>The same for negative numbers

Are you sure negative numbers are not "real" in the same way natural numbers are?

That would mean particles like electrons with a negative charge are not "real", too. And someone can argue that his negative bank balance is not "real".

As for complex numbers, I fail to see how they are less real than real numbers. Real numbers are points on the real axis, while complex numbers are points in R^2 plane. If we question the reality of real numbers, we can also question the reality of the plane.

Mr Euclid would frown to hear that.