To extend what I meant about infinitesimals, just reading back through the thread again has a good post by Indistinguishable thoroughly describing what I just noted above.
Really, most of the objections coming up were dealt with back in 2000 and the zombie revivification in 2012 (which itself is a rehash of other threads on the same topic). There’s nothing new or particularly insightful being added.
This has already been beat to death upthread, but I get a sneaking suspicion that you haven’t bothered to read the earlier posts, so I’ll summarize it for you.
If two real numbers are NOT equal, then you can find a number that lies between them. So, if
1 ≠ 0.99999…
Then can you name a number that is less than one, but greater than 0.99999…?
It seems that you would like to resort to various tricks to prove me wrong and
ignore what I already said. So I have to repeat what I said:
70.142857 = 0.999999
70.142857142857 = 0.999999999999
70.142857142857142857= 0.999999999999999999
.
.
.
70.142857142857142857142…= 0.99999999999999999999999…
You must notice and understand that no matter what you do, you cannot force the number 0.999999999999…magically change equal to 1.
Although according to your various “proofs” 1/7 = 0.142857142857… and also
71/7=70.142857142857…=1
What part of the proof do you not accept?
Do you not accept that 1/3=.333333333…?
Do you not accept that 3*1/3 = 1?
Do you not accept that 3*.333…= .9999… ?
or do you not accept that if 1/33=1 and 1/33=.999… then must 1=.99999… (by the transitive property of equality Equality (mathematics) - Wikipedia)?
7777777, are you familiar with the idea of modular arithmetic?
For example, mod 10 arithmetic. This is where we work with whole numbers, but decide two numbers count as “=” so long as their difference is a multiple of 10 (for numbers of the same sign, this amounts to them having the same last digit). So, for example, we’ll say 27 = 7, and thus 3 * 9 = 7, in mod 10 arithmetic.
You might object “3 * 9 isn’t 7! 3 * 9 is 27. 27 and 7 are totally different!”. And that’s true… as integers, 27 and 7 are very different.
But, “=” is just a symbol and we can use it however we want. We might choose to use it as meaning mod 10 equivalence rather than integer equivalence. And on that definition, 3 * 9 = 27 = 7.
Put another way, we might say we are working with a different notion of number than integers; we are working with mod10numbers, which are a lot like integers, but coarser-grained. As integers, 27 and 7 are different; as mod10numbers, they are the same.
There are lots of ways to think about what’s going on when we do modular arithmetic. You can think about it however you like. But, are you comfortable with that sort of thing?
Great, because I’m going to help you understand the language everyone else is using now!
You have a notion of number in mind under which 0.999… and 1 are distinct numbers, separated by an infinitesimal difference. There’s nothing wrong with this notion of number. And you can go ahead and say 0.999… is not = to 1 as a straight-up number.
But for whatever stupid reason, everyone around you is using “a = b” to mean “the difference between a and b is an infinitesimal”. Everyone around you doesn’t like to talk about straight-up numbers, but only likes to talk about mod-infinitesimals-bers.
And you wouldn’t deny that 0.999… = 1 when “=” is taken to mean “are equivalent up to an infinitesimal difference”. That’s a true claim, just as “27 = 7 (in mod 10 land)” is a true claim.
Similarly, you would likely agree that 1/7 is equivalent to 0.142857142857… up to an infinitesimal difference, and so on.
So to understand what everyone else is saying, just translate it into your head by understanding that they aren’t working in “true arithmetic”, but only in “mod infinitesimals arithmetic”. And this is an ok thing to think and talk about, just like mod 10 arithmetic is an ok thing to think and talk about.
It’s a pity they aren’t so interested in the finer distinctions between numbers which you are capable of appreciating in “true arithmetic” (really!), but they aren’t completely misguided. They’re just using words a little differently than you do.
Does any of that make sense?
This I don’t accept.
It is the same equation as 1=0.999999999…divided by 3
1/3*3 = 1 this is true
if 1/33 = 0.999999999…then must, as you said, 1=0.9999999999…
but 1/33 ≠ 0.9999999999…so 1≠0.999999999…
How about this: Do you accept that the sum from i=1 to infinity of 1/3^i is equal to 1/3 ?
Thank you. What does 1/3 equal?
I think you mean 3/(10^i)? Otherwise, this doesn’t make a lot of sense.
Again, can you show your work?
1/3 = 0.3333…
3*0.333… = 0.9999…
It works out fine. If we use an alternate, nonstandard real number system where infinitesimals exist, then 1/3 is also not 0.333… but infinitesimally different.
As mentioned repeatedly, there’s nothing inherently wrong with that, but it doesn’t really say anything about the standard real numbers most people like to talk about.
But I get the feeling you want to have infinitesimals AND still be operating under the standard real numbers. Unfortunately, it’s just not possible to have your cake and eat it too in this case.
Yes, in fact I do. Bad Leahcim!
However, it does. So it does.
I am not familiar with modular arithmetic. I have head about it sometimes, but it
did not sound useful for me. Perhaps it is not necessary to use it here.
I am not certain what do you mean. Are you suggesting perhaps that we
should try to make a compromise, that we are all somehow right if we just learn to use the same language?
Could this correspond to the situation that 1≥0.999999999999…
I used to think this way a long time ago.
No-one accepted this idea, so I decided to abandon it for a while. Then I began
working on the proof that 1≠0.9999999999…
I got inspiration from reading James Tanton’s paper
On this document Mr.Tanton is writing about how …999999=-1 and
I found how to use it in my proof.
This imo is the fundamental problem in your understanding/reasoning. There ARE NO 9s at the end of 0.99999999… because there is NO END of 0.99999999…
If there were an end, I could add say a 5 after the last 9 and get an elusive number between 0.99999999999… and 1.0. But I can’t.
It’s not a bad way looking at things, but, again, things depend on how we define them.
The problem is using a different definition from everybody else but expecting things to still work the same anyway. And that’s what’s happening here.
How exactly do you define …999? That’s something that doesn’t really exist in standard real analysis. So, clearly a nonstandard system is being applied. But if you have a nonstandard system, it doesn’t say anything about the standard system.
So, great, under some nonstandard system you like to use, 1 is not 0.9999…
But that still doesn’t apply in a standard system where infinitesimals and such aren’t defined, much less equal to the result of a subtraction. And where …9999 isn’t even defined.
If it makes you happy to operate under that nonstandard system and to play with concepts in it, great. That’s fine. That’s sort of how non-Eucliean geometry came about. But it’s not right to apply the results you get from that system to a different one (just as results from Euclidean geometry don’t necessarily apply to hyperbolic geometry). You have to re-establish results from scratch again. In the case of our standard, humdrum, everyday real number line, 1 is 0.9999…, pretty much by definition.
What gives you that feeling?
I think most posters taking 0.999… to be infinitesimally smaller than 1 have never been taught about the “real numbers” in any formal detail, and have also never really given much consideration to the idea that number systems could be formalized in various different ways. So they assume whatever preformal intuition they have for what numbers amount to (developed not necessarily unreasonably from their schoolchild experience with decimal calculations) describes the One True System of numbers; that there could be multiple systems of numbers, so that a numerical question has not one fixed answer, but only an answer relative to a particular system, never crosses their mind (again, this is just what we would expect, given the way they are taught in the first place).
So I don’t think there’s any desire to “still be operating under the standard real numbers”; I don’t think there’s even the conceptual language present to express such a desire.
There may also be some confusion caused by the term “real numbers”; if one does not recognize this as merely jargon, it sounds like one is claiming those to be the One True System of numbers, so that whatever it is that one perceives to be the most natural account of numbers must be the one that people intend to denote by the “real numbers”.
Perhaps, then, this is all you meant: it sounds like floor(70000000/9) wants all the rest of us to confront our mistake in taking our arithmetic system to be the “real numbers”, not understanding that this is a jargon term which needn’t impugn their own alternative arithmetic framework.
What are these ‘real number’ things you folks are talking about? I don’t believe they exist! I mean, have you ever seen a ‘.9’ in nature? Not hardly! And a .99 and a .999 and so forth are each even more ridiculous in turn! So the idea that .999999999… even exists is so far-fetched that it’s silly to even talk about whether it is equal to 1 or not! Both sides in this debate are wrong!
[j/k]
That is a good question.
Can I say that if 1/3=0.3333333333…then 1/3 equals to 0.333333…
But if 1/3≠0.333333333…then 1/3 does not equal to 0.333333…
Look, I am not trying to insist that we should accept any particular possibility
as a priori truth. Not at least without a proof. I am only interested in proving
things, either true or false.
Let’s accept what James Tanton writes that
…999999999 = -1
so that
…999999999 + 1 = 0 because -1 + 1 = 0
now if 1 = 0.999999999…
and insert 0.999999999…into Tanton’s equation we will get
…999999999 + 0.999999999… = …999999999.999999999… ≠ 0
In other words, we will get different result of the Tanton’s equation using either
1 or 0.999999999…If these numbers were the same, then we should
get the same result from Tanton’s equation.
You have not shown this, btw.
You are right, I have not shown that, yet.
I know that Tanton writes on his document that
…999999999.999999999…=…999999999 + 0.999999999…=
-1 + 1 = 0
He is assuming that 1=0.999999999…
He also writes that …99999999 + 1 = …00000000
meaning that …99999999 + 1.000000… = …00000000.000000000…
I think that
…9999999999.99999999999…
…0000000000.00000000000…
are not the same because every digit is different