Why there are irrational numbers? I do not quite get it. Why can't we represent all real numbers as a sum of rational numbers? Why do we need irrational numbers?
For example,


*

*$\pi=3.14159265358\cdots=3+10^{-1}+4*10^{-2}+10^{-3}+5*10^{-4}+\cdots$

*$e=2.71828182846\cdots=2+7*10^{-1}+10^{-2}+8*10^{-3}+2*10^{-4}+\cdots$

*And so on


Numbers that cannot be expressed rationally arise naturally as solutions of equations. The solution of $x^2=2$ & of $x(1-x)=1$, & any of an infinitude of specificable polynomial equations, cannot be expressed rationally. It can be proven directly, by a simple recipe, that no rational number can satisfy either of the two equations I have particularly cited; and the same can be proven similarly for the general case.
Moreover there is a class of numbers yet beyond this: the transcendental numbers, any of which can be shown to be not the solution of any polynomial equation comprising a finite number of terms having integer coefficients.
