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 $B$?
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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?
Best Answer
It depends on what you consider a “gap” in the rational numbers. As long as this is not a formally defined concept, we’re just talking about our everyday, geometrically informed conceptions of gaps.
The mere fact that a certain equation doesn’t have a rational solution doesn’t seem like a basis for identifying a “gap”. The equation $x^2=-1$ also has no solution in the rational numbers, and this fact also gives rise to an extension of the number system (to the complex numbers, in this case), but it doesn’t fit with our everyday notion of a gap to call this deficiency a “gap”. This corresponds to the fact that when we fill the need to solve the equation $x^2=2$ by introducing irrational numbers, we depict them on the same axis as the rational numbers, between rational numbers, whereas when we fill the need to solve the equation $x^2=-1$ by introducing imaginary numbers, we depict them along a different axis.
So the mere fact that some equation can’t be solved does not indicate a gap in the number system, if by “gap” we mean anything like what we mean by it in everyday language (where a “gap” would certainly be depicted along the same axis as the things between which it lies). By contrast, the fact that you can split the rational numbers into two sets, with all numbers in one set greater than all numbers in the other but without a number that marks the boundary, does seem to suggest that there “should” be a number at the boundary, so that, in a sense not too removed from our everyday use of the word, there is a gap at the boundary.