[kps]

  > However, the result of the multiplication of two n-bit numbers
  > could be as big as 2n-bits.

For unsigned multiplication. For signed two's complement multiplication, it's at most 2n−1 bits, and that is only when multiplying the two most negative numbers; otherwise it's at most 2n−2 bits.

~~~ wavy line flashback ~~~

A few years ago, I did a C code generator for a 2's complement 16-bit processor that had mostly adopted the convention that 0x8000 was an invalid sentinel value, so that integers would have a symmetric range around zero, ±0x7FFF. This was mostly a software convention (i.e. the ALU didn't trap or anything like that) except that, given that 0x8000 was 'unused', the multiplier only produced a 30 bit result. This made C 'long' multiplication interesting.