Why do we use $\mathbb{R}$? Since there are holes in the $\mathbb{Q}$ we have constructed $\mathbb{R}$ in order to fill in these holes. But I was wondering why we don't use $\mathbb{C}$ or some other number system that is even larger as the main number system. I was also wondering what are the implications of discovering a larger number system. Could someone briefly explain it to me? 
 A: As the previous answer said, it all depends on what properties you care about. The integers are best for modelling discrete things; the rationals are great for doing arithmetic; the reals and/or complex numbers are needed when you want to do analysis/calculus. 
The reals vs. the complex numbers have different benefits and hence different applications. Since $\mathbb{C}$ is algebraically closed (every polynomial has a root), algebra is really easy. The rich structure also lets you do complex analysis which enjoys a number of nice properties absent in the real case (see  spaces with complex coordinates  for some remarkable examples). 
So, why do we bother with $\mathbb{R}$ at all? Well the obvious answer is that real numbers are everywhere in physics and engineering and so forth. The real line fits an intuitive idea of the world we live in in a way that the complex does not. Perhaps this is why most of differential geometry and related fields focuses on spaces with real coordinates (although people do study  spaces with complex coordinates ). Mathematically, perhaps the most important advantage that the real numbers have over complex numbers is that they are ordered. This lets them be useful as a measurement of magnitude which is vital in many of their real world and mathematical applications. In fact, they are the unique complete ordered field, where complete means "no holes", unlike the rational numbers, and ordered field means that the order plays nicely with the algebraic operations (for example, if $a>b$ then $a+c > b+c$).
Edit:  as Sam brought up in his comment, the complex numbers fail the criterion of being ordered in a strong sense: not only is there no obvious or canonical order on them, but an ordering that plays nicely with the field operations is impossible. The linked Wikipedia article on ordered fields explains this, but it is instructive to work it out on your own. Try to find a contradiction by assuming $i > 0$ or $i<0$ and proving somehow that $-1 > 0$.
A: There is no "main" number system. We study the reals to say things about the reals, we study the complex numbers to say things about the complex numbers, we study the integers to say things about the integers, and so on. All of these sets are interesting and worth studying in their own right.
The reals enjoy a large application in engineering, along with the complex, but that doesn't make them "special" or the "main number system."
A larger number system would not be "discovered", it would be invented. An example of such a number system are the real Hamilton quaternions.
But again, there is no "main number system." There are many well defined sets of objects, and they are studied. That's it.
A: Note that larger number systems are known, for example the surreal numbers (as well as the superreal and the hyperreal numbers, which are subsets of the first), which also include the infinitesimal numbers as well as the transfinite ordinal numbers, as well as variations of them, for example $\omega+1.56$.
Why do we almost never use them?


*

*In elementary algebra, we don't use them since $\mathbb{C}$ is algebraically closed so they won't appear as solution. 

*In calculus infinitesimal numbers are sometimes used, but they shouldn't be treated as a (sur)real number. As well as $\pm \infty$ which shouldn't be treated as the surreal number $\pm \omega$.

*In set theory, where ordinal and cardinal numbers are frequently used, the infinitesimal are not used and things like $\omega^4+2.34$ aren't used either. Therefore it is not useful to consider surreal numbers, and they behave differently (see comments).

*In other fields of mathematics where I am aware of, it surely doesn't make sense to talk about this, because infinitesimal numbers or transfinite or infinite numbers are not used.


I believe that the superreal and the hyperreal numbers are used in abstract algebra. 
