show/hide this revision's text 3 explcit examples added, fixed numbering

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.

  • 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)

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

  • 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.

  • 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.

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

  • 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 ;).

  • 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.
    show/hide this revision's text 2 added second part

    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.

    show/hide this revision's text 1

    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.