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The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless there were a good conceptual reason for the contrary. The typical situation would be for a proof in finite combinatorics to be proven purely within the realm of finite combinatorics, a statement about integers to be proven using only the rationals (perhaps together with some formal symbols such as $\sqrt {2}$ and $\sqrt{-1}$), and so on. When the typical situation breaks down, the reason would be well-known and celebrated.

The prototypical field where things don't seem to work this way is Number Theory. Kronecker famously stated that "God invented the integers; all else is the work of man."; and yet, the real numbers (often in the guise of complex analysis) are ubiquitous all over Number Theory.

I am sure that this question is hopelessly naïve and standard but:

  1. What is the high-concept explanation for why real numbers are useful in number theory?
  2. What is the "minimal example" of a statement in number theory, for whose "best possible" proof the introduction of real numbers is obviously useful?

An alternative way of framing the question would be to ask how you would refute the following hypothetical argument:

"We know that calculus works well, so we are tempted to apply it to anything and everything. But perhaps it is in fact the wrong tool for Number Theory. Perhaps there exists a rational-number-based approach to Number Theory waiting to be discovered, whose discoverer will win a Fields Medal, which will replace all the analytic tools in Number Theory with dicrete tools."

(This question is a byproduct of a discussion we had today at Dror Bar-Natan's LazyKnots seminar.)

Update: (REWRITTEN) There has been some discussion in the comments concerning whether proofs and statements living in the same realm is "utopian". The philosophical idea underlying this question is that, in my opinion, part of mathematics is to understand proofs, including understanding which tools are optimal for a proof and why. If the proof is a formal manipulation of definitions used in the statement of the claim (e.g. proof of the snake lemma), then there is nothing to explain. If, on the other hand, the proof makes essential use of concepts from beyond the realm of the statement of the theorem (e.g. a proof of a statement about integers which uses real numbers, or proof of Poincare Duality for simplicial complexes which uses CW complexes) then we ought to understand why. Is there no other way to prove it?Why? Would another way to prove it necessarily be move clumsy? Why? Or is it just an accident of history, the first thing the prover thought of, with no claim of being an "optimally tooled proof" in any sense? For one think, if a proof of a result involving integers essentially uses properties of the real numbers (or complex numbers), such a proof would not work in a formal somehow analogous setting where there are no real numbers, such as knots as analogues for primes. For another, by understanding why the tool of the proof is optimal, we're learning something really fundamental about integers.
I'm interested not in "what would be the fastest way to find a first proof", but rather in "what would be the most intuitive way to understand a mathematical phenomenon in hindsight". So one thing that would make me happy would be a result for integers which is "obviously" a projection or restriction of some easy fact for real numbers, and is readily understood that way, but remains mysterious if real numbers/ complex analysis aren't introduced.

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    $\begingroup$ This is totally tangential but I just want to say I strongly disagree with the sentiment of the first sentence. If statements and proofs were typically "in the same world", math would be so dull. Dystopia. $\endgroup$ – Alon Amit Apr 14 '11 at 2:45
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    $\begingroup$ @Alon: "unless there were a good conceptual reason for the contrary". Are you arguing that statements and proofs should not live in the same world, and moreover, that the reasons for this should be unclear? $\endgroup$ – Daniel Moskovich Apr 14 '11 at 2:50
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    $\begingroup$ I think Alon is disagreeing with the sentiment of the first clause of the first sentence rather than the entire first sentence. $\endgroup$ – Qiaochu Yuan Apr 14 '11 at 3:05
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    $\begingroup$ The real moral of the story is that there aren't separate ``worlds'' in mathematics, but rather a beautiful and often mysterious all-embracing unity. $\endgroup$ – Lubin Apr 14 '11 at 5:26
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    $\begingroup$ What you suggest would be a utopia for people who don't want to learn new tools. We seem to have strong experimental evidence that fluency in one area is not a good way to make leaps in our ability to prove or understand mathematics. The question is somewhat related to the problem of whether tersely stated theorems should have short proofs - if the length of proofs were bounded by a computable function in the length of the theorem, we could check any mathematical statement in finite time. $\endgroup$ – S. Carnahan Apr 14 '11 at 8:33

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The Gödel Speedup Theorem provides some explanation why real numbers (and variants) are useful in proving statements in number theory.

Real numbers, complex numbers, and $p$-adic numbers are second-order objects over the natural numbers. Thus a proof of a number theoretic fact using such analytical devices is formally a proof of that fact in second-order arithmetic. The Gödel Speedup Theorem shows that there is a definite advantage to using second-order arithmetic to prove elementary number theoretic facts.

Gödel Speedup Theorem. Let $h$ be any computable function. There is an infinite family $\mathcal{H}$ of first-order (indeed $\Pi^0_2$) statements such that if $\phi \in \mathcal{H}$, then $\phi$ is provable in first-order arithmetic and if $k$ is the length of the shortest proof of $\phi$ in second-order arithmetic, then the shortest proof of $\phi$ in first-order arithmetic has length at least $h(k)$.

Since computable functions can grow very fast, this shows that there are true number theoretic facts that one can prove using second-order methods (e.g. complex analysis, $p$-adic numbers, etc.) but any first-order (a.k.a. elementary) proof is unfathomably long. Admittedly, the statements produced by Gödel to verify the theorem are very unnatural from a number theoretic point of view. However, it is a general fact that second-order proofs can be much much shorter and easier to understand than first-order proofs.


Addendum. This excellent post by Emil Jeřábek demonstrates another speedup theorem, which is in many ways more striking. The method of going from a first-order $T$ to a second-order $T^+$ is conservative, meaning that $T^+$ cannot prove more first-order theorems than $T$. However, the mere act of allowing sets to replace formulas and introducing the possibility of quantifying over such sets introduces speedups faster than any exponential tower. Introducing $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$ and so forth has a similar effect where one can package complicated ideas into conceptually simpler ones (e.g. replacing $\forall\exists$ statements by the higher-level idea of continuity) can lead to monumentally shorter proofs!

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    $\begingroup$ This doesn't say anything about why the real numbers get picked out as being particularly useful. Surely there's more to it than the fact that arguments involving real numbers correspond to arguments using quantification over infinite sets of naturals. After all, if it were just about infinite sets, we would use infinite sets directly and skip the real numbers. $\endgroup$ – Dan Piponi Apr 15 '11 at 0:13
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    $\begingroup$ Since the proof of this theorem, as you say, involves mathematically uninteresting examples, I am not convinced that this theorem helps us "understand" why proofs in number theory using R, C, Q_p, etc. can be efficient in ways that known proofs of the same results without them are not. There is not an actual connection made between this theorem of Godel and theorems of interest in number theory. $\endgroup$ – KConrad Apr 15 '11 at 5:33
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    $\begingroup$ The problem is not only that the statements are not interesting, but also that they're not number theory. For example, the Lovász-Kneser theorem is a standard "out of field" proof, because it uses functions of real numbers to prove a combinatorial result, but you could hardly call the result number theory. $\endgroup$ – Zsbán Ambrus Apr 15 '11 at 9:40
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A possible explanation is that $\mathbb R$ is one of the completions of $\mathbb Q$. And why does this matter? A reason is the Hasse, or local-to-global, principle. A minimal example of this is the Hasse--Minkowski theorem, which states that if $$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$ is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation $$Q(x_1,\ldots,x_n)=0$$ has a nontrivial solution in the integers if and only if it has a solution in the reals and all the $p$-adics.

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    $\begingroup$ I don't think that this answers the question. The $p$-adics are completions too. We may avoid $\mathbb R$ and $\mathbb Z_p$ by rephrasing: $Q=0$ has a non-trivial solution in the integers if and only if it has approximate solutions at any order for every valuation over $\mathbb Q$. $\endgroup$ – Denis Serre Apr 14 '11 at 6:04
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    $\begingroup$ Denis: But over R and the p-adics, methods of checking there is a nontrivial solution are much more streamlined than the corresponding methods that go out of their way to avoid directly mentioning those fields. (Talking about an equality in Q_p is conceptually simpler than talking about infinite sets of congruences mod powers of p.) $\endgroup$ – KConrad Apr 15 '11 at 5:36
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Consider the following

Theorem. The number of integral solutions of $x^2+y^2+z^2=4n+1$ tends to infinity as $n\to\infty$.

The only proofs (known to me) use a good deal of complex analyis. I will think of further examples as time permits.

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  • $\begingroup$ Hmm. My feeling is that it should be possible to prove this using quaternion arithmetic, although I am not prepared to back this up with details at the moment. $\endgroup$ – Pete L. Clark Apr 14 '11 at 17:54
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    $\begingroup$ @Pete I'd be very interested to see you try. This is essentially equivalent to showing that the class numbers of quadratic imaginary fields go to infinity (see eqn. 35 at mathworld.wolfram.com/SumofSquaresFunction.html .) I don't think there should be any easy way to prove that! But I've never understood how the quaternion picture ties in with the class number picture, so I'd love for you to turn out one of your notes explaining it. $\endgroup$ – David E Speyer Apr 14 '11 at 18:17
  • $\begingroup$ @Pete: The proofs known to me are closely tied with the Deuring-Heilbronn phenomenon of $L$-functions. @David: I highly recommend front.math.ucdavis.edu/1001.0897 $\endgroup$ – GH from MO Apr 14 '11 at 21:31
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    $\begingroup$ Already Gauss in his Disquisitiones observed connections between the class number and what later were realized to be quaternionic effects. Venkov used quaternions and Gauss's approach for proving the class number formula in most cases; for details, see Rehm's article in "Ternary Quadratic Forms and Norms" edited by Taussky. This was taken up in the last few years by a number of people. $\endgroup$ – Franz Lemmermeyer Apr 15 '11 at 10:55
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    $\begingroup$ @Junkie: Pintz confirmed that he had indeed proved Siegel's theorem in an elementary way: matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2458.pdf His paper combines rather simple facts from real analysis in very clever way. It also seems that Linnik (1950) found the first elementary proof. $\endgroup$ – GH from MO Apr 15 '11 at 22:36
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The question seems to assume or at least sympathise with something along the lines of

     God created the integers and the reals are derived therefrom. 

On the contrary, it seems just as plausible that the reals are our basic intuitive data. Meaning what?

  1. In the history of cognition, measuring precedes counting. For example a lion can measure the length of his leap more easily than he can distinguish between 29 and 39 wildebeests.

  2. The reals are the only Dedekind complete ordered field, if "cut" is taken in a set-theoretically naive way. See e.g., "Completeness of Ordered Fields" by Hall on arxiv. The integers are the unique discretely ordered subring of the reals. The induction principle for the ring of integers (and the uniqueness of the one and only discretely ordered subring of the reals) is a consequence of the completeness of the reals: An inductive set of integers that missed some positive integer would give rise to a cut with no boundaries. The notion of continuum, in other words, can be taken as the big defining concept here, and the main underlying intuition.

  3. Number Theory, or at least Diophantine Analysis, can be understood as the study of equations over the reals with solutions restricted to certain subrings. Does the question "What are the reals doing in number theory" then dissolve? Maybe.

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Two of the most basic facts of algebraic number theory, namely the finiteness of the class number and the structure of units in rings of integers of number fields do not seem to be provable without the use of real numbers (or some use of the archimedean nature of real numbers).

Added: In fact, most finiteness results in arithmetic geometry seem to use this archimedean prime; this is the case of Mordell-Weil theorem, Mordell conjecture, finite generation of Galois cohomology groups of number fields, etc.

Afterthoughts: There are many branches of number theory and in some of them you can't even state the results without using real numbers as was pointed out elsewhere. Now, in algebraic number theory or arithmetic geometry, which does not suffer from this problem, the analogy with function fields is a quite powerful tool to try to guess what can be true, and if you look at the problem from this angle you realise that real numbers are more of a nuisance than a help : all the statements above can be proven for function fields where real numbers play no role, and many others like the Riemann hypothesis or the global Langlands correspondence still elude us in the number field setting. The fact that you have to use them to prove the above results seem to indicate that you cannot ignore this nuisance so easily... (despite the product formula that makes you believe that the information that you can extract from the archimedean prime should be readable from the others).

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A possible candidate for a "minimal" result about integers that is a "projection" of a result about reals: the group structure of the solutions of the Pell equation $x^2-dy^2=1$ for $d$ a nonsquare positive integer.

The (positive branch of the) curve $x^2-dy^2=1$ has parametric equations $x=\cosh t$, $y=\frac{1}{\sqrt{d}}\sinh t$, and hence it is isomorphic to the real numbers $t$ under addition. If we let $x_t=\cosh t$, $y_t=\frac{1}{\sqrt{d}}\sinh t$ then the sum of points $(x_s,y_s)$ and $(x_t,y_t)$ is $(x_s x_t+dy_s y_t,x_s y_t+y_s x_t)$.

It follows that the sum of integer points is an integer point (and likewise the difference). Hence the integer points form a subgroup of the points on the curve. Moreover, we can see that this abelian group is infinite cyclic (assuming a nontrivial integer point), generated by the integer point with smallest nonzero $t$ value.

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  • $\begingroup$ It's not clear to me that you need to use the real numbers to see this. The group structure can be deduced from an algebraic isomorphism between the curve $x^2 - dy^2 = 1$ and the curve $xy = 1$ (the multiplicative group scheme) over a suitable extension of $\mathbb{Q}$. $\endgroup$ – Qiaochu Yuan Apr 17 '11 at 23:45
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    $\begingroup$ I agree that that the real numbers can be avoided in this result. However, I think they give a particularly simple proof, and they point to a group structure which is otherwise not obvious. $\endgroup$ – John Stillwell Apr 18 '11 at 0:13
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    $\begingroup$ +1 for a nice example. $\endgroup$ – user9072 Apr 18 '11 at 0:20
  • $\begingroup$ The fact that the group of integer points is infinite cyclic (or at least the torsionfree part is) is nicely seen from the fact that you get a discrete subgroup of the positive branch that is an isomorph of $\mathbb{R}$. $\endgroup$ – Todd Trimble Nov 15 '19 at 9:49
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I think the key intuition is that reals are easy to work with and integers are hard. Linear programming can be done in polynomial time; integer programming is NP-hard. Real closed fields are decidable, but not their integer equivalent. Diophantine equations are intractable in general, but complex analysis (which can be formulated in real terms, if desired) gives us pretty much all the information we'd want about (real) polynomials, the easiest sort of holomorphic function. Etc.

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Someone once suggested on MO that this is because on the one hand Matiyasevich's theorem shows that no algorithm can solve Diophantine equations over $\mathbb{Z}$ (and the corresponding result is not known over $\mathbb{Q}$), and on the other hand this is possible over $\mathbb{R}$ because of quantifier elimination. It is also possible over $\mathbb{Q}_p$ for every $p$, and this suggests that one would really like to have some sort of local-to-global principle so that special types of Diophantine equations can be solved, and if not, to understand how it fails...

Anyway, I am still not sure I totally agree with the sentiment of the first sentence. Think of it this way: if I want to understand a category $C$ and I understand a category $D$ very well, and moreover I have a functor $F : C \to D$, then it stands to reason that I can learn something about a problem in $C$ by translating it to a problem in $D$, where I understand what is going on very well. Nothing is mysterious about this process other than possibly the construction of the functor $F$, and in the case of number theory $F$ is something like the analytification functor from varieties over $\mathbb{Z}$ or $\mathbb{Q}$ to varieties over $\mathbb{R}$ or $\mathbb{C}$ and the existence of this functor is not so hard to understand.

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  • $\begingroup$ If you're finding a proof to a new result, this technique makes sense. But if you're trying to trully understand a (known?) mathematical result, you should know why F is useful, and exactly what category D is contributing. This in itself would probably be the key point for most such proofs, the one which is most important to understand. Quibble: an algorithm over R isn't a practical algorithm, because it can't be programmed. Could you tell me more? $\endgroup$ – Daniel Moskovich Apr 14 '11 at 3:44
  • $\begingroup$ @Daniel: I don't know the details, but there should be precise statements in this survey article by Poonen: www-math.mit.edu/~poonen/papers/aws2003.pdf $\endgroup$ – Qiaochu Yuan Apr 14 '11 at 3:51
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The Prime Number Theorem is a statement in Number Theory which can't even be stated, let alone proved, without real numbers.

EDIT: How about the statement that $\sum^N\phi(n)$ is asymptotic to $(3/\pi^2)N^2$ (where $\phi(n)$ is Euler's phi-function)?

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    $\begingroup$ @Daniel, I'll take your word for it. But you asked about "best possible" proofs, and whether introducing real numbers is useful. Do you think the C-D proof is better than the classical ones using complex analysis (or the Erdos-Selberg, using real analysis)? It may be possible to prove (something equivalent to) PNT without reals, but history suggests introducing the reals was useful. $\endgroup$ – Gerry Myerson Apr 14 '11 at 5:27
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    $\begingroup$ Can't you asymptotically replace $\log(n)$ by harmonic numbers, which can be defined in the rationals? So that stating PNT doesn't require the reals? $\endgroup$ – Todd Trimble Apr 14 '11 at 11:33
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    $\begingroup$ @Daniel and @Todd: Of course even the $x/\log(x)$ estimate in PNT is just an approximation to the better asymptotic $\pi(x) \sim \operatorname{Li}(x)$, which is even more difficult to discuss without real numbers. $\endgroup$ – Mark Meckes Apr 14 '11 at 13:32
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    $\begingroup$ I agree with Todd; it's far from clear that there is any version of the PNT that cannot be stated without using the real numbers. Usually statements involving the growth rate of explicitly defined real functions can be converted into statements involving only rational numbers, using standard tricks familiar to logicians. Ugly and inelegant, of course, but not impossible. $\endgroup$ – Timothy Chow Apr 14 '11 at 16:30
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    $\begingroup$ Yes, "can't" is certainly an overstatement. The real (no pun intended) -- and increasingly less mathematical -- question is whether such statements can be made without appealing to the reals and still be comprehensible to human beings. $\endgroup$ – Mark Meckes Apr 14 '11 at 16:51
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In algebraic number theory, the class number formula, which involves real numbers, is a fundamental tool in order to understand the arithmetic of a number field. It provides a purely analytical formula for the class number, which is a fundamental invariant.

This formula is not just a nice result, it also enables to compute the class number with a computer (say for quadratic fields). I won't enter into details here, but the point is that if you know in advance that some quantity is an integer, then it suffices to approximate it within less than one half in order to determine it. Thus a computer is able to determine the class number exactly.

Of course, one can argue that a computer always works with a finite approximation, rather than with an actual real number. But it definitely seems that working with (approximations of) real numbers is useful.

More generally, it seems some number-theoretic functions can be computed more efficiently if one allows the use of real numbers (I am not an expert, but I think that the computation of Bernoulli numbers is an example of this phenomenon). It would be interesting to make this rigorous in some sense.

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Second addition in view of the EDIT of the answer:

Some examples now focused towards understanding. How valueable/inevitable real and complex numbers are for them, I don't know.

  1. Q: What is a good way to understand that $\sum_{i=1}^n i$ is (up to small error) $n^2/2$. A: Integrate $x$ from $1$ to $n$. (Of course, this precise example is easily handled otherwise but for illustration)

  2. Q: What is a good way to understand the size of $\sum_{i=1}^n i^{-1}$. A: Integrate $x^{-1}$ from $1$ to $n$.

  3. Q: What is a good way to understand the size of $\sum_{i=1}^n d(i)$ where $d(i)$ is the number of divisors of $i$. A: Count the points 'under' the hyperbola $xy=n$. See here and here.

  4. Q: What is a good way to understand units in rings of algebraic integers. A: Consider the points (in $R^k$) obtained by taking the logarithms of the absolute values of their $k$ (essentially) different imbeddings into the complex numbers.

  5. More generally, results on linear forms in logarithms are a main tool in the study of certain Diophantine equations.

  6. Q: What is a good way to understand the number of primes below $x$. A: Observe that the probabilty of a number $y$ to be prime is $y / \log y$ and integrate, getting $Li(x)$. Some details regarding a proof of the precise quality of this approach still need to be established ;).

  7. In various places one will stumble somewhere in number theory over a $Q$-linear map or linear recursion. To undertand them one wants a convenient way to handle the roots of the attached characteristic polynomials, not just as formal constants.

And the list could go on and on.

I am not sure anything of this is what you are looking for.


There are already many answers, still I believe I have something to add.

The question talks about Number Theory (without any qualification). However, the following result is a number-theoretic one.

Let $a$ be an irrational number, then the sequence of numbers $ap$ where $p$ runs through the primes is equidistributed modulo $1$.

How to even state this without the reals, let alone prove it?

Actually, it is my impression, that many (if not most) people working on things like measures of irrationality of certain numbers, proofs of transcendence results, and so on, self-identify as Number Theorists (also look through the Number Theory MSC classification, in particular 11Jxx 11Kxx). And, results of this form are also frequently published in number theoretic journals.

Thus, without starting any philosophical discussion, but purely based on the everyday practise of Number Theory, I would say, obviously one needs reals numbers in Number Theory.

Now, returning from the defence of certain fields of Number Theory to answering your question more in its spirit:

As said by many, the reals are one completion of the rationals, sometimes it is useful to work in a complete structure.

And, since somewhat frequently in this thread the objection comes up that this or that example of an application of the reals could be avoided by doing something inconvenient, I would just like to mention that this seems to be true in many other (than Number Theory) contexts too.


To elaborate on the last statement a bit, and going down a slightly different line of argument:

if one wants to understand questions only involving extremely classical notions in number theory, say something like:

How does the set of divisors of a (typical) integer look like? (of course, made precise in some form)

Then, it turns out that answers to these 'discrete' questions can be obtained by 'approximating' the discrete real world by a 'continous model', think of distributions much alike as one finds them someplace else. Key word: probabilistic number theory, see for example a book by Tennenbaum or the book 'Divisors' by Hall and Tennenbaum. (And, this is not, or not mainly, the approach where one replaces the integers by some model of the integers with nice probabilistic properties, say Cramer's model; but by contrast very concrete statements about the true natural numbers that are best expressed using continous approximations.)

Also, the classical questions (mentioned by Gerry Myerson) of counting primes and other numbers defined by arithmetic properties, or derived quantities, can be consider as part of this.

So, now, could it be the case that one can replace all this by a purely discrete approach?

In some sense this seems as good a question to me as to ask, why one does, say, statistical physics rather than tracking each particle individually. Or, why all these differential equations in fluid dynamics, wouldn't it be possible one could get more insight tracking the particles constituing the liquid individually using some discrete model?

Perhaps one answer to the number theory question is that also the real world of integers (as the real real world) is simply way too complicated to allow for exact (discrete) answers, and therefore one has (at least for now, but basically I'd guess 'forever') use continous approximations to gain some insight. (Whether you encode them by reals or somehow else seems besides the point.)

And, to answer the final question, whether it would be possible that somebody comes along and sees that actually the integers are not as complicated as we think they are, well, what is truly impossible? But some of the most famous open problems in mathematics are way below such an insight; for example, what is the Riemmann Hypothesis, just an imprecise statement on a crude parameter of the set of primes.

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  • $\begingroup$ To clarify: Using the reals to make a proof more convenient is a legitimate use of the reals, I think. That's a distinct thing from "are strictly necessary", and something I would like to understand just as much. For example, say we know that the PNT has a proof over Peano Arithmetic (Cornoros-Dimitracopoulos)- is there any reason to believe that any such proof must necessarily be artificial/ less-direct/ clumsy? $\endgroup$ – Daniel Moskovich Apr 14 '11 at 12:09
  • $\begingroup$ Well, as a meta-reason, it took about a hundred years to get from the first proof to the other. $\endgroup$ – Cam McLeman Apr 14 '11 at 12:29
  • $\begingroup$ @Daniel Moskovich: I expanded my answer; while I started doing so before your comment, I think it is addressed to some extent. The PNT (with which error term, btw) is a very weak result; not that it is not a great result, but it is weak in the sense that it falls way short of describing the 'reality of prime numbers.' Even RH is weak, we would like to know way more. And regarding your other question, 'strictly necessary' I am not qualified to answer this, but why restrict to number theory? $\endgroup$ – user9072 Apr 14 '11 at 12:55
  • $\begingroup$ Well, in number theory it's clear what I mean by "in the same realm" and "outside it", and that "most techniques", contrary to my intuition are outside it. But definitely, I could have asked the same question just as well about many mathematical fields which draw heavily from results outside their realm. I imagine that understanding why, at least in a few prototypical cases, would increase my big-picture understanding. $\endgroup$ – Daniel Moskovich Apr 14 '11 at 13:05
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    $\begingroup$ Let me try one more example, which might or might not be better: you have a process that you know follows exactly a binomial distribution; you have many many events. Eventually, you will have to give up on considering the exact binomial distribution and consider instead of it the continous normal distribution, which is a constinuos approximation to the (in this situation) true discrete binomially distributed situation. But the normal one stays managable, when the binom. one is already infeasible to use. $\endgroup$ – user9072 Apr 14 '11 at 17:27
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To continue on a theme of a couple answers thus far:

The real numbers, along with the $p$-adics for each rational prime $p$ are the completions of the rational numbers. Regarding arithmetic objects (such as quadratic forms or algebraic varieties in general, among others) defined over $\mathbb{Q}$ as being defined over these completions endows them with additional structure that one can exploit.

This "exploitation" comes in various forms, including the Hasse principle mentioned in other posts (points over $\mathbb{Q}$ beget points over completions and the latter are some kind of evidence for the former).

Then there is also the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves and its various generalizations. These purport to relate global ("over $\mathbb{Q}$") things like values of a global $L$-function to a host of data that is germane to the various local completions of $\mathbb{Q}$. The period that turns up in the BSD conjecture, for example, is really a facet of the elliptic curve regarded over the reals where one can integrate forms in the usual sense. The Tamagawa numbers, on the other hand, come from the various $p$-adic completions of $\mathbb{Q}$.

I guess the theme in all of this is that global fields lack structure that their various completions possess, and embedding them in these completions gives extra insight and "handles" to grab onto in order to study the global field and the objects defined over it.

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  • $\begingroup$ This is a nice answer! I don't know almost anything about the BSD... could you tell me more about how the period is really a facet of the elliptic curve regarded over the reals? $\endgroup$ – Daniel Moskovich Apr 14 '11 at 21:41
  • $\begingroup$ The period that turns up in the leading term formula in BSD is literally the integral of the invariant differential of the curve (or perhaps a minimal model thereof) over its locus of real points. It is in regarding the elliptic curve as a real algebraic curve (and therefore a smooth real 1-manifold) that allows you to make sense of this sort of good old-fashioned integral of a differential form. $\endgroup$ – Ramsey Apr 14 '11 at 23:33
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I would like to say that at least to some extent we need the real numbers and not just rational approximations.

Suppose we were being completely formal and we asked the question, ``Is there some algebraic number $\beta>0$ such that for all $n$ there exist $p_n , q_n \in \mathbf{Z}_{> 0}$ with $\beta \ne \frac{p_n}{q_n}$ and there exists a fixed $\delta >1$ such that $|\beta - \frac{p_n}{q_n}| < \frac{1}{q_n^{1+\delta}}$'' ?

The answer of course is no by the Thue-Siegel-Roth Theorem. But moreover, we know that we can only discover a measure zero subset of the transcendental numbers in this way. So while we can approximate real numbers by rationals all day long, the amount of information we get out of a rational approximation can vary wildly.

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  • $\begingroup$ This answers "why do we need the real numbers?", but not "why might real-number techniques be optimal for problems which are purely about integers?" $\endgroup$ – Daniel Moskovich Apr 14 '11 at 16:34
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    $\begingroup$ @Daniel Moskovich: You can transform the question in this answer into one involving only integers. $\endgroup$ – Charles Apr 14 '11 at 19:26
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I'll admit I'm not number theorist but here is my take on why the field has to embrace the completions of $\mathbb{Q}$. It is a simple reason: $e$. Other branches of math have run across interesting numbers that aren't algebraic which deserve studying in their own right. So not only should number theory as a whole (not necessarily every practitioner) should take the reals as valid objects of study.

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Some historical context may be useful here.

In the late 19th and early 20th centuries, many mathematicians informally categorized mathematics into three tiers: arithmetic, analysis, and set theory. Roughly speaking—with caveats that I'll mention later—these correspond to what we nowadays call first-order Peano arithmetic (PA), second-order arithmetic (Z2), and Zermelo-Fraenkel set theory (ZF or ZFC). As an illustration of this mindset, note that after Gerhard Gentzen proved the consistency of arithmetic, he devoted considerable effort to trying prove the consistency of analysis. As another illustration, recall that in the first half of the 20th century, there was a lot of interest in an "elementary proof of the prime number theorem," by which was meant a proof of the prime number theorem (which was regarded as a result of arithmetic) that did not use complex analysis.

In terms of this three-tiered view of mathematics, I would rephrase the OP's question as, why is analysis useful for proving theorems in arithmetic? Or what is a minimal example of a theorem of arithmetic that requires analysis for its proof?

If we accept the aforementioned identification of arithmetic with PA and analysis with Z2, we might further rephrase the question as, what is an example of a theorem of Z2 that is unprovable in PA? This is the interpretation that (for example) François Dorais seems to be tacitly assuming. Under this interpretation, it then seems natural to look to reverse mathematics for examples of theorems that can be stated in the first-order language of arithmetic but cannot be proved from the axioms of PA.

However, I would argue that this interpretation does not really capture the spirit of the original question. If one examines carefully the sorts of things that are provable in PA, it turns out that much more is provable in PA than one might naïvely expect. In particular, it turns out that lots of invocations of the real or complex numbers can be logically eliminated by replacing the "continuous" reasoning with "discrete approximations." For example, Angus Macintyre's appendix to Chapter 1 ("The Impact of Gödel's Incompleteness Theorems on Mathematics") of Kurt Gödel and the Foundations of Mathematics: Horizons of Truth sketches how all of the superficially infinitary machinery used in the proof of Fermat's Last Theorem can probably be replaced by suitable discrete approximations, and carried out in PA. While the possibility of such discretization is certainly interesting in its own right, I think it shows that formal provability in PA is not really what people mean when they talk about an "elementary" number-theoretic proof. Suppose a certain argument uses infinitary machinery in a way that we instinctively regard as "non-elementary"; if we are then told that technically, there is an algorithmic but artificial way to convert all the arguments about natural infinitary structures into finitary but unintuitive arguments that make no mention of real numbers, we are not likely to change our minds and declare the proof to be elementary after all.

So I think that a better way to phrase the intended question is to avoid talking about formal systems and reverse mathematics and just ask, "Why is analytic number theory so powerful? What is a minimal example of a theorem of number theory that seems to require analysis?" (and here "require" is not to be taken in a strictly logical sense). Standard answers to this question include Dirichlet's theorem and the prime number theorem, and I say something about those theorems in an answer to another MO question. The fact that there now exist elementary proofs of the prime number theorem does not, in my view, invalidate these examples, which I believe do illuminate why real numbers and complex numbers turn out to be useful in number theory. In a nutshell, the point is that studying generating functions (notably zeta functions and L functions) is a very powerful technique for studying arithmetic sequences, and analysis is a very powerful technique for understanding the behavior of a generating function.

Finally, I think it's worth mentioning that Granville and Soundararajan have been spearheading a somewhat contrarian approach to analytic number theory, which they call a pretentious approach to analytic number theory. To quote Granville:

Since 1859 the only coherent approach to these problems has been based on Riemann's idea connecting the distribution of prime numbers to the zeros of the Riemann zeta function, which are the zeros of an analytic continuation. Some might argue that this is "unnatural" and ask for an approach that is less far removed from the original problems. Recently Soundararajan and I have proposed a different approach to the whole subject of analytic number theory, based on our concept of pretentiousness—recently we have realized our dream of being able to develop the whole subject in a coherent way, without using the zeros of the Riemann zeta function.

In other words, while they're not denying that the standard approach is useful, they are challenging the conventional wisdom about the extent to which zeta functions are really "necessary." Now you might look at the pretentious approach and still think that it "uses real numbers" in some sense; I don't know. But even if you feel that what I've said here still doesn't answer your question, I hope it will at least give you some context, and perhaps help you frame your question more precisely.

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  • $\begingroup$ I'm happy with theorems in $Z_2$ that are provably (or conceptually "obviously") harder to prove in $PA$. I would therefore agree with your restatement... and wish for some sort of "speedup theorem", or at least a plausible heuristic argument for such, for a "useful" theorem in number theory... $\endgroup$ – Daniel Moskovich Nov 16 '19 at 20:41
  • $\begingroup$ @DanielMoskovich : I doubt that there exist such examples. Certainly none of the main examples of theorems that are provable in (say) ATR$_0$ but not ACA$_0$ are what one would call "number theoretic." Even if you just want something that has a shorter proof using stronger axioms, I doubt that there are any (non-artificial) examples. I just don't think that reverse mathematics is the right place to look for an answer to this question. Much as we'd love to think that the value of a new concept lies in its logical strength, the reality is that this is rarely if ever true. $\endgroup$ – Timothy Chow Nov 16 '19 at 20:51
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This has been discussed in previous answers, but I want to emphasize it. A lot of what is considered at the center of number theory and that has been so considered for centuries, if not millennia, is now understood to deal with properties of the integers which are reflective of the supplementary structures which arise when integers are studied with their embedding in all the completions of $\mathbb Q$, including $\mathbb R$, as well as in all algebraic field extensions of $\mathbb Q$.

Among the branches of arithmetic and number which belong to this category, one can certainly count the multi-secular strand which starts with Fermat's assertion that a prime number $p$ may be written $x^2+ny^2$ with $n\in\{1,2,3\}$ if and only if $p$ satisfies a given linear congruence, leads to quadratic reciprocity, algebraic number theory, class field theory, complex multiplication, Artin theory, the Langlands program etc., the strand which starts with the computation of the sum \begin{equation} 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \end{equation} by Madhava's school in the 14th century, leads to Euler's computation of $\zeta(1-s)$ and continues with Gauss's and Dirichlet's studies of $L$-functions, then to the Birch and Swinnerton-Dyer conjecture, conjecture on special values of $L$-functions, Iwasawa theory etc., or the one which starts in the Arithmetica of Diophantus, passes through the tangent and chord method and leads to Mordell's conjecture, the theory of heights, Chabauty's method, the Weil conjectures, Faltings's work etc. (with innumerable interplays between these three strands).

The supplementary structure I am referring to can roughly be classified (to the best of our current very conjectural understanding) in the following way:

• The theory of motives, that is to say the idea that there is an absolute rational cohomology theory whose realizations in the various completions of $\mathbb Q$ (including $\mathbb R$) and the relations between them elucidate the arithmetic properties of the original motive.

• The interplay between automorphic and Galois representations, that is to say the idea that infinite-dimensional complex representations of adelic points of reductive groups are intimately linked with the algebraic properties of equations.

• Algebro-geometric methods which translate problems of integers into intersection problems in geometric spaces.

Seen from that perspective, the answer to question 1. certainly should be

The high concept explanation of why real numbers are useful in number theory is that the properties of integers we got interested in are the shadows in the world of integers that cannot even begin to be described without constant reference to the completions of $\mathbb Q$, including at the real place.

In fact, retrospectively, I would even invert question 1. Because there are such supplementary structures, integers satisfy remarkable properties that we got to notice, so in some sense, I would say that the question of why $\mathbb R$ plays a role in number theory is the wrong one: a theory in which $\mathbb R$ would play no role could not be linked with any of the three structures above, and so it would not exhibit any of the interesting properties that created number theory in the first place. If Fermat had asked himself when is a number the sum of a square and a prime times a cube, he would have not found anything noticeable, and number theory would have had to wait another couple of decades to get (re)born (in a sense, this is not an hypothetical, because alongside questions like $x^2+y^2=p$ and $x^n+y^n=z^n$ which proved incredibly fruitful and somehow at the very core of the intersections of the three structures outlined above, he also studied the question of whether $2^{2^{n}}+1$ was prime, and that did not prove to be nearly as interesting, probably because that question is not reflective of any supplementary structures that we know of at the moment).

Moving to question 2, the adjective minimal carries a technical meaning I am not at all equipped to address, but certainly I can give a purely number theoretic statement which Euler would have perfectly understood and recognized as number theoretic in the mid-18th century and for which the best proof certainly requires the rich interplay with the theory of $L$-functions alluded to in the first and second bullet above (so in particular, real numbers) because the only proofs known do so (to the best of my knowledge). Namely, Dirichlet proved in 1839 the following fact:

If $p\equiv 3\operatorname{ mod }4$, the quadratic excess between $1$ and $p/2$ is strictly positive, that is to say there are strictly more numbers which are squares modulo $p$ than numbers which are non-squares in that interval.

The statement is elementary enough so that it can explained to a clever and motivated elementary school kid, but requires (again to the best of my knowledge) relating the quadratic excess to the special values of the complex $L$-function of a quadratic extension whose positivity can then be showed.

"We know that calculus works well, so we are tempted to apply it to anything and everything. But perhaps it is in fact the wrong tool for Number Theory. Perhaps there exists a rational-number-based approach to Number Theory waiting to be discovered, whose discoverer will win a Fields Medal, which will replace all the analytic tools in Number Theory with dicrete tools."

We may yet discover completely new tools and ideas that will transform our understanding of number theory, but that will not show that the current approach is wrong.

The supplementary structures are there. At the heart of them, one finds harmonic analysis on some $L^2$ spaces, tauberian theorems, sheaves of differential forms, motivic integration and differential equations attached to $p$-adic Galois representations (among many much more surprising things). For the people whose contributions to number theory have been the greatest in the last three centuries, this has been number theory, almost by definition. Let us imagine that future mathematics contain purely discrete solutions of problems dealing with integers. Suppose someone were to find an elementary proof of \begin{equation} |\pi(x)-\operatorname{Li}(x)|\leq\frac{1}{8\pi}\sqrt{x}\log(x) \end{equation} for $x$ sufficiently large (something I would consider about as surprising as discovering that Mayan calendar were based on general relativity, but who knows). Or suppose, to take a more purely number theoretic statement, that someone could prove by elementary arguments that the primes $p$ such that $x^3-x-1$ splits in factors of degree 1 in $\mathbb F_{p}$ are precisely the primes $p$ such that the coefficient of $x^p$ \begin{equation}\nonumber x\left(1-x\right)\left(1-x^{23}\right)\left(1-x^{2}\right)\left(1-x^{46}\right)\left(1-x^{3}\right)(1-x^{69})(1-x^{4})(1-x^{92})\cdots. \end{equation} is 2 (note that the infinite product is purely a notation, for any $p$ you can decide by purely finitely means if the coefficients is $2$ or not). Well that would a surprising and obviously extremely significant development in the history of mathematics, but thousand of number theorists (including your very humble servant) would still pursue with the same passion the study of the subtle interplay between structures which intrinsically involve $\mathbb R$ (among many other things) and their consequences on numbers.

From the paradise, created for us by Euler, Gauss, Eisenstein, Dirichlet, Kronecker, Jacobi, Abel, Galois, Hilbert, Artin, Hasse, Weil, Tate, Serre, Shimura, Grothendieck, Langlands, Deligne, Fontaine, Faltings, Kato, Wiles..., no-one shall be able to expel us.

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  • $\begingroup$ So are you saying that many techniques were originally developed for complete fields, after which completeness could be dropped and we were still left with useful techniques? $\endgroup$ – Daniel Moskovich Nov 16 '19 at 20:34
  • $\begingroup$ @DanielMoskovich I don't think I'm saying that (but I'm not sure I understand your question). What I'm saying is that a lot (really a lot) of what is and has historically been classified as interesting number-theoretic phenomenon retrospectively appears to be shadows cast on integers by structures which are only meaningful over complete fields. The "rational-number-based approach to Number Theory" is the study of the shadows, but at some point we learned to escape the Platonic cave and we caught glimpses of the actual structures. I don't expect we are ever going back. $\endgroup$ – Olivier Nov 18 '19 at 14:57
  • $\begingroup$ @DanielMoskovich In fact, I think that the statement you ask us to refute and your comment indicates where the crux of the matter lies. You seem to believe that real numbers are tools for number theorists ("We know that calculus works well, so we are tempted to apply it to anything and everything") but to me, in addition to tools, they are primarily part of the very fabric of the phenomenon I study (which is why I'm not surprised that conceptual approaches of these phenomenon requires them heavily). $\endgroup$ – Olivier Nov 18 '19 at 15:03
  • $\begingroup$ @DanielMoskovich In some sense, even if I had a purely number-theoretic proof of some purely number-theoretic problem I enjoy, I would look for more, because I would feel that the purely number-theoretic proof would not illuminate the role the motivic or automorphic structures play (Dirichlet's theorem above would be a baby example, if you were to find a very clever combinatorial argument, it would be a remarkable achievement, because I think a number of quite clever people tried and failed, but it would not necessarily be a conceptual step forward). $\endgroup$ – Olivier Nov 18 '19 at 15:06

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