Algebraically, What Does $\Bbb R$ get us? In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$?
Say we start with $\Bbb N$:  $\Bbb N$ is closed under addition and multiplication.  But then we decide we'd like a number system that's closed under negation as well, so we construct $\Bbb Z$.  Great.  But then we decide we'd like to extend this number system further to be closed under division and so we construct $\Bbb Q$.  The next step is closure under exponentiation - but when we construct that number system, we don't get $\Bbb R$, we get a subset of $\Bbb C$ which I'll call $\Bbb Q_{\exp}$.
Now clearly when constructing $\Bbb R$ from $\Bbb Q$ we do gain completeness, but our gain is then analytic, not necessarily algebraic.  Do we gain any algebraic advantage in constructing $\Bbb R$ from $\Bbb Q$ similar to what we get at each of the other constructions I mentioned?

I thinks the motivation for $\mathbb{R}$ is not algebraic, but rather it corresponds to our geometric intuition about which numbers are possible. Also it was founded at a time when we viewed the universe as continuous space.
