int-n * int-n -> int-n (n > 1) is fundamentally the wrong type logically for a multiplication, though, so something has to give. Sure, some languages have chosen for the arithmetic to be broken in order to maintain type, but that's certainly not clearly the right choice when others have chosen auto-promotion and most hardware has chosen a (limited selection of) int-n * int-n -> int-2n multiplication primitives to work with.
The rule int<N> * int<N> —> int<2*N> breaks one of important expectations from a type, which is the property on being “closed” with respect to certain operations. Even if an arithmetical operation increases the length of the result, it has to stop somewhere, so there’s little justification for such increase in the first place, if one is to be logical about this.
BigInts are closed under addition/subtraction/multiplication. Limited-size integers are not. That's "a truth that may hurt" and all programming languages have to deal with it. Most of them decided that using modular arithmetic and silently throwing away the most-significant digits is fine.
But again, all your arguing about "numbers should be closed under arithmetic operations just as they are in C/C++" is kinda pointless because in C/C++, they are not! When you add/multiply two signed char's, or two shorts, you get an int, not a signed char/short, so only ints and longs are actually closed under arithmetic operations. Whoops!
Nobody expects subtraction to be closed on the naturals, nor division on the integers. Why would be expect multiplication defined to be closed over fixed-width integers to coincide with the general arithmetic operation?
Well, by your definition even division of reals is not closed - which means that your definition is not a correct one. A closed operation is allowed to have an "undefined" result. (And, by the way, division of integers is perfectly closed, if you ignore the remainder.)
