In the spirit of
Kill it because it's poorly conceived. In what sense do real numbers "exist" anyway? Do ordinary people even understand what is meant by cauchy-completeness? The ordinary examples of real nonrational numbers are still zeroes of polynomials with coefficients in Q (algebraic). In what sense does the quantity $\pi$ "exist"?
The idea that the real numbers are a totally concrete concept is really just a misunderstanding of the facts. In the same way that $pi$ is defined to be a solution to the pair of simultaneous equations cos(x)=-1, sin(x)=0 up to integral multiplication, $i$ is defined to be a solution to the equation $x^2+1=0$.