I'm the originator.  In fairness to Jan, I must say that I only showed
him the code and output with no other comment than that I hoped that
the students would not ask me to explain it.

I've in the meantime found a way of illustrating the point I meant to
make more clearly.

sage: z=1.+sqrt(-1)
sage: base_ring(z)
Symbolic Ring
sage: base_ring(real(z))
Symbolic Ring
sage: base_ring(imag(z))
Real Field with 53 bits of precision

Dirk

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