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I am interested in theorems whose. . .

  1. . . . hypotheses and conclusions are 'discrete' and mention only rational numbers.

  2. . . . most natural proofs invariably require an understanding of the continuous nature of the real number line, as opposed to the rational number line. In particular how it is connected.


For measuring points on a ruler, the real number line is inaccurate model. Every measurement we can make in the real world has finite precision. Every point you can mark on the line is in fact a very short interval. Looking too closely at the ruler reveals its granular -- not continuous -- nature. Looking closer still reveals the space between atoms and the uncertainty principle that makes it impossible to say exactly where we are.

To model the ruler $\mathbb Q$ is just as good and much easier to understand than $\mathbb R$. Every element can be given by finite data. We could go smaller: The set $\mathbb D$ of dyadic rationals is just as good of a model. The biggest we might want to go is the algebraic closure $\overline{\mathbb Q}$ of the field of fractions.

However the real number line has very nice topological properties. For one it is connected. For two it better captures the continuous nature of polynomials. A map can be continuous over the rationals but not over the reals for example.

But even if we are only interested in rational numbers -- henceforth called just numbers -- we still might want to pretend our polynomials are defined over the real line. Because then we can use topology to derive facts like the following.

Suppose the polynomial $P$ is negative at $0$ and positive at $1$. There exists a sequence of numbers $q_1,q_2,\ldots \in [0,1]$ such that $P(q_n) \to 0$.

This is just Bolzano's theorem. We introduce the real number line, prove a topological result, then discard the extra structure. We could of course prove a variant of Bolzano's theorem without defining the real numbers. But it would hurt!

If we want to remove the 'infinite' part of the above result we could have:

Suppose the polynomial $P$ is negative at $0$ and positive at $1$. There exists a number $q \in [0,1]$ such that $|P(q)| < 1/2$.

A much harder result is the Fundamental theorem of algebra. This is usually proves using a lot of complex analysis and a limiting argument on the plane -- which is two copies of the real numbers.

Let $P$ be a polynomial of odd degree. There exists a sequence of numbers $q_1,q_2,\ldots$ such that $P(q_n) \to 0$.

Or if we want to be even more discrete:

An $n$-th degree polynomial has no more than $n$ roots.

In all cases we introduce the topological structure, use it, then rephrase what the results tell us about numbers.

What is your favourite example of such a result? Where the topology is necessary as an intermediate step, but nowhere claims to be a realistic model of physical space or length?

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closed as unclear what you're asking by Lord Shark the Unknown, Leucippus, José Carlos Santos, user370967, user91500 Jul 16 '17 at 8:45

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  • $\begingroup$ Do not down-vote without an explanation $\endgroup$ – Daron Jul 16 '17 at 2:21
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    $\begingroup$ At first glance I thought you were asking for a convincing explanation of why the real numbers are useful. After reading through the question, it sounds more like you're asking for a "big-list" of theorems that don't explicitly mention the real numbers, but which have an easy proof if the real numbers are used. Can you clarify what you're asking for? $\endgroup$ – Tanner Swett Jul 16 '17 at 2:40
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    $\begingroup$ (1) The rational numbers are not granular, either. (2) There is no intermediate value theorem for continuous functions from the rationals to the rationals. (3) There are lots of nice and important functions, line $\sin x,\cos x,\sqrt{x},\log(x)$ which do not take rational values on the rational numbers. $\endgroup$ – Thomas Andrews Jul 16 '17 at 2:43
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    $\begingroup$ I find it amazing that you ask what good is the real number line... It requires extraordinary powers of non-observation to not see its usefulness. $\endgroup$ – Mariano Suárez-Álvarez Jul 16 '17 at 3:21
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    $\begingroup$ math.stackexchange.com/questions/1078593/… and the linked questions there. $\endgroup$ – Asaf Karagila Jul 16 '17 at 5:37
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The probability that two randomly chosen integers are coprime is $\dfrac{6}{\pi^2}$. Without transcendentals we lose results to very natural questions based purely on integers.

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  • $\begingroup$ Does this mean the probability of choosing uniformly from the first big $N$ numbers? $\endgroup$ – Daron Jul 16 '17 at 5:33

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