David Laight <david%l8s.co.uk@localhost> writes: > On Sat, May 15, 2010 at 09:22:39PM +0000, Joerg Sonnenberger wrote: >> >> Log Message: >> Follow the Fundamental Theory of Algebra. Disallow factorising of >> numbers less than 2 as it is not >> - naturally unique (negative numbers) >> - finite (0) >> - non-empty (1) > > The 'Natural numbers (N)' are the positive integers, for which both addition > and multiplication are defined. This heavily depends on school. At least set theorists define natural numbers as finite cardinals, many other logicians and some algebraists do the same, this makes zero natural number. > To make an additive group you need to include zero, zero is also needed to > make a number field. > If you include the subtraction operator then you need to include the > negative numbers - this gives the 'Integers (Z)'. > The notion of 'primes' is valid in Z - the definition of a prime is a > number that has no non-unit factors. > The units of Z are +1 and -1, so both +2 and -2 are primes (etc). This depends on definition of unit. Yours isn't canonical but it may be reasonable under some assumptions. > So nothing about algebra stops you factoring negative numbers. > However, since the 'prime factors' should be prime numbers, they > shouldn't include -1, but maybe the smallest factor should be negative. While it may be controversial whether to count 1 and -1 as prime, it is perfectly sensible to do so. Again, this boils down to difference in definitions and it should be discussed rather than rejected ex cathedra. Following the logic Joerg uses, one should reject all arguments to sqrt, asin, acos, atan, clog, casinh, cacosh, and other inverse functions just because they have more than one branch. In "fundamental theory of mathematics" be it geometry, real or complex analysis, or anything else, this approach is found counterproductive. -- HE CE3OH...
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