Why do we study real numbers?

I've often asked myself the same thing, and this is what I tell myself. $\mathbb R$ is (up to order-preserving field isomorphism) the only totally ordered, complete field. This is pretty big news, because these two nice structures lead to so many others we find useful to study in math. $\mathbb R$ (and more generally $\mathbb R^n$) is so great because a plethora of these fundamental "structures" studied in math are present in (at least some subset of) $\mathbb R$. When we learn of new concepts, it's natural (crucial) to seek examples, and we often find solace in the usual first stop -- $\mathbb R^n$.

Here's a poor-at-best survery of some of the aforementioned structures that $\mathbb R$ has.

Algebra

• Group -- we can combine elements, i.e., $a + b$, invert them, i.e., $a^{-1}$.
• Field -- we get more ways to combine elements, $+, -, \times, \div$.
• Ordered field -- we get to do things like transitivity, i.e., $a < b \wedge b < c \implies a < c$, and "add inequalities", i.e., $a \leq b \wedge c \leq d \implies a + c \leq b + d$.
• Vector space -- linear algebra's pretty important. Arrow-like addition is very physical.

Analysis

• Completeness -- analysts love sequences... to converge. This allows for a lot of "take a sequence..." arguments which start with a probably-desired sequence that ends up being Cauchy.
• Compactness -- we always want to exploit compactness in analysis, and $\mathbb R^n$ has a particularly nice characerization of it.
• Hilbert Space -- we all love Hilbert space. Orthogonality is a useful tool. So is the spectral theorem.
• Measure space -- measuring is very physical, and crucial for integrating! $\mathbb R$ is the natural setting for the famous Lebesgue measure, and all measures map into the "subset" $[0,\infty]$ of $\mathbb R$. For Riemann integration, the (Darboux) definitions hinge on the least upper bound property of $\mathbb R$.

Geometry

• Metric Space -- we can measure distances $d(p,q)$ between points. This is very physical. The triangle inequality is here too, which is even more useful in normed spaces, where it reads $\|u + v\| \leq \|u\| + \|v\|$, because it leads to many useful estimates in analysis.
• Manifolds -- things that by definition locally look like $\mathbb R^n$. Many "objects" that we deal with early in math are manifolds (we just didn't know it at the time).
• All of the separation axioms (Hausdorff, regular, normal, ...).
• All of the countability axioms (separable, Lindelof, ...).

Not shown (for the sake of space and the inevitable lack of completeness) is the interrelatedness between many of these properties for $\mathbb R$, which is another indispensable virtue of $\mathbb R$.

There's no way to do justice to "Why is mathematics about real numbers?" within the length constraints of a Math.SE post, but here are some relatively philosophical observations and opinions (meant to be a bit provocative, in the spirit of answering a soft question).

First, as multiple people have commented, the real numbers are not universally regarded as the be-all/end-all number system. In The Road to Reality, for example, Penrose argues that complex numbers are more fundamental for physics.

Setting that aside, why do we count, and where did natural numbers, integers, and rational numbers come from? I'm not a historian, so everything below should be regarded as a parable, biased by modern mathematical training.

Counting (both the possibility and the ability) arises from the tension between variation and uniformity in the natural world:

• Thanks to variation, there are "different types of thing": that piece of granite, that oak tree over there, the pine tree next to it.... If we look closely at the natural world, we find it to be made of unique objects, to occur in unique, irreproduceable events. In fact, the notion of "event" is our way of cutting the solid stream of existence into temporal and spatial chunks. As Heraclitus said, you cannot step in the same river twice.

• Thanks to uniformity, there are recognizable "classes of things": rocks, trees, snowflakes, stars, sunrises.... No two rocks (or trees, or snowflakes...) are exactly alike. At the very least, they're "in different places" or "at different times" (otherwise they'd be identical).

Once the natural world is observed to contain classes of things, "counting" is a reasonable way to measure "how many/how much". In (a paraphrase of) Kronecker's famous quotation, The integers alone were created by God. All else [in mathematics] is the work of Man. To the contrary, the natural numbers (and therefore the integers) were created by us, as well, an abstraction for enumerating distinct objects similar enough to group together for some purpose.

To make a long story short:

• An integer is a measure of additive comparison between two natural numbers. That is, it's a thing comprising a relationship between two other things. The standard construction of the integers in set theory merely formalizes this: An integer is an equivalence class of ordered pairs $(m_{1}, n_{1})$ of natural numbers, with $(m_{1}, n_{1}) \sim (m_{2}, n_{2})$ if and only if $m_{1} + n_{2} = m_{2} + n_{1}$.

• A rational number is a measure of multiplicative comparison between two non-zero integers. The standard construction of the rationals blah, blah, blah.

It's unsurprising that both abstractions were invented: If two people have flocks of sheep (say), it's natural to ask "who has more sheep, and by how many?". It's natural to represent debts as negative integers. If a flock of sheep (well...) must be divided among several people, it's natural to ask "What is each person's portion?", and to use rational numbers to represent the answer.

The real numbers obviously arose many centuries later, under pressures of Archimedes' method of exhaustion (for which one needs numbers representing "limits of rational sequences"), and were formalized two millennia after that in order to put calculus on a solid logical footing.

I won't even touch the complex numbers, partly for last of time and space (heh), but mostly because Penrose (and many others) do so far more competently, with the depth the subject deserves.

A nice property of real numbers is that they are complete: every Cauchy sequence converges.

In analysis, mathematicians like to study spaces that are complete. People study Banach spaces rather than ordinary normed spaces; study Hilbert spaces rather than ordinary inner product spaces. A space that is not complete does not have as nice properties as complete spaces.

Ever since Pythagoras' disciple discovered that $\sqrt 2$ is irrational, that was the beginning that signals that the rational numbers $\mathbb{Q}$ is not enough to represent all quantities.

However, there are some "problems" with the real numbers, and I know at least one professor (not to mention his name here) who does not believe in the real numbers. His reason, if I remember correctly, is that once you go deeper, a real number is not just a string of decimals: it is a equivalence class of Cauchy sequences. Not only is each Cauchy sequence infinite, each equivalence class is infinite (uncountable I think). This is the price to pay for dealing with real numbers.