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Re: Primes Are Fundamentally Additive/Subtractive, Not Multiplicative/Divisive
Posted:
Mar 8, 2009 8:30 PM


On Sun, 8 Mar 2009, Osher Doctorow Ph.D. wrote:
> Date: Sun, 8 Mar 2009 19:11:14 0400 > From: Osher Doctorow Ph.D. <mdoctorow@ca.rr.com> > To: MATHHISTORYLIST@ENTERPRISE.MAA.ORG > Subject: Re: Primes Are Fundamentally Additive/Subtractive, > Not Multiplicative/Divisive > > To understand what is actually happening, arguably, recall that the natural logarithm is the integral from 1 to x of dt/t: > > 1) ln(x) = I[1/t)]dt, where I...dt is the integral from 1 to x with respect to t. > > Since t > 1, the argument of this integral is a number between 0 (noninclusive) and 1, quite similar and in a sense isomorphic to y/x in Conditional Probability where y is P(AB) and x is P(A) not equal to 0, or in its variant y = P(B) and x = P(A) with y < = x. So we have division fundamentally involved from an integral viewpoint in the natural logarithm and thus in all logarithms on the real numbers.
Eh? That definition for ln(x) is also valid if 0 < x <= 1 if you're restricting yourself to real variables, and x having any finite complex value except 0 is OK if not. Of course the infinitely many possibilities for the path of integration lead to the infinitely many values of the complex logarithm.
 John Harper, School of Mathematics Statistics and Operations Research Victoria University, PO Box 600, Wellington 6140, New Zealand email john.harper@vuw.ac.nz phone (+64)(4)463 6780 fax (+64)(4)463 5045



