Two's Complement and Universal Algebra
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."
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 − 1", i.e., "2X = X − 1", and thus X = ...111 = −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. It is meaningful either as part of a two's-complement notation for all integers, as a typical 2-adic number, or even as one of the generalized sums defined for the divergent series of real numbers 1 + 2 + 4 + 8 + ···. 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. Continuity of binary arithmetical and bitwise operations in 2-adic metric also has some use in cryptography.
Read more about this topic: Two's Complement
Famous quotes containing the words complement, universal and/or algebra:
“A healthy man, indeed, is the complement of the seasons, and in winter, summer is in his heart.”
—Henry David Thoreau (1817–1862)
“It is long ere we discover how rich we are. Our history, we are sure, is quite tame: we have nothing to write, nothing to infer. But our wiser years still run back to the despised recollections of childhood, and always we are fishing up some wonderful article out of that pond; until, by and by, we begin to suspect that the biography of the one foolish person we know is, in reality, nothing less than the miniature paraphrase of the hundred volumes of the Universal History.”
—Ralph Waldo Emerson (1803–1882)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)