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The way that my real analysis book (Abbot) seems to motivate the theory, is that we get the integers and the rationals so that we can solve all linear equations, and that we then need to get the reals so that we can solve equations such as $x^2-2=0$. So does this mean that we are completing the rationals to the reals so that we are allowed to solve such equations? If this is so, why would need to prove that $x^2=2$ has a solution in the reals if that was what we constructed $\mathbb{R}$ to do?

If completing the rationals to the reals is not for the purpose of solving such equations, then what is completion for? What's the motivation behind it? Is it so that we're able to do calculus rigorously, and that fact that $x^2=2$ has a solution in the reals is only an incidental result? I guess my main question is, given that the reals are complete, why do we have to prove that there exists a real solution to $x^2=2$? Shouldn't the existence of $\sqrt{2}$ be implied from the fact that the real line has no "holes," that the reals are complete?

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  • $\begingroup$ In my opinion we don't really care about algebraic completeness. Otherwise that would be algebra and field extensions, a different field altogether. What we cate is that the rationss a dense yet we have sequences that don't converge. We care about the reals and completeness as the reals have the least upper bound property. $x^2=2$ only serves as a concrete example of the failure of the reals. $\endgroup$ – fleablood Sep 27 '17 at 5:39
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The notion of "completeness" that you seem to be considering is algebraic completeness (or the algebraic closure of $\mathbb{Q}$). In this context, you don't actually get the real numbers—instead, you get the complex numbers (or, at least, the algebraic closure of $\mathbb{Q}$, which is (in some sense) a fairly large subset of $\mathbb{C}$). In the complex numbers, the root of every polynomial can be found, including the roots of $x^2 - 2$ (i.e. the solutions of the equation $x^2=2$).

The notion of "completeness" that is useful in analysis (i.e. calculus) is metric completeness (or Cauchy completeness, or Dedekind completeness—these are all subtly different, but in this context, they all amount to the same thing). Essentially, we want to have a space that is large enough to ensure that certain kinds of sequences (specifically, Cauchy sequences) to converge in our space. Once we know that these sequences converge, it is reasonable to talk about limits, which is the entire game of calculus: the fundamental objects of calculus—the derivative and the integral—are both defined in terms of limits.

Once we have the reals as the completion of the rationals, it is not obvious that $\sqrt{2}$ (i.e. one of the solutions to the equation $x^2 = 2$) is a real number. That requires a little bit of extra work. In essence, we have to connect the algebraic structure of the rationals (and $\mathbb{C}$) with the metric structure of the rationals (and $\mathbb{R}$). It is not a priori obvious that $\sqrt{2}$ is real---indeed, I can change the equation slightly, and get something that isn't real: $\sqrt{-2}$ (one of the solutions to the equation $x^2 = {\color{red}-}2$ is not real).

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  • $\begingroup$ That basically completely answers my question. But in the back of my mind I still have a slight confusion. I feel like, since $\sqrt{2}$ is clearly on the real line, that it should be implied that $x^2=2$ has a real solution. Why is this not the case? Why do we have to prove, in detail, that $\sqrt{2}$ is a real number? $\endgroup$ – Wesley Sep 27 '17 at 4:04
  • $\begingroup$ The main reason to construct the reals is to get sequences to converge. This allows us to take limits, which in turn allows us to get continuity. Once we have a notion of continuity on the reals, we can prove the intermediate value theorem (IVT). We can then show that $x^2 = 2$ has two real zeros by invoking the IVT. Since we "already" know that $x^2 = 2$ has two solutions in the algebraic closure of $\mathbb{Q}$, we can connect the dots between the two notions of completeness. Also note: a lot of math is about proving things that we know in our hearts are true are actually true. $\endgroup$ – Xander Henderson Sep 27 '17 at 13:39
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Here is an argument as to why the reals are used in analysis instead of something smaller. One of the main machines coming out of analysis is limits, and the use of limits to rigorously define functions, such as the exponential: $$ \exp(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ Notice that the above definition really only uses polynomials and rational numbers, along with an argument that the series converges because it is Cauchy. However, without all the transcendental numbers, this function is mostly undefined, since for example $\exp(1) = e$, which is not a solution to any polynomial. If we move up to $\mathbb{R}$, then the above series does indeed define a function on the whole of its domain $\mathbb{R}$.

The discrepancy here is that you can show that the sequence defining $\exp(x)$ for any $x \in \mathbb{Q}$ is Cauchy, and so should converge, but if you don't have a complete metric space like $\mathbb{R}$, then there is no guarantee that the value actually exists.

If instead of sequences and series and convergence, you were only interested in solving polynomials with rational coefficients, you could instead use the field $\overline{\mathbb{Q}}$, the algebraic closure of the rationals. This doesn't contain any transcendental numbers, but it is large enough that any polynomial with coefficients in $\overline{\mathbb{Q}}$ has all its solutions in $\overline{\mathbb{Q}}$.

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That real line has no holes is because of the convention that real line is in bijective relationship with the reals, you cannot prove that, we agree to take it like that.

That besides rationals there are some other numbers is known at least from the time when ancient Pythagoreans discovered incommensurability of some numbers, for example $1$ and $\sqrt{2}$ are not commensurable so we even without any mentioning of the equations have the need for non-rational (irrational) numbers.

To introduce numbers that are not rational with the help of equations is one way of introducing them and it can be very useful since you can go from reals to complex numbers in the same fashion, by showing that reals are not enough if we want to solve all quadratic equations.

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