Chaos in Newtonian iterations: Searching for zeros which are not there

We show analytically that Newtonian iterations, when applied to a polynomial equation, have a positive topological entropy. In a specific example of an attempt to ``find'' the real solutions of the equation $x^2+1=0$, we show that the Newton method is chaotic. We analytically find the invariant density and show how this problem relates to that of a piecewise linear map.