Significance Arithmetic: The Carrying Algorithm

It has been remarked that, in mathematics, a notational deficiency is often a symptom of conceptual obscurity. The starting point of this work is one such deficiency, and such a common one as to be often overlooked or taken for granted. In ordinary binary arithmetic, a number ending in a string of ones, such as 0.0110111 .*. is equal to the number obtained by replacing the digit zero next to the string of ones by the digit one, and the string of succeeding ones by zeros; in the example, 0.0111000 ... . Except in the simple case 0.111 ... = 1 BOO ..., there is no specific notation in the binary system-or in the decimal system-to indicate this seemingly trivial fact that, under certain circumstances, two distinct strings of zeros and ones represent the same real number. And, indeed, if one relies upon the “magnitude” interpretation of real numbers made palatable by Dedekind and unquestioningly followed by mathematicians since, then the lack of uniqueness in the binary representation is an irrelevant oddity; but, then, from such a point of view, any representation of real numbers is irrelevant. To be sure, there were mathematicians of note, Euler among them, who were puzzled by this seeming imperfection, and devised ingenious ways of avoiding it. Our point of view is opposite: instead of considering this redundancy as a bothersome deficiency, we accept it as inevitable; introducing a suitable notation for it we draw the ultimate algorithmic consequences it entails. We are thus led to a new representation and, we would like to believe, to a different concept of the real number system.