Re: naive question from a non-mathematician


quat a écrit :

Are a real number x and a complex number whose real part is x and whose
imaginary part is zero mathematically eqivalent? For example, is
(real).123 mathematically eqivalent to (complex).123 + 0.0i?

The short answer is:
Yes, they are equal.

Isn't that almost like saying the vector (x, 0) = x? But this can't be true
because a vector is not a real number, so you can't compare apples and
oranges? Of course, in some sense you can upcast a real number to a complex
number in a natural way and vice versa.

Yes. It is usually implicitly assumed when you are talking about the
field of complex numbers, that you have the inclusions N in Z in Q in R
in C. It probably would be more correct (and horribly unpractical) to
talk about injective mappings in each case.

.