## Is 0.9999... really the same as 1?

What is the basis for reason? And mathematics?

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## Is 0.9999... really the same as 1?

Yes 0.9999... = 1
5
31%
No 0.9999... is slightly less than 1
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44%
Other
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25%

wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

### Re: Is 0.9999... really the same as 1?

Arising_uk wrote: I was interested in the later bits on skepticism and the alternative number systems where it doesn't equal 1.
Oh I see. Well this fits in with something I keep saying. If we defined the symbol .999... to mean 47, then since 47 is different than 1, it must be that .999... ≠ 1. Of course all these alternate systems are inherently interesting, so if this discussion has turned people on to the hyperreals or the p-adics or the surreals, then it's valuable that way.

But my thesis is that none of these alternate number systems bear on the question of whether .999... = 1 or not. It is unquestionably the case that if you give the symbols their standard mathematical interpretation, .999... = 1 just like the knight in chess moves the way it moves. There is no more meaning to one than the other. If someone asked, "But how does the night REALLY move?" or "How do we know the knight really moves that way," the question would be a category error. Chess is a game played according to formal rules and so is math.

All the problems come from trying to imagine .999... = 1 "means' something. It's not a statement about physics, or philosophy, or computer science. It's just a legal position in the game of formal math.

And even if one is not actually a mathematical formalist, when we are doing math, we should think like formalists!

That's my thesis in a nutshell, for what it's worth. So by that logic, even though the alternate number systems are interesting in their own right, they don't actually bear on the question.

Now if I'm missing some philosophical point, I'd like to know. Someone could say, "Oh, it's meaningful in physics because math reveals the truth about physics," well that's simply not true. Math is quite independent of physics.

That's my story and I'm stickin' to it!
Arising_uk wrote:Sorry unclear, I meant I found the discussion between the two of you interesting.
Oh. Well I'm glad. But I'm also sad. If you find this thread of interest I must not have made my point. Because my thesis is that this subject is as pointless as arguing over how the knight "really" moves. In other words not only is .999... = 1 in standard math; and not only is this simply irrefutable; but it is also, and especially, a pointless conversation

Thanks arising_uk for giving me this opportunity to be as clear as I could possibly be about my own opinion of this. But tell me, am I wrong? Am I missing some deeper meaning of all this? Is math required to mean something? Just because the physicists like to use math to build things, why is that math's problem?
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

### Re: Is 0.9999... really the same as 1?

wtf wrote:
Arising_uk wrote: I was interested in the later bits on skepticism and the alternative number systems where it doesn't equal 1.
Oh I see. Well this fits in with something I keep saying. If we defined the symbol .999... to mean 47, then since 47 is different than 1, it must be that .999... ≠ 1. Of course all these alternate systems are inherently interesting, so if this discussion has turned people on to the hyperreals or the p-adics or the surreals, then it's valuable that way.

But my thesis is that none of these alternate number systems bear on the question of whether .999... = 1 or not. It is unquestionably the case that if you give the symbols their standard mathematical interpretation, .999... = 1 just like the knight in chess moves the way it moves. There is no more meaning to one than the other. If someone asked, "But how does the night REALLY move?" or "How do we know the knight really moves that way," the question would be a category error. Chess is a game played according to formal rules and so is math.

All the problems come from trying to imagine .999... = 1 "means' something. It's not a statement about physics, or philosophy, or computer science. It's just a legal position in the game of formal math.

And even if one is not actually a mathematical formalist, when we are doing math, we should think like formalists!

That's my thesis in a nutshell, for what it's worth. So by that logic, even though the alternate number systems are interesting in their own right, they don't actually bear on the question.

Now if I'm missing some philosophical point, I'd like to know. Someone could say, "Oh, it's meaningful in physics because math reveals the truth about physics," well that's simply not true. Math is quite independent of physics.

That's my story and I'm stickin' to it!
Arising_uk wrote:Sorry unclear, I meant I found the discussion between the two of you interesting.
Oh. Well I'm glad. But I'm also sad. If you find this thread of interest I must not have made my point. Because my thesis is that this subject is as pointless as arguing over how the knight "really" moves. In other words not only is .999... = 1 in standard math; and not only is this simply irrefutable; but it is also, and especially, a pointless conversation

Thanks arising_uk for giving me this opportunity to be as clear as I could possibly be about my own opinion of this. But tell me, am I wrong? Am I missing some deeper meaning of all this? Is math required to mean something? Just because the physicists like to use math to build things, why is that math's problem?
Math is whatever assumptions are fed into it. 0.999... means there are an infinite number of 9's people would agree on that point. If you want to assume 0.999... Is "equal" to 1 and also the same as the limit of an infinite sequence then that is your choice though no more valid than calling 0.999... the same as 47. Your arguments take the form of argumentum ad populum and equivocation. When we pinned down the justification for 0.999...= 1 we arrived at a few arbitrarily definitions. I wouldn't assume 0.999... is 1 merely because 0.999... approaches it. Defining something as equal does not change the reality of whether they are equal.
Arising_uk
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### Re: Is 0.9999... really the same as 1?

marsh8472 wrote:... I wouldn't assume 0.999... is 1 merely because 0.999... approaches it. Defining something as equal does not change the reality of whether they are equal.
What's 0.999... in reality?
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

### Re: Is 0.9999... really the same as 1?

Arising_uk wrote:
marsh8472 wrote:... I wouldn't assume 0.999... is 1 merely because 0.999... approaches it. Defining something as equal does not change the reality of whether they are equal.
What's 0.999... in reality?
This Zeno's paradox is the equivalent problem in the binary number system
Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
Which is this series 1/2+1/4+1/8+1/16+.... in binary it's 0.1+0.01+0.001+0.0001+...=0.11111... the distance covered in an infinite series of steps
Arising_uk
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Joined: Wed Oct 17, 2007 2:31 am

### Re: Is 0.9999... really the same as 1?

Not sure of your point? As in reality Homer reaches the end of the path.

"Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
1. Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.
2. Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".
3. Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999…

Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'". ..."
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

### Re: Is 0.9999... really the same as 1?

Arising_uk wrote:Not sure of your point? As in reality Homer reaches the end of the path.

"Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
1. Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.
2. Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".
3. Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999…

Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'". ..."
True in homers case we can go from point A to point B while potentially traveling at an infinite number of points per second. Being able to traverse an infinite number of points at a finite number of points per second is the issue at hand I'd say.

The objections you've listed are valid reasons for rejecting the knowledge claim that 0.999... = 1 without any confusion about it. I haven't seen sufficient proof to justify accepting 0.999... = 1 as a true statement, just narrowly defined definitions that prove 0.999... = 1 with circularity. Intuitively a series of numbers added together on a calculator will never produce a fixed value until the equals button is pressed. Pretending an infinite series of numbers is magically added up by using limits is a pretend answer to the question.
Arising_uk
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### Re: Is 0.9999... really the same as 1?

marsh8472 wrote:True in homers case we can go from point A to point B while potentially traveling at an infinite number of points per second. Being able to traverse an infinite number of points at a finite number of points per second is the issue at hand I'd say. ...
Not sure what the issue is other than that Zeno was demonstrably wrong which points to maths not being really about the world.
The objections you've listed are valid reasons for rejecting the knowledge claim that 0.999... = 1 without any confusion about it. ...
And yet those objections were given as examples of how the maths student is misunderstanding what is going on in the maths?
I haven't seen sufficient proof to justify accepting 0.999... = 1 as a true statement, just narrowly defined definitions that prove 0.999... = 1 with circularity. ...
Not my field but it seems to me that you are misunderstanding proof in mathematics and from the sounds of it what this 'limit' thing is in maths.
Intuitively a series of numbers added together on a calculator will never produce a fixed value until the equals button is pressed. Pretending an infinite series of numbers is magically added up by using limits is a pretend answer to the question.
I think you want all maths to be about the real world but I think it probably isn't. I think you might be able to work this way with whole numbers but these fractional decimals don't exist(but don't quote me on this as like I say, not my field).
osgart
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### Re: Is 0.9999... really the same as 1?

how many .9's can you have before 1 absolutely had to click in.
osgart
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### Re: Is 0.9999... really the same as 1?

math manipulates reality. it doesn't explain it
surreptitious57
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Joined: Fri Oct 25, 2013 6:09 am

### Re: Is 0.9999... really the same as 1?

One needs an infinity of . 9 before I actually clicks in

Maths does explain reality as the laws of physics are written in mathematical form
Arising_uk
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Joined: Wed Oct 17, 2007 2:31 am

### Re: Is 0.9999... really the same as 1?

surreptitious57 wrote:One needs an infinity of . 9 before I actually clicks in ...
Apparently not according to the mathematicans.
Maths does explain reality as the laws of physics are written in mathematical form.
Modelling is not necessarily the same as the thing.
surreptitious57
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### Re: Is 0.9999... really the same as 1?

Were it not an infinity of . 9 then there would be at least another number between that and I and that would falsify the proposition
Eodnhoj7
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### Re: Is 0.9999... really the same as 1?

marsh8472 wrote: Fri Mar 03, 2017 6:24 am I've gone back and forth on this issue and am curious what the general public thinks about the answer given my reservations about it.

Firstly there's something called the Archimedean property or Archimedean axiom which states that there's no such thing as infinity or an infinitesimal real numbers. In that sense there is no difference between 1 and 0.99999... If we take 1-0.99999... = 0.0000....1 but since the 0's never end and there's no such thing as infinitesimal real numbers this is the same as 0.

Other proofs exist such as 1/3 + 2/3 = 3/3 or 0.33333... + 0.66666... = 0.99999... = 3/3 = 1

Let x = 0.9999999...
then 10x = 9.9999999...

Then subtract the x from 10x:

10x = 9.9999999...
x = 0.9999999...
=================
9x = 9
x = 1

however, I've come to notice the "=" sign is being used in a slightly different context than something like 1+1=2. It's behaving more like the equal sign in a limit

Like the example lim (x->infinity) 1 - 10^(-x) = 1
We can see that lim (x->infinity) 1 - 10^(-x) approaches 1 here:
f(1) = 0.9
f(2) = 0.99
f(3) = 0.999
etc...
f(x) may approach 1 as x increases but still will never reach 1. But the "=" sign in this context means "approaches" and not "of the same value", or is it?

Considering that 1 in its unity is represent through the curvature of the point, .99999999 exists only if it is rounded.

What we observe is fundamentally a duality where 1 is a stable entity and .9999999 is an entity continually "rounding" itself through perpetual flux. 1 and .999999 are equal in the respect they are symmetrical duals.

1 is the stable axel upon which .999999 as a gradation of 1 "spins". This "spin" in turn manifests through the stability of the axel. The question breaks down to the observation of cyclical duality of stability and flux, symmetry (as unity and stability) and assymmetry (as continual particulation and flux).

1 and .99999 are fundamentally approximates of eachother and in this respect maintain a unity.