Do we really need reals? It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.
Since almost all real numbers are not computable, it seems that real numbers are a set too big and most of them are unnecessary.
So why do mathematicians love this plethoric set of numbers so much?

Suppose you have a square of area 1 cm.You want to calculate length of the diagonal.If you don't have notion of real number then you can not find length of the diagonal.Another way to say this that in a countable system you can always find some equation that has no solution in that system.For example in rational number system there exist no solution of the equation $x^2-2=0$.
Note:
Mathematician always concern with mathematical structure in which every equation has solution.In natural number system there exist no solution of the equation $X+2=1$ .So mathematician extended system of natural number to set of integers.Again there exist no solution of the equation $x^2-2=0$ in rational number system.So mathematician extended the rational number system to real number system.Again the equation$X^2+1=0$ has no solution in real number.So mathematician extended real number system to complex number system.Indeed in complex number system every equation has solution.So they stopped...:)
Hope this helps.
