## 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
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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?

marsh8472 wrote: I've understood everything you've said. I think he's just saying it's interesting. Within real numbers and with limits 0.9999... = 1. But on another level is 0.9999...=1 true too? Using criteria like these https://en.wikipedia.org/wiki/Criteria_of_truth
I'm thinking Greta is a she but of course handles are arbitrary self-selected labels.

I have actually been thinking that this is the best .999... convo I've ever been in. They usually devolve terribly.

I hope I misunderstood that post in fact.

Anyway you are two or three posts ahead of me at this point but you've said some very interesting things I'd like to respond to in the next day or so. One thing I did want to say to you is that when you said you're an applied mathematician, I cracked up laughing. You are exactly the person I'm talking about I hope you understand my sense of humor about that. You raised several specific mathematical points that I'll try my best to address. There's nothing circular about the definition of the real numbers and the definition of an infinite sum. In fact they do that in calculus class. The sum of an infinite series is the limit of the sequence of partial sums. That's actually a clever trick. There really is no such thing as an infinite sum 9/10 + 9/100 + 9/1000 + ... Rather, that notation is defined as the limit of the SEQUENCE 9/10, 99/100, 999/1000, ... and you yourself provided the definition that the limit is 1.

But this is another good example of how in math we finesse things but forget to explain it to the general public. There is no such thing as an infinite sum of real numbers. The sum itself is simply defined as the limit of the sequence of partial sums. And sequences are kind of more believable than series, don't you think? People wonder, What does it mean to add up infinitely many real numbers? And the answer in math is that we never actually do that. Instead we redefine the sum as a sequence, and we define limits of sequence in terms of epsilon and N and a couple of quantifiers. We never actually added infinitely many real numbers at all! Does that make sense?

I do agree I might be a little obsessed with this topic. About two years ago I promised myself never to engage in another .999... thread online. A year ago I broke my rule, spent a few weeks arguing with a crank, and regretted every minute of it.

This is the best .999... thread I've ever been in. I'm getting to articulate a lot of things that are on my mind about this subject. I can see how it all seems a little over the top.
surreptitious57
Posts: 4225
Joined: Fri Oct 25, 2013 6:09 am

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

http://www.ilovephilosophy.com/viewtopic.php?t=190558

Only you know whether this is actually the best . 999 thread you have ever been in but the one from over
at I Love Philosophy from last year is arguably better and at 20 pages and 500 posts it is also much longer
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

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

wtf wrote:
marsh8472 wrote: I've understood everything you've said. I think he's just saying it's interesting. Within real numbers and with limits 0.9999... = 1. But on another level is 0.9999...=1 true too? Using criteria like these https://en.wikipedia.org/wiki/Criteria_of_truth
I'm thinking Greta is a she but of course handles are arbitrary self-selected labels.

I have actually been thinking that this is the best .999... convo I've ever been in. They usually devolve terribly.

I hope I misunderstood that post in fact.

Anyway you are two or three posts ahead of me at this point but you've said some very interesting things I'd like to respond to in the next day or so. One thing I did want to say to you is that when you said you're an applied mathematician, I cracked up laughing. You are exactly the person I'm talking about I hope you understand my sense of humor about that. You raised several specific mathematical points that I'll try my best to address. There's nothing circular about the definition of the real numbers and the definition of an infinite sum. In fact they do that in calculus class. The sum of an infinite series is the limit of the sequence of partial sums. That's actually a clever trick. There really is no such thing as an infinite sum 9/10 + 9/100 + 9/1000 + ... Rather, that notation is defined as the limit of the SEQUENCE 9/10, 99/100, 999/1000, ... and you yourself provided the definition that the limit is 1.

But this is another good example of how in math we finesse things but forget to explain it to the general public. There is no such thing as an infinite sum of real numbers. The sum itself is simply defined as the limit of the sequence of partial sums. And sequences are kind of more believable than series, don't you think? People wonder, What does it mean to add up infinitely many real numbers? And the answer in math is that we never actually do that. Instead we redefine the sum as a sequence, and we define limits of sequence in terms of epsilon and N and a couple of quantifiers. We never actually added infinitely many real numbers at all! Does that make sense?

I do agree I might be a little obsessed with this topic. About two years ago I promised myself never to engage in another .999... thread online. A year ago I broke my rule, spent a few weeks arguing with a crank, and regretted every minute of it.

This is the best .999... thread I've ever been in. I'm getting to articulate a lot of things that are on my mind about this subject. I can see how it all seems a little over the top.
Yes that all makes sense. Formal decimal notation of real numbers represents a limit of a sequence. Should "represents" be included in our definition of "="? Is it appropriate to say 0.99999... = a decimal notation of a real number represented as a limit of a sequence? I'm sure I'll be butting heads with the law of identity sooner or later if I haven't been butting heads with it already. But 0.9999... is not very formal either, people don't usually write that. If someone uses 0.9999... to represent the final element of an infinite sequence 0.9, 0.99, 0.999, ... that would not be a real number since no final element exists per archimedean property. But if 0.9999... were represented that way, as a non-real number, would 0.9999... < 1 in that case?
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

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

I think I came up with a possible way to prove 0.9999... < 1 with limits in the real number system. Let the set Compare = { LessThan, Equals, GreaterThan }

CompareReals( x, y) is a function with a domain of x and y in the set of real numbers and range in the set Compare.

CompareReals(x, y) = LessThan if x<y, Equals if x=y, and GreaterThan if x>y.

Let ninebar(n) be the sequence 0.9, 0.99, 0.999, etc... where ninebar(1) = 0.9, ninebar(2)=0.99, etc..

Let the limit as n approaches infinity of ninebar(n) = 0.9999...

or in otherwords

Let 0.99999... = the limit of the sequence 0.9, 0.99, 0.999, etc. .

Limit of CompareReals(ninebar(x), 1) as x approaches infinity = LessThan

Let "<" mean LessThan

Therefore 0.99999... < 1
wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

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

marsh8472 wrote: If someone uses 0.9999... to represent the final element of an infinite sequence 0.9, 0.99, 0.999, ... that would not be a real number since no final element exist
Right. There is no last element. A sequence is a function whose domain is the natural numbers 1, 2, 3, 4, ... Just as there is no last natural number, there is no "final" element of a sequence. You're about four posts ahead of me now
wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

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

surreptitious57 wrote:http://www.ilovephilosophy.com/viewtopic.php?t=190558

Only you know whether this is actually the best . 999 thread you have ever been in but the one from over
at I Love Philosophy from last year is arguably better and at 20 pages and 500 posts it is also much longer
I'm wtf over there too and that is the worst .999... thread of all time. The crank I referred to is the guy with 23,000 posts on that site, averaging 7 posts a day every day of the year for something like 8 years straight. He's completely full of shit I still feel dirty from trying to talk sense into him. I never went back to that site after that. I'm lost as to the virtues of that thread. Although I will say that was the thread that caused me to dig into the hyperreals and find out the mathematical truth about them.
wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

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

marsh8472 wrote:I think I came up with a possible way to prove 0.9999... < 1 with limits in the real number system.
Why are you devolving into this kind of discourse? If we're discussing gravity and I say, "I've found a way to prove that things fall up," you don't need to read further. Right? What's the metric on your compare set? What do you mean by limit? You started out insightful but you're veering into crankery. Is it something I said?
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

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

wtf wrote:
marsh8472 wrote:I think I came up with a possible way to prove 0.9999... < 1 with limits in the real number system.
Why are you devolving into this kind of discourse? If we're discussing gravity and I say, "I've found a way to prove that things fall up," you don't need to read further. Right? What's the metric on your compare set? What do you mean by limit? You started out insightful but you're veering into crankery. Is it something I said?
Not at all, the proof that can be challenged. I might have messed something up on one of the lines. In an earlier posts you used the definition of x<y to be if x-y have a difference of some [real] number greater than 0. I'm pretty sure < does not have a formal definition, that's the problem. The metrix I'm using is the function called CompareReals(x, y) which compares finite numbers without limits but still comparing real numbers with <, >, and = operations.

0.9<1
0.09<1
0.009<1
0.0009<1
etc..
Even if we grant that 0.99999... = 1 with the limit approach where we could be swapping the least upper bound with the actual value of 0.9999... in order to present the illusion that they are equal. This proof uses a similar technique except with the comparison operation:

0.9<1, 0.99<1, 0.999<1, ... 0.9999... = 1 may be true with the least upper bound comparison technique
but the limit of "<", "<", "<", "<", "<", ... "=" is "<" because "=" will never be reached for the same reason 0.0000...1 is considered 0 as a real number. 0.9999... and 1 may not differ by a real amount but we should be able to show a difference exists with this Compare set. The CompareReals should be compatible with real numbers too.

Here's the claim:
For all real numbers (CompareReals(x,y) = "LessThan" and x<y) or (CompareReals(x,y) = "Equals" and x=y) or (CompareReals(x,y) = "GreaterThan" and x>y)

To disprove this I would expect the only counter-examples that would exist are ones like appealing to 0.9999... = 1 but that's what we're trying to disprove. What's wrong with this?
wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

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

marsh8472 wrote:...for the same reason 0.0000...1 is considered 0 as a real number.
There is no such number. What on earth could it mean? Didn't I explain that a sequence is a function whose domain is the natural numbers? Why are you going down this rabbit hole? Please, get out your old real analysis book and feel free to ask me any questions about what you find in there that's not clear.

"I'm pretty sure < does not have a formal definition, that's the problem. "

Look it up, it most certainly does.
surreptitious57
Posts: 4225
Joined: Fri Oct 25, 2013 6:09 am

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

There is no real number that exists between I and . 999 ... which means that they are the same. However were there a real
number difference no matter how small it would be proof that I and . 999 ... were different numbers. The set of all the real
numbers is an infinite set so for there to be no other number between them is another proof that they are actually the same
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

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

wtf wrote:
marsh8472 wrote:...for the same reason 0.0000...1 is considered 0 as a real number.
There is no such number. What on earth could it mean? Didn't I explain that a sequence is a function whose domain is the natural numbers? Why are you going down this rabbit hole? Please, get out your old real analysis book and feel free to ask me any questions about what you find in there that's not clear.

"I'm pretty sure < does not have a formal definition, that's the problem. "

Look it up, it most certainly does.
I'm using more abstract math for defining a custom set.

Where is the definition of "<"? I couldn't find one for the equal sign. In language "less than" is a semantic primitive. Which means there is no way to define it without referring to some definition of itself. I suspect the same problem exists in math.

https://en.wikipedia.org/wiki/Real_numb ... c_approach
Definition

Main article: Construction of the real numbers

The real number system ( R ; + ; ⋅ ; < ) can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.
wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

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

marsh8472 wrote: Where is the definition of "<"? I couldn't find one for the equal sign.
Equality is defined by the axiom of extensionality. That has a Wiki page.

When you define the natural numbers via the successor operation, you can define < via succession. You use that to define < on the rationals, and from that the reals. It's in your real analysis book. The construction of the rationals from the integers is done in abstract algebra, and the construction of the integers from set theory is typically done in set theory class. I don't understand why you're not saying, "Oh, I see, these things are clarified in classes I didn't take," rather than trying to prove established math wrong.

You're correct that < doesn't seem to have an explanation on Wiki. That's not definitive proof there's no definition in math, you understand that right?
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

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

wtf wrote:
marsh8472 wrote:...for the same reason 0.0000...1 is considered 0 as a real number.
There is no such number. What on earth could it mean? Didn't I explain that a sequence is a function whose domain is the natural numbers? Why are you going down this rabbit hole? Please, get out your old real analysis book and feel free to ask me any questions about what you find in there that's not clear.

"I'm pretty sure < does not have a formal definition, that's the problem. "

Look it up, it most certainly does.
I was trying to use that to explain. 0.0000...1 can be like 1-0.9999... or 1 - 0.9 - 0.99 - 0.999 ...
wtf
Posts: 981
Joined: Tue Sep 08, 2015 11:36 pm

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

marsh8472 wrote:
I was trying to use that to explain. 0.0000...1 can be like 1-0.9999... or 1 - 0.9 - 0.99 - 0.999 ...
There is no such number and no such expression in math. If a decimal expression (which is just a notation for a particular infinite series) is a function whose domain is the natural numbers, when what could the "last" digit possibly be?
marsh8472
Posts: 54
Joined: Sun Oct 19, 2014 3:06 pm

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

wtf wrote:
marsh8472 wrote: Where is the definition of "<"? I couldn't find one for the equal sign.
Equality is defined by the axiom of extensionality. That has a Wiki page.

When you define the natural numbers via the successor operation, you can define < via succession. You use that to define < on the rationals, and from that the reals. It's in your real analysis book. The construction of the rationals from the integers is done in abstract algebra, and the construction of the integers from set theory is typically done in set theory class. I don't understand why you're not saying, "Oh, I see, these things are clarified in classes I didn't take," rather than trying to prove established math wrong.

You're correct that < doesn't seem to have an explanation on Wiki. That's not definitive proof there's no definition in math, you understand that right?
But you defined x<y as when x is smaller than y by some number in order to prove 0.9999... = 1. Why can't a different definition be constructed to prove 0.9999 < 1. Is that not just as valid as what you did to prove 0.9999... = 1?