If you're having math problems, I feel bad for you, son...

[one_true_leroy]

10. "God created the integers"

And when "God" created the integers, she said they were of finite length. There are no integers of infinite length, and there is no "look[ing] at why," as you implore. The very careful definitions of integers, rationals, etc., cannot be muddled ( as you suggest) without crumbling the edifice upon which math is built.

<You've made an argument for 1/3.> {sorry- incorrect attribution} It is NOT an integer, but a ratio of integers, and its repeating nature places it in the rational numbers.

If you are to continue a MATHEMATICAL argument, you must accept the mathematical definition that there exist NO integers of infinite length. Now, there is no limit to the length of an integer, but this is not the same as admitting infinite length.

In other words, if you pick a number, and it is an integer, it WILL be finite in length. If it is not finite, it may be cyclic (rational) or not (irrational), but not an integer. You can dig around and find a proof that every ratio of ANY pair of integers will have a terminating or cyclic decimal representation.

If you are unwilling to accept these definitions, you are not making a MATHEMATICAL argument, but merely a semantic one. Math is the elimination of semantics in pursuit of truths.

If you are sincere, read Howard Eves Foundations and Fundamental Concepts of Mathematics. Best introduction to higher math I've ever read. If you insist on being ornery for ornery's sake, read something by Bukowski.

Peace and waffles,
leroy
On edit: it wasn't you that mentioned 1/3, and that post argued correctly about the decimal expansion.
Double edit: be more careful in math arguments. It is a VERY unambiguous field, especially at this level.