Friday, July 09, 2004
A more formal approach to defining the factoring problem
I've been thinking more (of course) about factoring. Here's some thoughts; I'm recording them here mostly so I won't forget what I was thinking (it's more portable and permanent than paper):
Let II equal the set of all "Information". I believe there's already a well-defined "universal" set, but I don't know it's symbol. Anyway, all sets of numbers are subsets of this "Information" set. Since this is the set of all "Information", for kicks we'll throw into this set all possible logical operations, all math functions, etc. Z, the set of positive integers, is a subset of this universal set. R, the subset of rational numbers, is also a subset of this set. It seems to me that most apporaches to factoring are based on techniques applied mostly to the set Z. What if we include the set R? In R, a number like 7 has factors, because it can also correspond to 70, 700, 7000, etc. Namely, 7/2 = 3.5, which is equivalent to 7 * 5 / 10, or 35/10. It could also be equal, for that matter, to 700/20, and so on. Thus while 7 has no factors in Z besides one and itself, it _does_ have factors in R. Perhaps there is some logical or functional subset of II which, when given a member of I, determines whether it has factors in R as well as I.
I'm trying to formalize this because I've found that by symbolically representing things I sometimes see stuff that I otherwise wouldn't see. Anyway, some food for thought.
Let II equal the set of all "Information". I believe there's already a well-defined "universal" set, but I don't know it's symbol. Anyway, all sets of numbers are subsets of this "Information" set. Since this is the set of all "Information", for kicks we'll throw into this set all possible logical operations, all math functions, etc. Z, the set of positive integers, is a subset of this universal set. R, the subset of rational numbers, is also a subset of this set. It seems to me that most apporaches to factoring are based on techniques applied mostly to the set Z. What if we include the set R? In R, a number like 7 has factors, because it can also correspond to 70, 700, 7000, etc. Namely, 7/2 = 3.5, which is equivalent to 7 * 5 / 10, or 35/10. It could also be equal, for that matter, to 700/20, and so on. Thus while 7 has no factors in Z besides one and itself, it _does_ have factors in R. Perhaps there is some logical or functional subset of II which, when given a member of I, determines whether it has factors in R as well as I.
I'm trying to formalize this because I've found that by symbolically representing things I sometimes see stuff that I otherwise wouldn't see. Anyway, some food for thought.
