honestly i didn't get the _whole_ point of this message but i have some comments on it Thomas wrote on Sunday, March 27, 2011 at 3:21:04:
...However, XInt and other bigint libraries took the opposite decision, so there is strong circumstantial evidence against using two's complement representation for (iii)...
i wondered myself why chad didn't use two's complement because it seemed straightforward to me but i didn't ever mentioned it eventually chad stated that sign-magnituted implementation suits better for non-trivial operations and i have no point to protest
...So I tried to write the Quaternion group (the one with the 8 elements 1, -1, i, -i, j, -j, k, -k) as a semi-direct product. This didn't work, because the Quaternion group is not a semi-direct product...
perhaps you ment octonions since quaternions have {1, i, j, k} elements in contrast to what you mentioned
...This is also one of the reasons why the two's complement representation should be avoided for floating point numbers, because the relevant metric in this case is the normal metric of the real numbers.
afaik IEEE 754 / ISO/IEC 60559 standards describe just that sign-magnitude representation which is widely used in the majority of computer processors what i understood is that you suggest to carefully think of signed/unsigned behavior of integers when one develops an integer algebra library but i think that in C++ we have no choice but to follow the historical tradition and to treat integers like our parents did otherwise, such way may introduce confusion to users and render well known idioms unappliable however i do not object to begin treating integers in semantically more correct way even if it can cause consequences -- Pavel P.S. if you notice a grammar mistake or weird phrasing in my message please point it out