Quantum Gravity 398.7: Well-Defined Issues in y/x --> 1 + y - x by ERASE/ADD
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From: Osher Doctorow on 8 Jun 2010 06:26 From Osher Doctorow To convert y/x to 1 + y - x (both x, y in [0, 1] and y < = x) by ERASE followed by ADD (ANTIERASE) is conceptually simple: 1) ERASE / in y/x, then ADD (ANTIERASE) -, then ADD 1. The well-definedness or uniqueness of this is not a difficulty when y and x are in decimal form as separate probabilities (respectively either P(AB) or P(B), and P(A)) which gives a unique result for all real numbers. However, if x and y are proper fractions or in proper fractional form, then some non-uniqueness can arise without special care or even rules. The main thing required is this: 2) If y = a/b and x = c/d, then y/x = (a/b)/(c/d) --> 1 + a/b - c/d (where a, b, c, d are integers) which requires NOT USING (a/b)/(c/d) = (ad)/(bc) (here b, c are nonzero). If we used the latter equation, then we would get: 3) (incorrect change) (a/b)/(c/d) = 1 + ad - bc, which in general is not 1 + a/b - c/d The difficulty in (3) is that a double division is actually involved, because: 4) a/b - c/d = (ad - bc)/(bd) and comparing (2) and (3) with (4), the extra division by bd in (4) creates a difficulty. For similar reasons, if y = a/b and c/d where a, b, c, d are integers, then we do not divide out common prime factors or powers of prime factors from y and x in y/x, which ideally means that they have no common prime factors and a and b have no common prime factors and likewise for c and d. To avoid these difficulties, converting to decimal form always works uniquely for a/b and c/d. Osher Doctorow
From: Osher Doctorow on 8 Jun 2010 06:32
From Osher Doctorow In (3), = should be -->. Osher Doctorow |