Math makes me happy

[crono_logical]

Couldn't you consider .999... sticking a 9 after a series of infinite nines?

It'd be part of the infinite string of 9's, so you could simplify it back to 0.999... and not have the contradiction


SeedRankLou: Simplifying the final solution makes sense (and makes things more readable too), although I wouldn't say working in simplest terms all the time is necessary during the working out stage of your solution. For example, I might prefer not to simplify certain algebraic expressions in an equation because I might foresee the need to unsimplify it again later on to get some terms to cancel out more easily. Or when working out nth roots of complex numbers in (r,theta) notation, you want to take into account values which are not the simplest or principal value of your number if you want to get all solutions.

[Peegee]

the 0.9 = 0.999....9 is right, because the same digit is put in the end, so it really doesn't matter. To bring it back to SeedRankLou's point, it's just redundant to say it.

A slightly better example would be 0.127 (127/999). The result would be 0.127...12, or 0.127..13, if you were to round it, that is.

Of course that's due to rounding. You never put digits after the repeating digits, so it's logically wrong, not a matter of subjective thinking.

edit: Arche just answered it, so um...factor x² + 2x + 2 . Or show me the graph, I don't really care.

[omnitarian]

So, you're saying that if you wrote a 9 at the end of .999..., you'd get .999... . With the same logic, we could say that 999... + 1 = 999... or infinity + 1 = infinity.

Of course, writing a 9 at the end of an infinite string of nines makes no sense, and neither does the equation x + 1 = x.

If you can't have 0.000....1, then you can't have 0.999...

It comes down to which logic you use for infinity. And infinity isn't very logical to begin with. So I don't think there's a "real" answer here.

[Flying Mullet]

Yes, just think of the .999... as a line, and as long as there are only 9's, the line continues infinitely.

Now look at 0.000...1, and it's as long as there are only 0's. The line is not infintely long because as soon as we reach a one, it's like reaching the end of the line.

Also, saying that 0.000...1 is infinite is like trying to say that

(.999... + .999...) is greater than .999...

but when dealing with infinity is like a boolean, either it is or it isn't infinite. And as both numbers/equations are infinite, one can't be greater than the other.

[Peegee]

[q]Of course, writing a 9 at the end of an infinite string of nines makes no sense, and neither does the equation x + 1 = x.[/q]

x + 1 = x is true for { x | x є ∞ } and only that. x + 1 != x for { x | x є R }

and as for the concept of 999... , would that just be ∞ ? Or would it be the lim (x²) x -> ∞ ?

[omnitarian]

Now look at 0.000...1, and it's as long as there are only 0's. The line is not infintely long because as soon as we reach a one, it's like reaching the end of the line.

But you never reach the end of the line, because there are an infinite number of zeroes. Just like there is no final 9 in .999... there is no 1 in .000...1.

So, depending on what logic you use, you could say that .000...1 = 0, or that .000...1 = .000...1, or that .000...1 is illogical.

[crono_logical]

So, you're saying that if you wrote a 9 at the end of .999..., you'd get .999... . With the same logic, we could say that 999... + 1 = 999... or infinity + 1 = infinity.

Of course, writing a 9 at the end of an infinite string of nines makes no sense, and neither does the equation x + 1 = x.

Quoting BoB's proofless post doesn't proove or disprove anything though. Is there something you're trying to say?

Writing anything at the end of an infinite string doesn't make snese anyway so maybe such a question shouldn't have been asked or answered as such The equation makes sense though, it just has no solutions for x.


[q="Flying Mullet"]but when dealing with infinity is like a boolean, either it is or it isn't infinite. And as both numbers/equations are infinite, one can't be greater than the other.[/q]
Actually, you'll find that you can get infinites of different "sizes", as it were, so it's not quite boolean The infinity you get from counting all rational numbers is a smaller infinity than the infinity gotton from counting all irrational numbers, as one example. This'll require me to go back to my notes from 2 years ago on how to prove this one though

[Flying Mullet]

But you never reach the end of the line, because there are an infinite number of zeroes. Just like there is no final 9 in .999... there is no 1 in .000...1.

So, depending on what logic you use, you could say that .000...1 = 0, or that .000...1 = .000...1, or that .000...1 is illogical.

in·fi·nite - adj.
1. Having no boundaries or limits.
bound·a·ry n.
1. Something that indicates a border or limit.

If there's a 1 at the end it has a boundary and a limit as it can't proceed beyond the 1.

Actually, you'll find that you can get infinites of different "sizes", as it were, so it's not quite boolean The infinity you get from counting all rational numbers is a smaller infinity than the infinity gotton from counting all irrational numbers, as one example. This'll require me to go back to my notes from 2 years ago on how to prove this one though [/j]

Yeah, I'm curious to see that proof. I thought that all infinites are equal.

[Peegee]

0.00...1 is not illogical as much as it is incorrect. What you are thinking of is:

10^-10n , where 'n' is a large integer, or whatever.

[Flying Mullet]

Another way I like to view .000...1 vs. .999... is that .999... is infinite while .000...1 is infintely growing.

Imagine that the big bang is pushing out on some "padding", and that we can never go beyond the boundaries of the big bang's ever expanding borders.

.000...1 is like that border, it is always growing outward, but we will reach the end eventually. .999... is never-ending, we can't reach it's end.

Anyways, that's how I visualize .000...1 and .999...

[Dr Unne]

Yeah, I'm curious to see that proof. I thought that all infinites are equal. --Flying Mullet

You can look up Cantor, and the diagonal method. http://en.wikipedia.org/wiki/Cantor%...gonal_argument for example.

[Peegee]

Then prepare to think differently because they are both always growing.

.999... is the limit of (1/x + 1) (I think) as x -> ∞

0.000..1 or whatever is the limit of (1/x) as x -> ∞

[Flying Mullet]

Well yeah, both are infinitely growing, but .000...1 has a limit where .999... doesn't.

I'm just trying to put visuals to it.

[omnitarian]

in·fi·nite - adj.
1. Having no boundaries or limits.
bound·a·ry n.
1. Something that indicates a border or limit.

If there's a 1 at the end it has a boundary and a limit as it can't proceed beyond the 1.

See? You're using the logic that .000...1 is illogical, Or incorrect, or whatever term you want to use. Personally, I don't see it any different.

The equation makes sense though, it just has no solutions for x.

Yes, that's basically what I meant. Ner cow touched it good enough.

Quoting BoB's proofless post doesn't proove or disprove anything though. Is there something you're trying to say?

I guess what I'm trying to say is that when you work with numbers that defy logic, you get answers that defy logic. As hard as we try to shackle it in, infinity is still one really crazy number.

That's all I have to say on the issue, really. So it makes no sense to hang around this hornet's nest any longer. *flees*

[Peegee]

actually 0.999... = 1 :D

I was thinking that the lim (1/x + 1) x-> ∞ was greater than one, but wrote it as less. MY MISTAKE :D