abo wrote: > Patricia Shanahan wrote: >> I am very familiar with the difficulties of arithmetic with a bounded >> range. You lose all sorts of useful properties, such as associativity >> and commutativity of addition. Adding up numbers in one order may lead >> to an overflow, adding them in a different order doesn't. > > Order doesn't matter if one is working in the natural numbers, of > course. In the natural numbers both commutativity and associativity > hold, in these forms: > > (x)(y)(z)(x + y = z ==> y + x = z) > (u)(x)(y)(z)(u + (x + y) = z ==> (u + x) + y = z) >
Yes, I was wrong in a couple of ways. Commutativity is not affected by gaps and bounds, and bounds do not affect unsigned addition associativity.
However, associativity is lost for signed arithmetic if there are bounds, and even for unsigned arithmetic if there are gaps.
For example, u, (x+y), and u+(x+y)=z could all exist, but u+x could fall in a gap, and not exist.