Why `Chop` does not make the real part of machine complex numbers an exact zero?

Question

The documentation says

Chop does not make the real part of machine complex numbers an exact zero:

In[2]:= Chop[10.^-12 + 2. I]

Out[2]= 0. + 2. I

Why does Chop leaves such a useless 0.?

Answer 1

Complex numbers are atomic in Mathematica. Their real and imaginary parts cannot be separated. Thus both must have the same type: either floating point (inexact) or exact number.

1. is an inexact Real, not Complex. It has no imaginary part. Thus Im[1.] can give 0.

But 2. I is an inexact Complex, which does have a real part even if it's zero. Since this is an exact type, its real part is an inexact zero.