Why do the reals need to be constructed? Do they somehow "span" the rationals, the roots, and the transcendentals like e and pi?

If you like, and some people do, you can forget about any construction of the reals from the rationals (or anything else) and instead define them axiomatically. One such axiomatization is Tarski's.

This approach will avoid any weird feeling you might have about a real number being an equivalence class of whatnot.

Usually, the reason to provide an explicit construction of something from a simpler things is that it proves that that something exists (mathematically). Moreover, it allows you to study properties of that something in terms of the simpler things that you presumably know better.

Nobody thinks of real numbers as equivalence classes of anything. Once the construction is done you can just forget about it if you like. Having a construction just means that the model of the real numbers that you fantasize about is at least as consistent as a model you might have of the simpler things. To some people it gives reassurance, to others a headache.

As for your attempt to define the read as something spanned by those things we have names for, together with some operations on there. The problem is that there are only countably many such things while there are uncountably many real numbers (at least if you believe that every real numbers admits at most two decimal representations). So this can't work. It might be strange to think about there being more reals then potential names or ways to approximate reals but it's a real fact (pardon the pun).

Constructing the reals is important if you want to do analysis. If you want to talk meaningfully about sequences or continuity, you need to fill in the "holes" in your space. You're coming from the perspective that we built the reals because we need "more stuff", but that's not the case. The reals are designed to fit together a certain way, and it just so happens that you need a lot of stuff to do that. If all the interesting analysis we wanted to do could be done with a smaller, countably infinite structure, it's possible that's what we'd call "the real line". In fact, I think some people do try and do analysis with the computable numbers.

You can get away as follows: You demand axiomatically that there exists a complete ordered field. It can be shown that any two such fields are canonically isomorphic and thus whatever someone assumes to be his personal idea or mental representation of $\mathbb R$, it is essentially the same as other people's idea as long as they agree to talk about a complete ordered field. (You than rather obtain $\mathbb Q$, $\mathbb Z$, $\mathbb N$ as subsets instead of constructing the other way round)

The vector space of $\mathbb R$ over the field $\mathbb Q$ is an infinite dimensional vector space. The reason is that $\mathbb Q$ is a countable set therefore $\mathbb Q^n$ is also countable, but $\mathbb R$ is not countable. So, we will need an uncountable basis to constuct real numbers from rational numbers.

You can actually do math without explicitly constructing the real numbers, although you end up constructing them implicitly.

If you accept that the number "1" exists, as well as the basic operations plus, minus, multiply, and divide, you can construct the rationals.

Even though you can't picture fractions like 3559/3571 in your head, you can certainly see how they could be constructed.

Sadly, there are several problems you can't solve with rational numbers:

• x = x + 7
• x*0 = 5
• x^2 = 2
• x^2 = -1

Why does not having a solution to "x^2=2" bother us more than not having a solution to the other problems above?

Answer: you can find rational numbers p and q such that p^2 < 2 and q^2 > 2 AND make q-p < epsilon, for any rational value of epsilon, no matter how small.

In other words, you can "squeeze" rational number squares as close to 2 as you want, without actually touching it. This offends our intuition, although Zeno claims it's quite normal (we try to punch him, but can get only arbitrarily close).

How do we solve this problem? Several possibilities:

• Create a new number "s" and declare that s^2=2. Of course, this doesn't help with problems like "x^2=3" or "x^3=2".

• Declare that every polynomial with rational coefficients is now also a number, namely the number that solves the polynomial itself.[1]

• This seems to work fairly well, until someone points out the ratio of a circle's circumference to its diameter is not part of your number system. Again, you can arbitrarily close to that ratio, but never quite hit it.

• So, how do you solve this new problem? You declare every set of rational numbers to be a number. Notice that you still haven't explicitly constructed the real numbers: your number system consists only of rational numbers and sets of rational numbers (we throw out the solutions to polynomial equations with rational coefficients since it's redundant).

• With a little cleverness, you can define rules for adding, subtracting, multiplying, and dividing these new numbers you've created, both with each other, and with the rational numbers themselves.

• How do these new numbers (ie, arbitrary sets of rationals) solve "x^2=2" and similar problems? You declare that a set S of rational numbers is a solution to f(x) = y, provided that:

  • For all r in S, f(r) <= y

  • For any rational epsilon, there exists r in S such that |f(r)-y| < epsilon

• You have now implicitly constructed the reals, simply using rational numbers and sets. No real numbers anywhere.

• Of course, there are a few problems with declaring any set of rationals to be a number. For example "x^2=2" now has an infinite number of solutions.

• At this point, you might want to declare two sets to be equivalent under certain conditions (eg, the "least upper bound" condition), but this isn't really necessary: if you're OK with having infinite solutions to problems like "x^2=2", you can stop here.

There you have it: mathematics without explicitly constructing the real numbers!

DISCLAIMER: I realize this probably has some errors (eg, removing unbounded rational sets), and I intend it solely as a general guideline.

[1] In traditional mathematics, polynomials have multiple solutions, so declaring a polynomial to be a single number is admittedly a bit odd. However, I'm using this as a throwaway example.