[dan-k]

Actually, from a math perspective, it's kind of nonsensical to talk about type conversion at all. That is, 3, 3.0, and 3/1 are purely notational differences, and all three represent exactly the same object. Since math is theoretically infinite precision, the way you write a number has no impact on the way operations like division act on it, making whether it belongs to Z, or just Q or R a moot point. Now, if you restrict your problem domain to Z, then that affects the operations you're able to do to a number and remain in that domain. That's more like what happens with types in programming; we restrict our domain by default because leaving the integers changes the computer's representation of the number in a way that affects its behavior.

[SamReidHughes]

Kind of. But you might say that rational numbers are ordered pairs of integers and positive integers, real numbers are Cauchy sequences of rationals, and complex numbers are ordered pairs of real numbers, and the real number 3 can't be added to (2+i) without converting it to a complex number first.

[dan-k]

Those are all valid ways of conceptualizing those sets, but I don't think it changes the point I was making. The real number 3 doesn't need to be "converted" to a complex number to be added to 2+i. 3 is always both a real number and a complex number, which may be represented as either 3 or 3+0*i, and either way gives 5+i when added to 2+i. All the latter notation really does is clarifies what domain you're currently working in, and even so, I've never seen anyone write it out explicitly.

Type conversion is more like if you had the written number 3 and a picture of the point 5 on a number line and someone told you to add them. Naturally, you would write 5 as a number first, because you don't have a useful way to add a number and a picture. But this doesn't change the results of adding the quantities 3 and 5; it's purely an artifact of the way the information was presented to you.

[orra]

A nice thing about John Conway's construction of the Surreal numbers is that integers are really a subset of the rationals, which are really a subset of the reals, which are really a subset of the surreals. I find this much more elegant than having "merely" finding an embedding of the smaller structure inside the larger one.