TITLE: Why do we study real numbers?
QUESTION [42 upvotes]: I apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\mathbb{R}$ and its subsets (as opposed to any other algebraic structure)?
Is this because $\mathbb{R}$ is "objectively" more interesting in that studying it allows one to gain deep insights into mathematics, or is it sort of "arbitrary" in the sense that we are inclined to study $\mathbb{R}$ due to historical reasons, real-world applications and because human beings have a strong natural intuition of real numbers? 
Edit: Note that I am not asking why $\mathbb{Q}$ is insufficient as a number system; this has been asked and answered on this site and elsewhere. Rather, why, in a more deep sense, are $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ so crucial to mathematics? Would we be able to construct a meaningful study of mathematics with absolutely no reference to these sets, or are they fundamentally imperative?

REPLY [0 votes]: Because of settling on late 19th century and early 20th century formalization of the idea that R is supposed to represent. Plenty of people study variations on R that may look more like what for example physicists may be used to, like for example variations with infinitesimals.
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