.9~=1

[Rici]

riciardos, it may interest you to know that I actually attempted to comprehensively develop a "super-infinite" decimal place algebra. I'm sorry to say that it is not a closed group under multiplication (what's .0~1*10?), and thus can't really be taken into consideration

You got a point there.
But saying that a concept equals 1 number can't be right either.
Because infinity equals infinity.
Just because infinity means all the numbers doesn't mean it equals just one of them.
Like I said earlier
68xinfinity=infinity
infinity/infinity=1

So every number would be equal to 1 -_-'

 

[rounder_nk]

someone ignores rounder's limit proof too many times. So limits say that if a number approaches another number then it takes on the value of that number. So in a mathematical equation it would equal one. It might not exactly be one if we delved deeper into that area but in math today it equals one.

 

[bornfidelity.com]

It might not exactly be one if we delved deeper into that area but in math today it equals one.

So math today is not able to deal with infinity correctly.
Thank you

 

[Uncle Meat]

Riciardos, infinity isn't a number so you can't divide it by anything.

 

[Killer Tree]

You got a point there.
But saying that a concept equals 1 number can't be right either.
Because infinity equals infinity.
Just because infinity means all the numbers doesn't mean it equals just one of them.
Like I said earlier
68xinfinity=infinity
infinity/infinity=1

So every number would be equal to 1 -_-'

Infinity / infinity does not equal 1, it is undefined. Like dividing by zero.

It might not exactly be one if we delved deeper into that area but in math today it equals one.

Auctually it IS EXACTLY one. When we dig the deepest we get it will always be exactly one.

So math today is not able to deal with infinity correctly.
Thank you

Auctually math CAN deal with infinite.

http://en.wikipedia.org/wiki/Infinity

 

[Doraki]

bf.com, please, stop posting the same stupid thing over and over in this thread especially since you already said

sorry sometimes I find joy in talking crap and seeing if people buy it or not

 

[bornfidelity.com]

cut me some slack. rounder just gave me open goal for that one.
besides its not stupid, but its not mathematically correct. and since you're dealing strictly with mathematical means here, its just that we approach stuff differently.

i've had it with this topic anyway.
stupid math fanboys

 

[Killer Tree]

but its not mathematically correct

wrong

If you STILL don't understand READ THIS NOW!

http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1

Solved by proofs as in MATH PROOFS

 

[rounder_nk]

why is it that people are impervious to facts? Another thing is why is it that nobody has noticed that i have been over everything that this page talked about in the last few pages.

 

[bornfidelity.com]

wrong

If you STILL don't understand READ THIS NOW!

http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1

Solved by proofs as in MATH PROOFS

No, read again please, I was saying that MY reasoning was mathematically incorrect (I reason from another point of view) and that therefor me and Doraki will never agree on this matter.

 

[Killer Tree]

..................Not this **** again...........................
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No, read again please, I was saying that MY reasoning was not mathematically incorrect (I reason from another point of view) and that therefor me and Doraki will never agree on this matter.

This is mathematically correct, only because mathematically there's an end to infinity.
This is in real life incorrect, because in real life infinity is infinite.

Auctually you are wrong mathematically because for some strange reason you think there's an end to infinite. Which is wrong. ANY point of view that thinks infinite is not infinite is... wrong.


Years of work my Mathmeticians > some random persons opinion based on nothing but what the think is right without any facts/reason.

 

[Killer Tree]

MORE PROOF


.4242424242... = 42/99
.402402402402402... = 402/999
.000... = 0/9
.111... = 1/9
.222... = 2/9
.333... = 3/9
.444... = 4/9
.555... = 5/9
.666... = 6/9
.777... = 7/9
.888... = 8/9
.999... = 9/9
9/9 = 1
.999... = 1



.999... = sigma(.9*[.1]^[n-1]).





.999... = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)
Let a, 1st term = .9
Let r, common ratio = 10 ^ -1
Infinite geometric progression: a + ar + ar²... (ad infinitum)
Sum to infinity = a / (1 - r) = .9 / (1 - .1) = .9 / .9 = 1
.999... = 1
http://www.geocities.com/admiralmussina/SigmaNinev2.rtf
--- OR ---
sigma(n = 1, n -> inf.) 9/(10^n) = Definition of geometric series.
9 * sigma(n = 1, n -> inf.)(1/10)^n = Property of a series.
9 * 1/9 = (r/[1 - r], r = .1)
3/9 = 1/3
.999... = 1
--- OR ---
.999... = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)
sigma[-1:0 -> inf.](.9 * .10^-1)
omega -> sigma = .9 / (1 - .1) = .9 / .9 = 1
.999... = 1





Let S = .999...
S = .9 + .09 + .009 + .0009 + .00009 + .000009 .... (ad infinitum)
S = 0.9 + (1/10)S
(9/10)S = .9
1 = S
.999... = 1




Thanks to Blizzard for this proof:
lim(m --> inf.) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus, x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9




.333... = sigma(n = 1, n -> inf.) 3/(10^n) = Definition of geometric series.
3 * sigma(n = 1, n -> inf.) (1/10)^n = Property of a series.
3 * 1/9 = (common ratio, r/[1 - r], r = .1)
3/9 = 1/3
1/3 = .333...
1/3 * 2 = 2/3
.333... * 2 = .666...
2/3 = .666...
.333... + .666... = .999...
1/3 + 2/3 = 3/3
3/3 = .999...
3/3 = 1
.999... = 1
--- Corollary Proof ---
.999.../3 = .333...
1/3 = .333...
.333... = .333...
.999.../3 = 1/3
.999... = 1




For two numbers to be different, there must be a number between them.
.999... < N < 1
Find N. Because .999... has infinite nines, there are no numbers between them, thus N cannot exist. .999... = 1.



http://www.math.lsa.umich.edu/mmss/courses/infinity/Equals1.shtml
http://www.cut-the-knot.org/arithmetic/999999.shtml
http://www.mathmojo.com/interestinglessons/999repeating/999repeating.html
http://mathquest.com/discuss/sci.math/a/t/174695
http://www.spiraleye.250x.com/rave1/rave3.htm
http://www.beliefnet.com/boards_mini/index.asp?boardID=28680
http://www.myrkul.org/docs/infinite.htm
http://www.notmuch.com/Speak/BBS/Odds-And-Ends/509x2.html
http://mathforum.org/dr.math/faq/faq.0.9999.html
http://mathforum.org/library/drmath/view/57174.html
http://mathforum.org/library/drmath/view/55746.html
http://mathforum.org/library/drmath/view/53387.html
http://www.shout.net/~mathman/html/katiec.html
http://mathforum.org/library/drmath/view/53388.html
http://jwgh.org/miscwritings/math/division.html

 

[Rici]

Notice that

is not equivalent to

. If the second were true, it would have to be true for every x, and, by transitivity of the equals relation, all numbers would be equal.

Well I got close with that .

Sorry guys for being such a donkeyhead. Now I see why, so it's official, you guys are right, I'm wrong, I lost. There I said it.
And all the time I thought I was right .

One more thing:
[quote="Uncle Meat]
Riciardos, infinity isn't a number so you can't divide it by anything.[/quote]
But what I think is strange is that you can multiply by it. Because if you can't divide with it, how can you multiply with it.

But that's just my opinion. I lost so I'm not gonna argue anymore.

 

[bornfidelity.com]

....Not this again......

Wow. I said that you lot were right (well, that Doraki was, and since you a pretty much saying the same thing...). And then I get stuff like this.



EDIT: Ohhhhh wait. I reread my previous post and it said 'not mathematically incorrect' the NOT should be removed. Bad timing for typos.

EDIT EDIT: And besides, no matter how much mathematical proof you'll give me doesn't matter because I'm not saying your proof is wrong, I'm saying mathematics are wrong. And since all that proof is mathematical it means nothing to me.
Simply put, 1 = 0.999~ is not true, because there will always be a difference, a difference that is so small that math can't comprehend the smallness hence we can't word it or write down either. And since math can't comprehend that small difference, there are all these ways of prooving mathematically tat 1 = 0.999~.

Well, joy to the world.

 

[Killer Tree]

Wow. I said that you lot were right (well, that Doraki was, and since you a pretty much saying the same thing...). And then I get stuff like this.



EDIT: Ohhhhh wait. I reread my previous post and it said 'not mathematically incorrect' the NOT should be removed. Bad timing for typos.

EDIT EDIT: And besides, no matter how much mathematical proof you'll give me doesn't matter because I'm not saying your proof is wrong, I'm saying mathematics are wrong. And since all that proof is mathematical it means nothing to me.
Simply put, 1 = 0.999~ is not true, because there will always be a difference, a difference that is so small that math can't comprehend the smallness hence we can't word it or write down either. And since math can't comprehend that small difference, there are all these ways of prooving mathematically tat 1 = 0.999~.

Well, joy to the world.

More like infinity is an abstract idea and you are trying to make it stract? o_O that even a word stract lol

 

[Doraki]

bf.com, your "I'm saying mathematics are wrong" is sig-material.
I think I've never ever seen someone say something that obviously false, you just broke a new record. I think it's dumber than someone saying "I don't exist" (which is unthinkable, according to Kant and Descartes lol)
Congratulations.


LOL @ KillerTree's proofs though they're basically all the same.

Riciardos I'd have liked if you had commented on what I wrote for you somewhere >_>

 

[Rici]

What post?
That about 0.999~0?
Well I was trying to say that:
x=0.999
10x=9.990
9x=9.990-0.999=8.991
x=0.999

Only the 9's in the numbers are infinite, but meh...

 

[bornfidelity.com]

Doraki, math is extracted from the world by humans wereas our existance is not. So no go there
I told you *why* IMO math today isn't correct:

Simply put, 1 = 0.999~ is not true, because there will always be a difference, a difference that is so small that math can't comprehend the smallness hence we can't word it or write down either. And since math can't comprehend that small difference, there are all these ways of prooving mathematically that 1 = 0.999~.

Oh wait I said I had enough of this topic
well its addictive

 

[Office_Shredder]

But what I think is strange is that you can multiply by it. Because if you can't divide with it, how can you multiply with it.

You can't multiply with it....

People just like to think you can. Inifinity in and of itself isn't a number, it's like trying to multiply by apple.

 

[Doraki]

Riciardos,

Do you agree that when you put a 0 at the end of a number's decimals, that number doesn't change ?
Then 0.999~ is also equal to 0.(infinite number of 9s)0.
But then 0.999~ - 0.999~ = (0.(infinite number of 9s)9) - (0.(infinite number of 9s)0)
= 0.(infinite number of 0s)9
though it should be exactly 0 ?

bf your posts always make me cry out of pity. (for your (ok, non-existent) math teacher, not for you, of course)

 

[Uncle Meat]

What math teacher?

 

[Killer Tree]

You can't multiply with it....

People just like to think you can. Inifinity in and of itself isn't a number, it's like trying to multiply by apple.

But you CAN divide it.

42/apple ftw

 

[Killer Tree]

bumpo.........

 

[Mr.GAW]

Ricardos and BF.com are smart.

EDIT: Doraki's smart too.

I hate when people aren't open minded.

 

[Crimson King]

I hate when people aren't open minded.

Isn't that just as closed minded though? I've always wondered. If you call someone closed minded because they don't believe what you believe, that is just as closed minded, in my opinion.

 

[R0D4N]

You can't multiply with it....

People just like to think you can. Inifinity in and of itself isn't a number, it's like trying to multiply by apple.

But apple*apple=more apples. Let me explain.

Apples have seeds, which grow into apple trees. And IIRC it takes two fruit trees (a male and a female) to take the pollen from the male tree to fertilise the female tree so it can produce more apples.

EDIT: Although your right about the multiplication of infiniti.

 

[Mr.GAW]

Isn't that just as closed minded though? I've always wondered. If you call someone closed minded because they don't believe what you believe, that is just as closed minded, in my opinion.

But you wouldn't call someone closed minded just for not believing in what you believe. You call people who stubbornly persist one thing (whether or not it's right is irrelevent) and refuse to explore other possibilitys or hear out other people. People who don't try to understand somebody else viewpoints because they already think (or maybe they know) that they have the right answer.

Example:

People who already know that .9~=1 and don't address/ consider viewpoints of those who have not reached that conclusion yet, or those who have perhaps thought FURTHER than the close-minded person on the subject.

Also, people who INSIST that .9~ does NOT equal 1, and ignore all of the proofs in front of them that clearly state it's true.



Only people who acknowledge that it is generally accepted that .9~=1 and understand the reasoning behind it are can be open-minded on this subject IMO.

These people may also continue to try and prove this false, or yearn to learn more about it. These people can also sit content and not try to disprove or prove anything. The thing is, these open-minded people have to consider the viewpoints of others and perhaps remember when they were going through that thought process. Maybe they understand why somebody thinks something that may be wrong and helps them go down that path until they realize it's wrong.

They have to continue to look at things expecting them to disprove the equation, until they can 100% prove it's wrong, even if they already know they have the right answer.

I thought that those three people were all of that in this thread, and thus I called them "smart" when what I really meant was "open minded".

(Even though they are definetly smart.)

 

[AltF4]

Lol, are people actually debating this? Anything divided by nine is itself repeating.

So yes, it's true. BUT! Be wary of those proofs you see. A lot of them are bad.

From the first post:

A simpler but not shure if it is legal is.
1/3=.3~
1/3 X 3 = 3/3
3/3=1
.3~ X 3 = .9~
.9~ = 1


You can't assume that 1/3 = .3~ in this proof. The property of equality between repeating decimals and fractions is what is being proven. You can't use it as a premise.

This is mathematically correct, only because mathematically there's an end to infinity.
This is in real life incorrect, because in real life infinity is infinite.

Lol, that's almost sig-worthy.

.4242424242... = 42/99
.402402402402402... = 402/999
.000... = 0/9
.111... = 1/9
.222... = 2/9
.333... = 3/9
.444... = 4/9
.555... = 5/9
.666... = 6/9
.777... = 7/9
.888... = 8/9
.999... = 9/9
9/9 = 1
.999... = 1

This one was on page 7. It's pretty demonstrative of the point here, but isn't a proper proof.


The rest of Killer Tree's proofs are good. I think I'm going to put together a funny math thread together if you guys liked this one...

Lol, I just realized this thread had gotten a bump from a year ago.

 

[Jayferd]

It's definitely true. Here's another proof, using the geometric series law that
\sum_{n=0}^\infty ar^n = a/(1-r) whenever -1 < r < 1:

0.9~ = 0.9 + 0.09 + 0.009 + ...
= 0.9*(0.1^0) + 0.9*(0.1^2) + 0.9*(0.1^3) + ...
= \sum_{n=0}^\infty 0.9*(0.1^n)
= 0.9/(1-0.1)
= 0.9/0.9
=1 !

For those who aren't 1337 enough to know TeX (lolz), the funky "\sum_{n=0}^\infty" just means "the summation as n goes from 0 to infinity"

http://en.wikipedia.org/wiki/Geometric_series

Cool stuff.

mmm... fried brain...

 

[GoldShadow]

It's definitely true. Here's another proof, using the geometric series law that
\sum_{n=0}^\infty ar^n = a/(1-r) whenever -1 < r < 1:

0.9~ = 0.9 + 0.09 + 0.009 + ...
= 0.9*(0.1^0) + 0.9*(0.1^2) + 0.9*(0.1^3) + ...
= \sum_{n=0}^\infty 0.9*(0.1^n)
= 0.9/(1-0.1)
= 0.9/0.9
=1 !

Huh... that's pretty cool, I never tried using sigma notation and the sum of an infinite geometric series to prove .9~=1, instead opting for the easier algebraic route

edit: although now that I look at it, this is the easier route.

 

[MVPaintballer]

1/3 > .333~
2/3 > .6666~
3/3 > .9999~

 

[Killer Tree]

1/3 > .333~
2/3 > .6666~
3/3 > .9999~

Why do you think this.

 

[Rici]

Lol, I remember this xD
Now I've reread it all, I see that we were actually saying the same things, only on different ways. I was trying to explain that you can't use infinity in an equation(but I didn't explain it very well) since it's a concept and not a number.

Though, I still think I was right on the point that those equations are wrong and, how surprisingly, 0.999~=1 still is kind of BS. Like Office Shredder said, saying that infinity=1 is like saying 5=apples, it just doesn't make sense.

And now I think about it again, you guys we're saying I was wrong for thinking that infinity ends, but YOU were the ones that put it in an equation in the first place, meaning it's a solid number that can't be changed.

Not trying to argue again, but just wanted to point that out . In the end, everybody's right.

**** I'm stubborn.

 

[GoldShadow]

Riciardos: I have immense respect for you because you really like Scrubs.

However I firmly believe you don't have a strong grasp of such mathematics!

 

[Rici]

I just graduated from high school, so you may be right, but that doesn't mean that I'm wrong. We all agree that infinity is not a solid number, so 0.999~ is actually, not a solid number. That means it can't be treated in such ways, meaning it can't be in an equation.

Really, I know I can be stubborn, but in this case I'm pretty sure of myself.
Please teach me. I do wanna learn, but if it is the way as I see it now, it's kinda... not right for me.

Edit: Although that proof that Jayferd gave has started me thinking.

 

[AltF4]

yes, .9999~ does have a set value. It is in fact a, as you said, "solid number". What is that value? 1

The problem in your thinking is that you're assuming they're two different numbers right from the start because they look different. It has already been proven that they are equal, so I'm going to try and explain qualitatively how they can be equal:

Imagine the number .9, it is obviously less than 1. What about .99? Closer, yes. How about .999? Not quite there, but closer still. What about .9999999999999999999? Still not there. Using this method, you'll never reach 1, because you'd need an infinite number of 9's to reach 1.

But wait! What if you DID have an infinite number of 9's! That's what .9999~ is representing. For any finite number of 9's, it falls short of 1. But with an infinite number of 9's, it finally reaches 1.

This is one of the early concepts you learn in calculus. You learn that an infinite series of numbers can add to a finite number

Just like the series 1/2 + 1/4 + 1/8 + 1/16 + ... adds up to 1. Try doing it on a calculator. The more numbers you add, the closer and closer to 1 you get without ever reaching it. But if you have in infinite sum, you finally reach 1.

 

[Mr.GAW]

Very nice example.

I like your avatar, btw.

 

[Rici]

So, okay, I get that now. As I said, Jayferd's proof has started me thinking and what he said is about the same as what you(AltF4Warrior) explained right?

But, what I wonder is, are those equations allowed as proof(I mean those on the first page and stuff)? Because to me it seemed like there are mathematically wrong as in multiplying and dividing with infinity is kinda wrong, I believe.

 

[Jayferd]

Just like the series 1/2 + 1/4 + 1/8 + 1/16 + ... adds up to 1. Try doing it on a calculator. The more numbers you add, the closer and closer to 1 you get without ever reaching it. But if you have in infinite sum, you finally reach 1.

Very nice. In fact, that's just another special case of the same thing. There's nothing special about base 10 (or base 2, in the case of AltF4's example), except that we might be more used to seeing it. In fact, for any base b > 1, we can represent "0.(b-1)~" (spoken: "Zero point b-1 repeated"), in base b as

(b-1)*b^-1 + (b-1)*b^-2 + (b-1)*b^-3 + ...

just as we did with

0.9~ = 9*10^-1 + 9*10^-2 + 9*10^-3 + ...

We can also write our representation of "Zero point b-1 repeated" in summation notation, as

\sum_{k=0}^\infty (b-1)*b^(-k - 1)

or, equivalently,

\sum_{k=0}^\infty (b-1)/b * (1/b)^k

(look closely at this ^. Nothing has happened, just exponents moving around.)

Now we plug into the geometric series formula a/(1 - r), where a = (b-1)/b, and r = 1/b. Weeelll, we have that 1 - r = 1 - 1/b = (b - 1)/b = a, so that a/(1 - r) = 1!

Hence "Zero point b-1 repeated" always evaluates to 1, whether b is 10, 2, 8, 16, or 645.

[/proof].


Anyways, we mathematicians do welcome the sort of intuitive notion that "0.9~ just ought to be different than 1". It's sort of the same sense that 0 + x just ought to be x, no matter what x is; we use this sort of intuition all the time in constructing the very thing you call "Real numbers".

It's certainly possible to construct a model of "Real numbers" in which 0.9~ is different from 1. The problem is that this one intuitive notion that "0.9~ just ought to be different than 1" is not consistent with our other intuitive notions of addition and multiplication. i.e. such a model would not even closely resemble what we think of as "Real numbers" anyways.

Am I making sense?

edit:

But, what I wonder is, are those equations allowed as proof(I mean those on the first page and stuff)? Because to me it seemed like there are mathematically wrong as in multiplying and dividing with infinity is kinda wrong, I believe.

There has been no multiplying or dividing by infinity, at least in any of the examples I have given, only infinite sums, which constitute limits to infinity (which is the very thing we mean by the "~" in "0.9~"). And multiplying by infinity is perfectly acceptable, so long as you don't try to multiply it by zero.

\infty * x
= \infty, if x > 0 (including x = \infty)
= -\infty, if x < 0 (including x = -\infty)
undefined, if x = 0

x / \infty
= undefined, if x = +\infty or x = -\infty
= 0, otherwise.

In analysis, you start working with something called the "Extended Real Line", which is the real numbers with +\infty and -\infty capped on the ends. You get used to funky rules like that. :D

edit edit: Oh! You mean with the whole "10x = 9.999~" bit? Yes, you can do that, but it's not as obvious as it looks. That's a pretty subtle fact that takes a little proving. You start with "x = 0.999~", which is essentially saying "the limit of a sequence is x", and you end with "10x = 9.999~", which is the same as saying "the limit of 10 times a sequence is 10 times the limit of the sequence, which is 10*x". Kinda weird, subtle, but it works.

edit edit edit: sorry about the TeX tags. \infty = "infinity", \sum = "Big Sigma"

 

[Rici]

Yes, I know there wasn't any multiplying or dividing in your proof, and as you've noticed, I meant the first examples.

Thanks for explaining that all, but there is one thing still bugging me.

I was thinking of this when I saw 0.999~ . I'm taught that 1/x will never reach 0, even if you'll go on infinitively(grammar?). As my teacher said: You can continue to draw this graphic as long as possible, you'll never reach 0, you'll only get closer.

So, my logic was that .999~ will never reach 1, even if you'll go on infinitively, you'll only get closer.

So, are these 2 things 2 whole different subjects or is my logic just a tiny bit justified?