You’re absolutely right in your understanding of it. I agree. However, just because you understand it doesn’t mean it’s not a paradox, right? From MathWorld:
As shown by cantara’s and ccwaterback’s posts, there are at least some people who consider this counterintuitive. I myself have no problem understanding why 0.999… = 1, but I still consider this a paradox.
I also don’t think it’s a problem with understanding the meaning of a limit. I personally actively oppose the idea of a limit as “after an infinite amount of time” or whatever, but I can still see whence the confusion comes. I think it’s a problem with understanding the relation between path and path length.
I also refuse to address any calculus problem stated in terms of apples. For personal reasons.
ccwaterback: No, the distance is the same, no matter how it’s traveled. Looking at your equation, I see that on the LHS you squared the distance, then doubled. But on the RHS you doubled, then squared. You need to be consistent. Either double first, then square, or square, then double. Either way, you’ll find that the distance is the same.
Achenar: yes, in the sense of “counterintuitive”, it can be considered to be a paradox. I don’t like the use of a paradox that way, but I guess that’s a bit snobbish of me.
I assumed that I could use the length of the diagonal on the right side of the equation. So, what I was assuming is the square of the sum of the two half diagonals is equal to the square of the original diagonal. This is not true.
Arrrgggghhhh … The sum of the squares of the two smaller diagonals DOES NOT equal the square of the original diagonal. That’s what I meant. I think … hehe. Later dudes.
Well, in the most strick sense there’s no such thing as a paradox and the word should be preceded by the word “apparent”, but that’s rather pedantic. I would consider the Russel paradox to be true paradox; it exposed a real flaw in naive set theory, and was only solved by defining it out of existence. In case you’re not familiar with it, the basic idea is to define R as the set of all sets which don’t contain themselves. The question then is: does R contain itself?
I recall many years ago in Vector Analysis(?) class, that components of a vector (x component and y component in this example) have a sum that is always larger (in magnitude) than the vector’s magnitude for all x > 0 and all y > 0. Damn, that was 20 years ago! I’ll have to dig out my books out of the garage. Forgive my rust…
So strictly speaking, an apparent paradox is a true statement which is apparently false, while an actual paradox is a true statement which is actually false. Yeah, I definitely meant the first one.
I do agree, though, that Russel’s paradox is the closest thing we have to an actual paradox.
