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Forums - General - 0.9999.... = 1.0

 

Are you convinced?

Yes 34 58.62%
 
No 20 34.48%
 
not sure 1 1.72%
 
Total:55
Jay520 on 30 April 2013
MDMAlliance said:
Soleron said:
MDMAlliance said:
...

Either way, 0.999... is not the exact same thing as 1 due to the fact that it is created by something of infinite "additions."  1 is 1, 0.999... is not.  It's the same logic as there's no "biggest number" so therefore infinity isn't the same thing as the "biggest number" because it doesn't exist.

It's not created by infinite additions. It's a fixed object with certain properties. Everything we do to test those properties (addition, multiplication, comparison) shows it is indistinguishable from 1. And the best way of thinking about it is that the number '1' has two valid decimal representations, being 1 and 0.999...


How is 0.999... a fixed object?  Is there some way you're getting 0.999... that I do not know of?  

Are we talking about two different 0.999...'s because I don't think it would be a fixed object.

If we talk about 1/9 * 9 or 1/3 * 3 (or whatever else way people use to get 0.999 and 1 being the same thing), to some extent I agree that those two numbers are representing the same thing.  I do not agree, however, that 0.999... as an infinity is 1. 



It's fixed because it is the sum of an infinite convergent series, which I've already told you equals a fixed point. It doesn't approach 1. It only approaches 1 if you count a finite number of the terms .9 + .09 + .009... But when you count an infinite number of 9s, and find their sums, (which is what .9999 represents) then you actually reach the number 1.

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drkohler on 30 April 2013
Netyaroze said:

 c is not c in real life because its impossible to reach.

Are you suggesting photons only travel at c in math, but not in real life ? I'm somewhat puzzled by your c argument....



Zkuq on 30 April 2013
drkohler said:
Netyaroze said:

 c is not c in real life because its impossible to reach.

Are you suggesting photons only travel at c in math, but not in real life ? I'm somewhat puzzled by your c argument....

He was talking about objects with mass. Photons are widely considered massless.



MDMAlliance on 30 April 2013
Jay520 said:
MDMAlliance said:
Soleron said:
MDMAlliance said:
...

Either way, 0.999... is not the exact same thing as 1 due to the fact that it is created by something of infinite "additions."  1 is 1, 0.999... is not.  It's the same logic as there's no "biggest number" so therefore infinity isn't the same thing as the "biggest number" because it doesn't exist.

It's not created by infinite additions. It's a fixed object with certain properties. Everything we do to test those properties (addition, multiplication, comparison) shows it is indistinguishable from 1. And the best way of thinking about it is that the number '1' has two valid decimal representations, being 1 and 0.999...


How is 0.999... a fixed object?  Is there some way you're getting 0.999... that I do not know of?  

Are we talking about two different 0.999...'s because I don't think it would be a fixed object.

If we talk about 1/9 * 9 or 1/3 * 3 (or whatever else way people use to get 0.999 and 1 being the same thing), to some extent I agree that those two numbers are representing the same thing.  I do not agree, however, that 0.999... as an infinity is 1. 


It's fixed because it is the sum of an infinite convergent series, which I've already told you equals a fixed point. It doesn't approach 1. It only approaches 1 if you count a finite number of the terms .9 + .09 + .009... But when you count an infinite number of 9s, and find their sums, (which is what .9999 represents) then you actually reach the number 1.

Actually, not really.  You never "reach" it.  That is part of the problem.  

Even with the theories that say 0.999... is 1, your sums never go to the point where it's like, "now I add this last part to finally get to X."  

These are all based on theories that people accept by faith.  I am arguing more on a semantic level, but also on a mathematical level.  For the most part, I am really trying to figure out as much as I can about why people say 0.999... equals 1, but to no avail since no one is really making an argument that details why.  All I see are flawed systems to say why 0.999... equals 1.  Why not someone argue to me, explicitly why these theories are correct and why they actually make 0.999... equal 1?  

Really, the hole I see is that if it eventually did hit one, there should also be an objective point where the switch happened.  I can also see why 0.999... is looked at "since it goes on forever, and there cannot be a number between 0.999... and 1, 0.999... has to be 1."  I understand that.  That doesn't change the fact that it isn't a fixed object in my perspective.  Infinity would have to be a fixed object, too.  



Zkuq on 30 April 2013
MDMAlliance said:

Really, the hole I see is that if it eventually did hit one, there should also be an objective point where the switch happened.  I can also see why 0.999... is looked at "since it goes on forever, and there cannot be a number between 0.999... and 1, 0.999... has to be 1."  I understand that.  That doesn't change the fact that it isn't a fixed object in my perspective.  Infinity would have to be a fixed object, too.  

Do you think 0.999... is a number at all? If you think it is, then it is definitely a fixed object. There's also a difference between an infinite number of periods and infinity. Also, what do you have against the proof using the sum of a geometric series? Can you tell where it goes wrong? If it goes wrong somewhere, you can point out the exact spot where the error is and you can do it without geometric intuition.



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MDMAlliance on 30 April 2013
Zkuq said:
MDMAlliance said:

Really, the hole I see is that if it eventually did hit one, there should also be an objective point where the switch happened.  I can also see why 0.999... is looked at "since it goes on forever, and there cannot be a number between 0.999... and 1, 0.999... has to be 1."  I understand that.  That doesn't change the fact that it isn't a fixed object in my perspective.  Infinity would have to be a fixed object, too.  

Do you think 0.999... is a number at all? Nope. If you think it is, then it is definitely a fixed object. There's also a difference between an infinite number of periods and infinity.  I do acknowledge their differences, but there are many ways where they function the same way.  Also, what do you have against the proof using the sum of a geometric series? Can you tell where it goes wrong? If it goes wrong somewhere, you can point out the exact spot where the error is and you can do it without geometric intuition.  I simply do not think that the equation is perfect, especially considering what its limitations are.  I have not taken math in over 3 years so I would really need to study this stuff and come back to it later in order to produce any realistic conclusions.





Jay520 on 30 April 2013
MDMAlliance said:

Actually, not really.  You never "reach" it.  That is part of the problem.  

Even with the theories that say 0.999... is 1, your sums never go to the point where it's like, "now I add this last part to finally get to X."  

These are all based on theories that people accept by faith.  I am arguing more on a semantic level, but also on a mathematical level.  For the most part, I am really trying to figure out as much as I can about why people say 0.999... equals 1, but to no avail since no one is really making an argument that details why.  All I see are flawed systems to say why 0.999... equals 1.  Why not someone argue to me, explicitly why these theories are correct and why they actually make 0.999... equal 1?  

Really, the hole I see is that if it eventually did hit one, there should also be an objective point where the switch happened.  I can also see why 0.999... is looked at "since it goes on forever, and there cannot be a number between 0.999... and 1, 0.999... has to be 1."  I understand that.  That doesn't change the fact that it isn't a fixed object in my perspective.  Infinity would have to be a fixed object, too.  


Yeah, it looks like you're arguing about semantics. When people say it is equal to 1, they don't mean it transfroms from a decimal into 1. It is still expressed as 0.999... It just so happens that 0.999... and 1 are of the same value. It doesn't "hit" 1 in the sense I think you're describing, but it is equal to 1 in terms of its value. The point is there is no difference between 1 & 0.999.... under any circumstances.  

The fact that you understand that there's no number between 1 & 0.999.... should be enough reason to believe they are of the same value. Whether or not you feel comfortable calling them the "same number" is really all semantics.

BTW, the sums do get to a point where it equals X. If you have a convergent geometric series, and take the sum of a finite number of figures, then yes you will only approach the limit. But with 0.999... we aren't adding a finite number of figures. We are adding an infinite number of figures (which isn't an objective point as you implied it was), in which case, the the sum will equal the "limit". This is accepted in mathematics.

I feel like I'm starting to repeat myself so I just post a link of this post which I think is more convincing, albeit quite long. Give it a read later if you want.



Netyaroze on 30 April 2013
drkohler said:
Netyaroze said:

 c is not c in real life because its impossible to reach.

Are you suggesting photons only travel at c in math, but not in real life ? I'm somewhat puzzled by your c argument....


Zkuq is right. I was talking about objects with mass. 



drkohler on 30 April 2013
MDMAlliance said:

Actually, not really.  You never "reach" it.  That is part of the problem.  

Even with the theories that say 0.999... is 1, your sums never go to the point where it's like, "now I add this last part to finally get to X."  

These are all based on theories that people accept by faith.

holy crap..... there is absolutely no faith involved here. This is solid math - you simply seem not to understand the concept of this particular math.

0.(period)9 is a number that has an infinite number of 9s. There is no "last part to finally get somewhere". There is no "let's add a few more 9s so it is even closer to 1", there is no "we never reach it becaiuse I can't write enough 9s"

0.(period)9 is a (rather stupid) way of writing 1, you could also write (4 + 6*3) / (7 + 3*5) which is also a stupid way of writing 1.



MDMAlliance on 30 April 2013
drkohler said:
MDMAlliance said:

Actually, not really.  You never "reach" it.  That is part of the problem.  

Even with the theories that say 0.999... is 1, your sums never go to the point where it's like, "now I add this last part to finally get to X."  

These are all based on theories that people accept by faith.

holy crap..... there is absolutely no faith involved here. This is solid math - you simply seem not to understand the concept of this particular math.

0.(period)9 is a number that has an infinite number of 9s. There is no "last part to finally get somewhere". There is no "let's add a few more 9s so it is even closer to 1", there is no "we never reach it becaiuse I can't write enough 9s"

0.(period)9 is a (rather stupid) way of writing 1, you could also write (4 + 6*3) / (7 + 3*5) which is also a stupid way of writing 1.


What I was saying was about the geometric series, not 0.999...

When I say "theories," I was referring to the Axiom of Choice theorem.