Do we really need reals?

It is much harder to avoid "unnecessary" real numbers than to accept them.

From a physics point of view, all you ever produce as the outcome of a measurement is a rational number. More accurately, any well-defined physics experiment is reproducible, and thus one can always produce a finite sequence of rationals out of a potentially infinite sequence rationals of outcome measurements per experiment. Each measurement is an approximation to the 'actual outcome', whatever it may be. Now, if all goes well, the entire countable sequence of rationals is a Cauchy sequence which is the actual outcome.

Now, mathematically it is convenient to be able to speak of this 'actual outcome' as a single object in a nice system of measurements. One way to do so is define the real numbers. Then the outcome of an experiment really is a real number. Now, when you do that you find that are actually created lots and lots of new real numbers, most of which can't ever be obtained as the outcome of anything. That's not a big deal, and its a rather small price to pay for having a really convenient system of numbers of great relevance to physics to work with. The fact that you only use a fraction of those numbers is not particularly relevant.

Another reason for introducing the real numbers is that they exhibit pleasant computational properties. Perhaps most importantly is that they form a complete metric space. Now, without that property most of analysis fails, so we really need the reals to be complete, and that necessarily means lots of numbers out of our reach. Again, this is a small price to pay for having a fantastic accompanying mathematical theory.

Finally, don't forget that it is not always a single number that is sought as the result of a computation. Sometimes what one needs to know is whether or not an integral converges, but the actual value where it converges to is immaterial, and whether or not it is a computable number in any sense is irrelevant. Then again it is crucial to have a system where you can use lots of tools and the reals provide that.

Real numbers include all the numbers you might want to use - including limits of convergent sequences of rational numbers. It is true that only countably many of these can ever be defined, but we don't know which we might want in advance.

These numbers are packaged in a convenient form which enables results to be proved for all the numbers we might encounter on our mathematical travels. It is particularly useful to know that the real numbers are essentially a unique model for the key defining properties, so when we prove theorems we know we are all talking about the same thing.

Admitting countable limits of rationals, rather than simply finite limits, turns out to have far-reaching consequences - for example, we can prove the intermediate value theorem. Without this, demonstrating the existence of a point relevant to the problem in hand may require a more complex case-by-case analysis.

A similar enriching of possibilities occurs in measure theory, where countably additive measures make a huge difference.

Almost all physical models use PDEs - partial differential equations (and ODEs). Without real numbers, how do you do PDEs? It is one thing to measure quantities in the laboratory, but without PDEs, our mathematical descriptions of nature would be very poor indeed.

Mathematicians don't compute things. Computers compute things.

Mathematicians are not limited by computation, much like Turing machines are not bounded by $2^{1024}$ gigabits of tape.

The real numbers are a tool, and it turns out to be useful. Since mathematicians are utilitarians, they like useful things. So they use the real numbers. If you want to ask why do physicists need the real numbers, you should ask that on Physics.SE or some other community centered around physics.

Why do mathematicians like the real numbers? Because they are closed under taking limits, which makes them ideal for talking about approximations by rational numbers, something we can easily comprehend (this is true at least in principle the notion of a rational number).

Since these are closed we don't have to worry and keep track whether or not our sequence is convergent, is the sequence itself is computable, do these properties hold or not. We have an arbitrary sequence, if we know it's a Cauchy sequence, then we know that it has a limit.

This is why mathematicians tend to embrace the axiom of choice when they work outside "the countable domain". When you want to make a statement about "all commutative rings with a unit", you don't want to start adding conditions on and on and on, that will help your proof go through. You want to say "Take any such and such object, then we can do this and that". Period. Utilitarianism galore.