0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1".
This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.
If you could step up to the board and actually expand the ... for us, we'd be much obliged.
You can keep writing 9 forever and you will never reach a whole number. But, it does reach a whole number and there is proof! So, they made a rule. The rule is, of course, silly. Infinity itself is silly. Not even atoms are infinite. Not even atoms can be infinity divisible.
It's a horrible idea, really. (I have karma to burn.)
By the way, it's a damned good thing infinity doesnt exist. If you had a tape measure that was infinitely long and folded it in half, we aren't really sure what would happen.
Hell, I wasn't using that karma for anything better.
While I'm here... When you're done finding an infinite tape measure, I'd like it if you could bring me -1 apple. No, not a slip of paper representing an owed apple, but -1 apple.
Math and reality are only loosely coupled. Mathematics is a language. It can tell lies. 1 + 1 = 74 See? That's pretty obviously a lie. It can tell stories, physicists are pretty fond of using it for that. It can be so very descriptive and elegant. It can be so deceptive, as we see with the 'well, let's just let that be represented by 9/10' silliness.
It's pretty horrible, really. Like I said, it's best not to think about it too much. It's kind of contrived in a few areas.
It seems you don't know what a sequence is. A sequence is just a function whose codomain is the natural numbers, i.e. you give any natural number you choose, and I give you the corresponding digit. I don't need to 'expand' the ellipses to define that function, any more than I have to write down all perfect squares in order to define what a perfect square is.
Seriously, this is basic mathematics. You should study it before publicly embarrassing yourself by calling mathematicians 'liars'.
The ... represents an infinitely repeating numeral, does it not? It is a shorter way of writing 0.999999999999 (into infinity).
Now, you can do that forever and you will never reach a whole number. Ever... Go ahead, try it. I'll wait.
So, we've all happily agreed on some very contrived rules that you'll quote verbatim and gleefully declare to be basic mathematics and I the embarrassment. You'll commit no more thought to the process and, as I've already stated, that's a very good choice.
(Karma for days...)
What in the world are you rambling about? Do you know what a limit is?
> I hold a Ph.D., you know.
Yeah, and I'm the king of France. If you really have a Ph.D., I'd wager it's not in mathematics, given the content of your posts.
But you don't know what a limit is?
We've effectively just said, 'Well, bugger it. We don't have to expand that.' We've all agreed it equals one. That's kind of silly.
I can quote the rules, the reasons, and even helped to author the early versions of the Wikipedia article on the subject. I understand the textbooks, and they are silly.
I haven't found the history of when this was discovered, so I have to speculate. I think it went a bit like this...
Bob, that's his name, was smoking opium one day and pondering mathematics. On a hunch, he sat down at his desk and, between nods, he discovered that 0.999... = 1.
This, of course, is absurd. After all, anyone would notice that the deception lies in the conversion to a fraction. But, we can't have that.
Bob chose to keep it a secret.
Except Bob was quite an opiate addict and only made it about a week before he started telling anyone who would listen.
Unfortunately, so many people heard about it that the higher order of mathematicians decided they couldn't just assassinate Bob and, after all, he was right - and he could prove it!
So, they thought long and hard. How could this be made true? How could this still be right and the order still be maintained?
Eventually, a bright young neophyte named Hank piped up, 'Wait! I have it! Let's just imply a limit!'
They all cheered and headed to the bar to buy Hank a round of drinks.
Something like that. ;-)
In an earlier comment you said you didn't have to expand it. You're right. You don't have to, but you can. There is no rule that prevents you from stepping up to the board and writing 9s until the heat death of the universe, only an imposition made to keep order. Kinda silly, isn't it?
As I said above, mathematics is a language. You can say anything you want. Some of it is trivially proven false, though it is still useful.
Much obliged.
If it goes on forever, it can't reach the whole number. But, I can prove it is equal to one. Horrible.