# Why does Rudin say "the rational number system is inadequate as a field"?

In the INTRODUCTION of chapter 1 of Baby Rudin, he says

The rational number system is inadequate for many purposes, both as a field and as an ordered set.

Addition and multiplication of rational numbers are commutative and associative, and multiplication is distributive over addition. Both 'zero' and 'one' exist.

Plus, as I recall, rational numbers are the smallest subfield of $\mathbb{C}$. So exactly what does Rudin mean by “inadequate as a field”?

• because it doesnt have limit points, so you cant safely do calculus with it, which is what that textbook is about. Aug 13, 2017 at 8:26
• Is it possible that another form of intelligent life in the universe is technically superior to us but finds the field of rational numbers more than adequate in extending their understanding of science? Aug 13, 2017 at 11:00
• @MikeMathMan no, we do not choose whether to use reals or rationals. The rationals simply don't have enough points for us to do calculus in. Aug 13, 2017 at 17:12
• Just in case it's a language issue, "inadequate" here doesn't mean that $\mathbb{Q}$ fails to be a field or an ordered set. It means that, despite being an ordered set and a field, it is not adequate for all purposes in which we need an ordered field. Aug 13, 2017 at 23:42
• If worded thus, can be better understood: "Q, being at the same time a field and an ordered set, is inadequate for many purposes." Aug 14, 2017 at 8:37

Rudin isn't questioning $\mathbb{Q}$'s status as a field — he's questioning it's suitability for doing algebra and analysis.

For the purposes of algebra, $\mathbb{Q}$ is not a very nice field to work with: for example, many (most?) polynomials don't even have a root, let alone factor completely, and dealing properly with this deficiency is the subject of the frontiers of mathematical research!

(for comparison, while not every polynomial over $\mathbb{R}$ has a root, it has a very simple relationship with the complex numbers $\mathbb{C}$, a field over which every nonconstant polynomial has a root)

Similarly, for the purposes of analysis, $\mathbb{Q}$ is not a very nice ordered set; for example, they form a totally disconnected space, making them nearly useless for capturing even basic familiar geometric notions such as continuity.

The author means that $\mathbb Q$ as an ordered field is incomplete, i.e. not every Cauchy sequence in $\mathbb Q$ converges in $\mathbb Q$, or equivalently not every nonempty subset of $\mathbb Q$ that is bounded above has a least upper bound in $\mathbb Q$. This makes $\mathbb Q$ "inadequate" for many purposes in analysis, where the least-upper-bound property is required. For example, when $a$ is a positive rational number and $n$ a positive integer, you want the equation $x^n=a$ to have a solution; this does not always happen in $\mathbb Q$.

$\mathbb Q$ can be made complete by enlarging it to the set $\mathbb R$ of real numbers. There are two main ways by which this can be achieved: either by considering equivalence classes of Cauchy sequences in $\mathbb Q$, or by means of Dedekind cuts. It can be shown that $\mathbb R$ as a complete ordered field is unique: any two complete ordered fields are isomorphic.

• Just for completeness :P There are other nontrivial ways of completing $\mathbb{Q}$, one for every prime $p$: the $p$-adic fields $\mathbb{Q}_p$, powerful tools for number theorists in which one can also do an analysis similar to that of $\mathbb{R}=:\mathbb{Q}_\infty$. These are all the nontrivial ways of completing the rationals, by Ostrowski's theorem. Sep 25, 2017 at 13:09

Your remark “Addition and multiplication of rational numbers are commutative and associative, and multiplication is distributive over addition. Both 'zero' and 'one' exist” is inappropriate. Of course all of this is true because otherwise $\mathbb Q$ would not be a field.

However, as far as Analysis is concerned, $\mathbb Q$ is inadequate because:

• in $\mathbb Q$, a Cauchy sequence is not necessarily convergent;
• a monotonic and bounded sequence of rational numbers doesn't have to converge to a rational number;
• a bounded set of rational numbers doesn't need to have a supremum or an infimum in $\mathbb Q$;
• a continuous function defined on a closed and bounded interval of $\mathbb Q$ may well be unbounded;
• a continuous function from $\mathbb Q$ into $\mathbb Q$ may not have the intermediate value property.

And so on.

• You mean "monotonic" sequence, not "monotonous", but it won't let me make an edit shorter than 6 characters. ("Monotonous" means "boring.") Aug 13, 2017 at 19:49
• Edited. Thanks. Aug 13, 2017 at 20:27

To add to the other (excellent) answers, the rationals are inadequate even for elementary mathematics. Starting with only a vague school-level idea of what numbers are, when we draw the graph of (say) $y=x^2-2$, it's visually obvious that the graph crosses the $x$ axis. With school arithmetic, we can even calculate to high accuracy just where the crossing points are. But the points do not exist if we are restricted to rational numbers. Similarly, under this restriction, no sense can be made of even simple geometric ratios, such as between the diagonal and side of a square or the circumference and diameter of a circle (albeit proof of the latter impossibility needing much more advanced mathematics).

Another way to look at it is there are a lot numbers of interest or importance in calculus that are not rational, namely irrational numbers such as $\pi$, $e$, $\sqrt{2}$... In fact, the set of irrational numbers is a far bigger subset of the real numbers than is the set of rationals.

Once I was surprised to see that many dozen pages in the Chapter on Riemann Stieltjes Integrals given in Apostole's Mathematical Analysis are valid even in the rational number system. So there can be a good part of analysis for rational numbers also.

• Are you sure? Isn't continuity required for integrals?
– user370967
Sep 10, 2017 at 11:01

The real numbers are a subfield of the complex numbers. Not every sequence of rational numbers converges to a rational number: rational numbers aren't a complete set.

• I was wrong! I thought in the biggest subfield. Thank you very much for your comment! Aug 15, 2017 at 1:59
• While I didn't downvote you, I can understand why someone would. The first point that you raise seems to be irrelevant and the other two points just repeat points that have already been made in other answers. Before posting an answer, ask yourself if your answer adds anything to what has already been said. Aug 15, 2017 at 2:04
• Indeed, not every sequence of rational numbers converges to a rational number. And not every sequence of real numbers converges to a real number. Besides, not every sequence of complex numbers converges to a complex number. So, what's peculiar with the rationals as far as this is concerned? Aug 15, 2017 at 12:34
• If a sequence of real numbers converges, converges to a real number; if a sequence of complex numbers converges, converges to a complex number. Aug 15, 2017 at 21:25