Editing Two's complement (section)

==Two's complement and {{nowrap|2-adic}} numbers==
In a classic ''[[HAKMEM]]'' published by the [[MIT AI Lab]] in 1972, [[Bill Gosper]] noted that whether or not a machine's internal representation was two's-complement could be determined by summing the successive powers of two. In a flight of fancy, he noted that the result of doing this algebraically indicated that "algebra is run on a machine (the universe) which is two's-complement."<ref>{{cite web |url=http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html#item154 |title=Programming Hacks |work=HAKMEM |at=ITEM 154 (Gosper) |archive-url=https://web.archive.org/web/20240224184437/http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html#item154 |archive-date=2024-02-24 |url-status=dead}}</ref>

Gosper's end conclusion is not necessarily meant to be taken seriously, and it is akin to a [[mathematical joke]]. The critical step is "...110 = ...111&nbsp;−&nbsp;1", i.e., "2''X'' = ''X''&nbsp;−&nbsp;1", and thus ''X''&nbsp;=&nbsp;...111&nbsp;=&nbsp;−1. This presupposes a method by which an infinite string of 1s is considered a number, which requires an extension of the finite place-value concepts in elementary arithmetic.<!--Does this interpretation take into account a sign bit?--> It is meaningful either as part of a two's-complement notation for all integers, as a typical [[p-adic number|2-adic number]], or even as one of the generalized sums defined for the [[divergent series]] of real numbers [[1 + 2 + 4 + 8 + ⋯]].<ref>For the summation of 1 + 2 + 4 + 8 + ⋯ without recourse to the 2-adic metric, see {{cite book |last=Hardy |first=G. H. |author-link=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id={{LCC|QA295|.H29|1967}} |pages=7–10}}</ref> Digital arithmetic circuits, idealized to operate with infinite (extending to positive powers of 2) bit strings, produce 2-adic addition and multiplication compatible with two's complement representation.<ref>{{cite book |title=On circuits and numbers |last=Vuillemin |first=Jean |year=1993 |publisher=[[Digital Equipment Corporation]] |location=Paris |page=19 |url=https://hplabs.itcs.hp.com/techreports/Compaq-DEC/PRL-RR-25.pdf |access-date=2023-03-29 |chapter=Chapter 7}} See especially chapter 7.3 for multiplication.</ref> [[continuous function|Continuity]] of binary arithmetical and [[bitwise operation]]s in 2-adic [[metric space|metric]] also has some use in cryptography.<ref>{{cite web |url=http://crypto.rsuh.ru/ |title=ABC Stream Cipher |last1=Anashin|first1=Vladimir |last2=Bogdanov|first2=Andrey |last3=Kizhvatov|first3=Ilya |year=2007 |publisher=[[Russian State University for the Humanities]] |access-date=24 January 2012}}</ref>