The is somewhat correct, except one could quibble that “a numeral” is better used, when discussing numeral systems, to mean a single digit. It’s just about language though. And yes, in the decimal system, and all positional systems with a rational base, pi cannot be represented fully with the limited set of symbols we usually restrict ourselves to when saying a mathematical construction consists a decimal representation of the number.
Mathematicians however do say that 0.333… represents a decimal representation of the number “one third”, and by extension, using only the tools allowed for in mathematics, 0.999… then is a quirky representation of the number “one”.
All your quibbles at this point boil down either to your disagreement with how the rest of the mathematical word has defined words such as “number”, “is” and “equals”, or your disagreement with how modern mathematics deals with infinities.
The notation for recurring decimals has three periods, and usually only uses three digits when all the digits are the same. like this 0.333… The notation is also defined to mean the limit of the infinite series.
I’d go on and, like others before me in this thread, try to find out how your brain juggles infinities differently when they’re given in an expression of a limit and when they’re in the form of an implied never ending row of digits, but from your replies so far in the thread that seems like a fool’s errand squared. If you accept that we can get to the number 1 with recurring decimals if we just slap a “limit” infront of the notation, by all means, add all the “limits” you want and we’ll call this a “yes” to the original question.
LOL. Ok !!
Google around for yourself. I could copy/paste direct quotes which show opposite definitions.
The limit of the series is a representation that calculates to 1/3 actually.
“by any legitimate mathematical definition it equals the number one.”
Yes, because limits, should they exist, are the number. In this case:
Limit Σ ar[sup]n[/sup] = L
n→∞
I am fully aware of the equals sign in this statement, and also fully aware of the limit concept.
L is the number here.
Limit Σ ar[sup]n[/sup] is a way to represent L.
n→∞
Not at all. It was other people’s understanding of 0.999…
They actually believe that you can complete an infinite sum, and that the sum = 1. Without using limits, that is, using only addition because they “understand infinity”.
They are using “an endless string of 9s” as “a number”, (which is an idea/concept of measure or magnitude), and then using some math definitions to turn it into the number one.
I’d like to know how they conjured up any magnitude for which you could assign infinite 9’s to and how they were able to do that.
Chacun son gout.
I didn’t “run around” anywhere.
I didn’t “hunt” for anything unless thinking = hunting.
I didn’t “grasp at straws” whatsoever
I didn’t fabricate anything either.
The only thing fabricated was that entire paragraph.
Trust me, not everyone in the world uses the same definitions, concepts, nor has the understanding of concepts and implied concepts as people in this particular thread which accounts for probably a very small fraction of 1% of math folks. So acting like you are on some sort of high horse just because you and a handful of other folks are all on the same page doesn’t impress me. But now that you got all that off your chest, will there be anything else constructive that you can add to the discussion and/or thread ??
The equality of .999… and 1 is not some fringe notion held by “a handful of folks” here, it is about as mainstream and accepted a mathematical concept as it gets (though it is admittedly difficult for some to grasp.)
More to the point, many people (including me) have already given valid proofs in this very thread. R has a specific definition; equality in R has a specific definition; .999… has a specific definition; and 1 has a specific definition. Real numbers are Cauchy sequences modulo sequences that converge to 0. The real number .999… denotes the sequence (.9, .99, .999, …). The real number 1 denotes the sequence (1, 1, 1,…). Since (1 - .9, 1 - .99, 1 - .999) = (0.1, 0.01, 0.001,…) converges to 0, we have 0.999… = 1.
That is the proof. There’s nothing about processes or limits, infinity, numeric representation, base-n representations, or any of the nonsense that’s cluttering up this rambling, 42-page thread.
No thanks. The real number 1 has many sequences which have that limit.
{9/10, 99/100, …} is one of them, however, “0.999…” already implies a limit concept.
limit of sequence {9/10, 99/100, …} = 1
limit of sequence (1, 1, 1, …} = 1
and many others.
You don’t start with a bunch of digits, and claim it is a number, and then do analysis, introduce a limit, and claim equality.
Start with your sequence {9/10, 99/100, …} and then find your real number. Not the other way around.
You are saying “infinite 9’s” is the “limit of infinite 9’s”
Sorry, but your list of every growing 9’s is always infinitely far from reaching an infinite string of 9’s.
You stated that 0.999… is a real number.
For what sequence is the “number” 0.999… a result of the limit concept ??
Oh, I “get” it just fine. I “got” it a long time ago. Like a monkey, I reproduced solutions for years during my studies.
0.999… as a “real number” is not the limit of anything, therefore not a “real number”
0.999… as a process, ie, the concept of limit with infinity involved, outputs the real number 1.
A) For what f(n) is Limit Σ f(n) as n→∞ is a string of 9’s the *solution *of ??
B) For what sequence is the “number” 0.999… the limit of ?
Solutions:
A) If f(n) = 0.999…, we know 0.999… equals limit of sequence {9/10, 99/100, …} which equals 1. therefore, if f(n) = 1, then we have Limit Σ (1) (n=1 to N) (N→∞)
I am sure you agree “0.999…” is not the limit of that.
B) Try the sequence {9/10, 99/100, …}
Each term is described by f(n) = 1 - 1/10[sup]n[/sup]
the limit of this f(n) is clearly 1.
Again, “the real number 0.999…” was not the result of of the series or the sequence, but rather some sort of starting point.
We don’t try and represent the sequence as a real number, but analyse the sequence and see if it converges, which is the real number.
How do you differentiate a process? It’s really important to physics that for f(x)= 0.999…, f’(x) = 0 … it’s a little off-topic … but 0.999… has to be a number, or some things don’t work like they should.
I’m sorry you don’t like the concept of a limit, but that’s too bad; the real numbers aren’t up for debate. As I’ve mentioned before, the point of the construction of R is that it’s the minimal field in which such limits exist. But I’m not talking about limits in Q, and limits aren’t necessary for the construction. An element of R is a seuquence a = (a_1, a_2, …) such that for all \epsilon > 0, there exists some N such that |a_n - a_m| < \epsilon for all n, m > N. Two elements a = (a_1, a_2, …) and b = (b_1, b_2, …) are equal (by definition) iff for all \epsilon > 0, there exists some N such that |a_n - b_n| < \epsilon for n > N. That is the definition of real numbers. It doesn’t matter whether you like it or not. If you want to define your own extension of Q, feel free to do so.
A real number is a sequence of rational numbers. Re-read the definition above. The definition is set up so that the real number a = (a_1, a_2, …) conveniently has lim a_n = a, but no one is talking about limits in R except you, and no one else seems to have as much difficulty understanding the concept as you.
You also don’t get to decide what an infinite sequence is: It’s simply a map from the positive integers to R. The fact that finite subsequences of an infinite sequence a are not equal to a is completely irrelevant.
We have already covered this many, many, many times. The scare-quoted number 0.999 is, in the notation above, the sequence a = (a_n) with a_n = 1 - 10[SUP]-n[/SUP]. That’s what decimal notation means: 0.abc… corresponds to the sequence (a/10, a/10 +b/100, a/10 +b/100 + c/1000, …).
The limit of 0.99999…is equal to 1 as can be seen.
The limit of 0.9999…is not equal to 0.99999…
If you still continue to insist that the limit of 0.99999…is 0.99999…
just check with the above formula
lim[sub]n -> ∞[/sub] S = a/(1 - q) = 0.9/(1 - 1/10) is not equal to 0.99999…
It is very clear that 0.99999…is not equal to 1. The limit of 0.99999…is equal to 1.
It doesn’t make sense to talk about the limit of a single number. Functions and sequences— even constant functions and sequences— can have limits.
0.999… is a real number. It is also a sequence of rational numbers, because that’s what a real number is, in the same sense that a rational number p/q is an ordered pair (p, q) of integers.
The real numbers are conveniently defined so that a = (a_1, a_2, …) has a = lim a_n (embedding Q into R in the obvious way), but this not particularly important to the current question.
0.999… is a real number: it is the real number corresponding to the Cauchy sequence (0.9, 0.99, 0.999,…). It also equal to 1, since the sequence (0.9, 0.99, 0.999, …) converges to 1.
This is nonsense. I mean that literally, not as an insult: I don’t know what you could possibly mean by ‘limit’ such that lim 0.999… makes sense and is not equal to 0.999… itself.
One more time:
Two real numbers a = (a_1, a_2, …) and b = (b_1, b_2, …) are equal iff for all \epsilon > 0, there exists some N > 0 such that |a_n - b_n| < \epsilon for all n > N. The real numbers 0.999… and 1 correspond to the sequences (0.9, 0.99, 0.999,…) and (1, 1, 1, …), which have difference (0.1, 0.01, 0.001, …), which tends to 0. Thus 0.999… and 1 are equal. End of proof.
I’ve given you the proof, as have several others. If you don’t understand the question that’s the topic of this thread or have anything to contribute besides repeatedly asserting that ‘0.999… is not equal to the limit of 0.999…’, I would respectfully ask that you please take up the matter elsewhere. I would be happy to talk more about the axiomatic construction of real numbers, real analysis, more details about any of the proofs of this thread, and similar topics, but the question in the OP of this 42-page thread has already been answered repeatedly and thoroughly many times over. There is nothing more to say.
You fail to see that 0.99999…is a convergent geometric series as well as
a number. Because of this, 0.99999…has a limit and the limit can be calculated
with the well known formula
lim[sub]n -> ∞[/sub] S = a/(1 - q) = 0.9/(1 - 1/10) which is equal to 1 but not
equal to 0.99999…
It is your mistake not to understand this.
You continue to insist that the limit of the series 0.99999…is 0.99999…
but that is wrong because
lim[sub]n -> ∞[/sub] S = a/(1 - q) = 0.9/(1 - 1/10) ≠ 0.99999…
The numbers 0.9999…and 1 are not equal. You don’t want to understand the concept of limits.
The difference between 1 and (0.9, 0.99, 0.999,…) just tends to 0 but it does not
become 0. Therefore we have the concept of limits. You are trying to treat the limits
as if they did not exist at the end.
It is you who have been given the proof. But you refuse to understand it.
Yeah, I’m going to go out on a limb here and suggest that I’m probably a lot clearer on the definition of a limit that you are. So, define a limit. Seriously: In your own words, but rigorously, define what a limit is. I’d also like to know what the “limit of 0.999…” means and how you define it.
Just saying “No, you’re wrong,” does not constitute a proof. Point to the mistake in this proof:
“Two real numbers a = (a_1, a_2, …) and b = (b_1, b_2, …) are equal iff for all \epsilon > 0, there exists some N > 0 such that |a_n - b_n| < \epsilon for all n > N. The real numbers 0.999… and 1 correspond to the sequences (0.9, 0.99, 0.999,…) and (1, 1, 1, …), which have difference (0.1, 0.01, 0.001, …), which tends to 0. Thus 0.999… and 1 are equal.”
Also, is anyone besides 7…7 and his ilk still reading this? It looked there was some interest earlier in general set theory and real analysis, and I’m happy to continue the thread for the benefit of anyone who cares. If it’s just me, though, I’ll take my own advice and start up a Pit thread.
It is a long time since I studied this stuff, I did set theory in 4th year pure - but I was only taking it as an elective next to my CS degree, so I didn’t work all that hard. I got out my old text book a while ago, and was astounded that some still made sense. ZF Set theory isn’t a total stranger.
So it is all fun, despite the troglodyte interjections.
Then again, maybe another thread is the right idea.
I’m still reading and I note that the thread reads is still going up at a rate of about 50 per post. I am going to be a bit selective on what I respind to though. FWIW I have been enjoying posts from both of you and the clarity with which you have given them - even if the intended recipients are deaf.
Lord only knows why.