Why do the reals need to be constructed? Do they somehow "span" the rationals, the roots, and the transcendentals like e and pi? Here is my question.  Why do the reals need to be "constructed" by this bizarre "Dedekind cut" or "equivalence class of Cauchy sequences" argument?  Why can't they simply be "observed" as consisting of all numbers that "span" some known sets of numbers?  
I am thinking here, in part, by analogy with linear algebra and with the complex numbers, where $i$, the square root of $-1$, is really all you need, in addition to the reals, to get all complex numbers as a spanning set of $1$ and $i$ over $\mathbb R$.  (Every complex number may be expressed as $a\cdot 1 + b\cdot i$ where $a, b\in\mathbb R$.)
We have a couple of known transcendental numbers, $e$ and $\pi$.  We have all the rationals.  We have all the square roots most of which are irrational.  We have all the fractional roots of $e$ and $\pi$. We have the $e$th roots of all the numbers that exist, and the $\pi$th roots. Maybe we also have some other sets of transcendental numbers out there that we can use?  
What I am trying to ask is, are we using these "Dedekind cuts" and "equivalence classes of Cauchy sequences" just because we don't "know enough real numbers yet", because their characterization hasn't occurred to us yet, or do we already have enough real numbers in our arsenal, like e and pi, to make a "spanning set" without using equivalence classes of infinite sequences and the like, or, is it the case that we really have to use these kinds of constructions of the reals, for some deep mathematical reason?
It just doesn't seem right.  Because you have to admit, by identifying "sets of numbers" like Dedekind cuts and equivalent classes of Cauchy sequences which are both sets of numbers, with actual numbers, mathematicians create (at least in my mind) some cause for doubt about what they are doing here with the reals.  A "set" seems like a strangely undefined term, which I understand, but not well, is subject to various kinds of paradoxes and levels of analysis problems.  (This last paragraph may be more of a separate question, about the validity of using sets of numbers as numbers, from the first question, which is more about why aren't there simpler ways to define or understand the real numbers in terms of numbers and operations we already understand.)
 A: 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.
A: 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)
A: 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.
A: 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. 
A: Lots of people have given good answers, but they seem to me not to be hitting your main question.  You asked why the reals have to be constructed, and is the reason perhaps that there are "not enough" numbers.  The answer is, if you are doing analysis, then no, there are not enough numbers, and the constructions you currently find bizarre are actually the way the need for the reals arise.
Say you just want to use the rational numbers.  They're a nice, ordered field, and they are the space where we in essence take all "real" measurements, like in science.  But in analysis, and its more well-known application calculus, the main new tool one uses is limit-taking: approaching points infinitesimally.  Suppose you say you'll only approach points "nicely", with sequences that never diverge off to infinity, don't oscillate wildly, etc.  A great class of "nice" sequences are the Cauchy sequences, where the points get "arbitrarily" close together the further out in the sequence you get.
Unfortunately, this idea won't work. Look at this sequence:
$3, 3.1, 3.14, 3.141, 3.1415,...$
It's monotone, bounded above and below, and so Cauchy, and it consists only of rationals, but it is obviously meant to converge to $\pi$.  So the logical thing to ask is: what is the smallest number of additional points I need to add to the rationals to make it so all rational Cauchy sequences converge?  But that's the whole real line.
This is why things like the computable numbers seem appealing, and may be needed in explicitly constructive applications.  But nothing short of the reals guarantees rational Cauchy convergence, and this is such a basic need in analysis that anything less is simply a huge headache; i.e. you'd have to condition any theorem involving limits on whether or not the limit exists in your chosen space that is not the whole real line.
I think most of the other answers here are better than this one, and give more interesting details, but I did not want this main point to slip.
A: A very good question, in part because it will be hard to say whether an answer is the right answer.  There are many good answers here already, many correct answers.  But the question has interesting ambiguities, whether intended or not.  Here I offer a few things, from my own perspective, which I think will contribute something to the set of answers.
First, "need."  In an extreme sense, we don't even need all the integers.  All the calculation we need to make things can be done on a finite machine.  A slightly less extreme position is that the computations in making things have always been done with finite precision.  I am not being facetious, because at some point the need comes down to what one is willing to assume just works and what one is unwilling to accept without justification.  Many people are willing to assume that there are numbers that work just fine, and they go out and design and build bridges and financial systems and so forth.  This sort of choice happens in mathematics as well. One may choose to work from a set of assumptions while another may choose to investigate how those assumptions may be justified.  While everyone recognizes the importance of sound footing, they also realize there are important problems someone should think about and not wait on the highly unlikely outcome that there are fundamental problems with our assumptions.
Second, there is the question of whether such a manner of construction of the reals is necessary.  The suggestion that the real numbers are "all the numbers that span a certain something" presupposes that some numbers have been constructed or otherwise exist; and among those numbers, some have a property that would distinguish them as real.  The supposition raises the question of how these other numbers came to exist, were they constructed or are they assumed to exist.  It would be simpler to construct the reals directly via cuts or sequences or, as has been suggested, to assume they exist with the necessary properties.
Third, to do analysis, it is not strictly necessary to construct the reals, for one could construct the surreal numbers instead.  They "work" (that is, could be used to do analysis), since they contain a field isomorphic to the reals. But the reals tend to be more convenient.
Fourth, there is the question of the purpose or a "deep mathematical reason" in constructing the reals.  Someone summarized one reason very nicely in a comment: for the sake of limits.  There is an older reason.  The basic idea in Dedekind's development of irrationals may be found in Euclid (Bk. V, Def. 5.).  It is attributed to Eudoxus.  The purpose was to develop a rigorous theory of similar figures that could handle irrational proportions.  Since the concept of a real (or even a rational) number was absent, a definition in terms of integral multiples was needed.  With it, Euclid proved the gem of Bk. VI, a generalization of the Pythagorean theorem, that if similar figures be constructed on the sides of a right triangle, the area on the hypotenuse equals the sum of areas on the sides.
Finally, another purpose was to develop calculus without appealing to any intuitive notions of what a geometrical line is like and or what an inifinitesimal is.  Sometimes one might "see" something a certain way but it turns out not to be that way.  Something like if $f'(0)=10$ then $f$ is increasing near $0$.  It turns out not to be true always, even though one sometimes (loosely) talks of the graph having an upward slope.  Such loose thinking might produce errors in analysis, and some thought it was important to show that the analysis of real-valued functions of real variables can be founded on a theory that depends only on numbers and not on geometric properties.
A: 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). 
A: To be honest, I find this question quite obscure - yet there have been a lot of very detailed, deep answers discussing different approaches such as Tarski's axiomatic introduction of the reals etc.
However, in my opinion, there is one very simple point about the question that is - mathematically speaking - incorrect and resolving this, we receive a much simpler answer.

We have a couple of known transcendental numbers, e and π. We have all
  the rationals. We have all the square roots most of which are
  irrational. We have all the fractional roots of e and π. We have the
  eth roots of all the numbers that exist, and the πth roots.

Now, as long as we're dealing with the set of rational numbers, we do not "have" any of these. We do not even have the vocabulary to say in any sensible way "the number $\pi$", because $\pi$ is not a number as long as we are "inside" $\mathbb{Q}$ . Neither could you say $3.1415926...$ since the limit process hidden behind the "..." does not have a limit (once again, in $\mathbb{Q}$). You might say "well, I'll just define it as the limit of this Cauchy-sequence and add it to $\mathbb{Q}$" - but then it is highly doubtful what you mean by the limit of a sequence that does not converge. The limit in a larger set? - you would need the reals in the first place!
You might want to take the expression "limit" purely formally. This will then identify your "number" $\pi$ with the sequence you chose to approximate it. For consistency reasons, you might want your "number" to be independent of the sequence you chose (remember that, for the limit, you can always forget about the first few numbers in the sequence) and you will quite naturally end up doing exactly the usual completion process using equivalence classes of Cauchy-sequences.
So, if you are talking about "the numbers $\pi$ and $e$", I kindly ask you to clarify what you mean by these expressions. And I am quite sure that you will actually run into either of these constructions. And if you find a new construction method - even better!
A: The real reason people have constructed the Reals from the Rationals is historical: For a long time people just assumed that Real numbers existed as they understood them, and could be manipulated using the usual arithmetic operations, without making any attempt to justify their reasoning.  This eventually led to some serious confusion when Calculus and Analysis were developed.  So mathematicians regrouped and acted to introduce rigor to the basic mathematics of Real numbers, in order to salvage what they had built on top of them.
One aspect of this, as mentioned elsewhere, is that someone finally sat down and wrote out a set of axioms for the Real numbers, so that anyone who reasoned about them could point to the relevant axioms at each stage of their proof and say 'this is what justifies what I just said'. But once you introduce axioms, a question arises: Is there really a mathematical object that satisfies those axioms?  Because if there isn't, then anything we say about them is nothing more than jaw exercise.
The standard way to address this issue is to construct a model.  I.e, we start with something we already know exists (here, the rationals), and use that to provide the 'bricks' of a set of constructed objects, then use the properties of the construction rules to show that those objects satisfy the axioms.  That is what these 'cuts' do.  If we accept that the 'bricks' exist, and accept that the construction rules produce things that exist from those 'bricks', then we can say that what we're doing is actually legitimate - e.g 'Yes, Reals exist!'.
A: To respond to your comment

Because you have to admit, by identifying "sets of numbers" like Dedekind cuts and equivalent classes of Cauchy sequences which are both sets of numbers, with actual numbers, mathematicians create (at least in my mind) some cause for doubt about what they are doing here with the reals

I would point out that the term "real numbers" is not intended in a philosophical sense (that would imply that these numbers are "more real" than some other more hypothetical numbers), but rather in a technical sense meant to describe a particular, and rather artificial, mathematical construct, whose antecedents happen to have been known in the literature under the adjective "real" at least since Descartes.
In fact given the choice I would have called them Stevin numbers in honor of Simon Stevin who was the first to envision a prototype of this number system, by seeking to represent each number by an unending decimal.
