There are several possible meanings to assign to the sentence A==B (mod C) when some of A,B,C might be non-integral. In the present Monthly context, the meaning is that A and B are rational, C is an integer, that the denominators of A and B are relatively prime to C, and that the numerator of A-B, after reduction to lowest terms, is an integer multiple of C. Reduction to lowest terms won't matter in this case, since the denominators of A & B are prime to C. When B is 0, this reduces to "C divides the numerator of A". Dan's first glance interpretation is a plausible one, but different: Treat the sentence as asserting that the expressions involved *are* integers, i.e., that the statement implicitly asserts "type" information about the ingredients. It is indeed true that Hn (n>1) isn't an integer; the "instant proof" is that, since there's always a prime q between [K/2] and K, the sum of the reciprocals to K will necessarily have q as a divisor of the denominator. The second interpretation, treating the sentence as being about numerators, is OK when B is 0, but breaks down when B is nonzero, since expected rules like A==B --> A+1 == B+1 will often fail. It's reasonable to demand that any use of A==B (mod C) notation other than the all-integer case should clarify what definition is intended. There's obvious uncertainty over what set of quotients is allowed for (A-B)/C. A similar problem turns up if I say x(x+1) = 0 (mod 2), which is true when x is interpreted over the integers, but false for polynomials. I don't know the history: I don't think the extended notations are used in Hardy & Wright, except maybe for (mod 1) to mean fractional part, or for stuff with algebraic integers. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Dan Asimov Sent: Thu 6/7/2007 12:27 PM To: math-fun Subject: [math-fun] Modular notation In an article in the June-July 2007 Monthly, it is stated that for any prime p >= 5, the harmonic number H_(p-1) == 0 (mod p^2) (where == means a triple equals sign). Huh??? I thought? H_n can't be an integer! Must be a typo. Until I noticed more of the same kinds of apparent errors in the rest of the article. For the first case, H_4 = 1 + 1/2 + 1/3 + 1/4 = 25/12. So apparently x == j (mod N) is defined for rational numbers x, and means that the numerator of x (in lowest terms) == j (mod N) in the old-fashioned sense. Is it common notation to apply modulo to rationals? Of so, when did it begin to be used? --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun