It "plugs a hole" in the rationals.  Contrast this with the imaginary unit $i$, which doesn't fit into the rational number system in any natural way—it lives in a place that is orthogonal to the rationals.
