Sure it is. Why wouldn’t it be? It’s not particularly interesting not related (except to 7777777), but it’s certainly fair - which is itself subjective.
Under the context 7777777 would like to operate, it’s salient. That’s relevant and fair enough in my book. It’s not salient to you or the parts of this topic that interest you, but whether or not that’s sufficient to drop the topic is again a subjective call.
See up top. “Fighting ignorance”.
Besides, trying to argue with other posters about why they shouldn’t argue with other posters is getting a bit meta for me. Also, pot meet kettle.
For what it’s worth, for all my mention of 10-adics above, those are not the only context in which “…999” would be taken to denote -1. In fact, “…999” also denotes -1 in the reals… on one natural account of how to interpret infinite decimal notation as specifying a real number. [Not the order-based one I’ve mentioned before, which would make …999 infinite; a different one which happens to agree with that one on decimals which terminate on the left]
Specifically, recall that decimal notation can be taken as describing a number through the coefficients of some power series to be evaluated at 10. If these power series are finite, there’s no debate as to how to carry out this evaluation; however, in the infinitary case, there are various summation methods we may use. There is, for example, the theory of absolute convergence, as well as the less conservative textbook theory of summation in terms of arbitrarily close approximation by sums of discrete initial segments of a series. I’m going to note another summation method by which we can interpret decimal values.
Specifically: given power series P and Q, let P * Q be their product (basically, this is multiplication without carries). It may so happen that Q(10) is absolutely convergent to 1, and (P * Q)(10) is absolutely convergent to some finite value. In that case, let us say that P(10) sums to that value [regardless of whether it is itself absolutely convergent].
This definition would appear to make the evaluation of P(10) depend on the choice of some other arbitrary power series Q, but it’s easy to check that any two such choices must give consistent answers. The notion of summation we get from this is very close to absolute convergence, and thus inherits most of its good properties, except those having to do with ordering [e.g., we lose the guarantee that a series with positive coefficients must have a positive sum].
In particular, we will find that …999 corresponds to P(x) = 9 + 9x + 9x^2 + … . Taking Q(x) = (-1 + x)/9, so that Q(10) absolutely converges to 1, we observe that (P * Q)(x) = -1, so that (P * Q)(10) absolutely converges to -1, and thus we have that …999 denotes -1 on this interpretation of decimal notation into real numbers. [In fact, this is just a reframing of the very proof given by Albert previously.]
So it’s not as though this claim depends on using some wacky number system other than the reals. It’s just a matter of how you choose to interpret decimal notation into the reals.
The reason I think it’s not salient is because, as I said, I don’t imagine getting them to say “Ok, …999 is not -1. It’s infinite, or meaningless, or some such thing” will help move them to conclude “Ok, .999 = 1”. They’ll still have all their underlying “Things aren’t equal when the digits don’t match” confusion to cling to. I guess I should clarify that by “salient”, I meant “salient to the apparent goal in such hammering of getting 7777777 to accept .999 = 1”.
As I said, the only reason 7777777 brought “…999” up in the first place is because they accept “…999 = -1”, and see that it will make “.999 = 1” equivalent to “…999.999… = 0”. And they refuse to accept “…999.999… = 0” because the digits don’t match; therefore, they consider “.999 = 1” disproven as well.
But even if you get them to abandon “…999 = -1” and thus lose their support for “…999.999… = 0”, they can still apply the digits-don’t-match-therefore-not-equal objection directly to “.999 = 1”. That’s why I say it’s not salient to attack them on these grounds if the ultimate goal is having them say “Alright, .999 does equal 1”.
As for why I say it’s not fair, it’s because it seems pretty clear that 7777777 doesn’t have the conceptual language available to say “I want to work in system so-and-so, rather than system such-and-such”, so when one thing we know for sure is that they do want us to take …999 = -1 by the Albert argument, why not at least grant them that context, instead of slamming them for that choice? (Remarks like “No, …999 is not a number in the conventional sense! It’s a construct which cannot possibly be resolved to any real or complex number (your failed proof which does something about as viable as dividing by zero notwithstanding)” and so on)
Although I said “Ignore them. Let it go.”, I didn’t mean to imply I have any right to tell you what to do. You shouldn’t stop arguing with **7777777[b/], if you really intrinsically enjoy it; I was just confused that people still seemed to be thinking they could actually convince 7777777 to change their mind. If your motivations are something other than accomplishing this, you may well have good reason to carry on.
I definitely am a hypocrite on the topic of stepping away from arguments, though.
So, yeah. I feel bad about attempting so directly to steer the conversation away from what I unfairly presumed everyone would agree was a quagmire. My apologies.
(Also, even the presumption that no one can get through to 7777777 is unfair. After all, I once thought this a remote but existent possibility too, even as others knew better; the Messianic stuff was a point of no return for me, but why should I demand it be for everyone else? More apologies…)
If you don’t see why, you must be blind:
…99999 = -1 is true
What do you assume will be the number 1 based on this truth?
Might the number 1 be 0.999999…?
Let’s see if we can calculate it:
…99999 + 2 = …00001 = 1
surprise! The result is not 0.99999…:eek:
What can you do now?
Should you perhaps attack seven sevens, might a new attack restore everything?
Or a very old trick may help, write that 1/3=0.333333333…and multiply it
by 10. Or invent a new trick, multiply it by 10 and then divide it by 10.
Prove a statement true by assuming it is true. Deny everything. Deny every fact,
especially if it threatens your position. Seek help from your friends. Sacrifice truth.
And all this at the cost of having to pay a very high price. The price whose value
is so high that I can’t tell yet.
Tell me why should number 1 have a decimal expansion, even though, for example, 1/5 has?
Tell me why the decimal expansion of 1 should be equal to 0.9999…even
though I have proved that these two numbers are never equal, the infinitesimal
difference never equals to zero, no matter if you say it does. There is no
infiniteth decimal place where the difference magically disappears. No, the
the difference just gets smaller and smaller…
What is wrong with writing 1/3=x ?
Why do you insist writing that 1/3=0.3333333…even though
1/3=1/3 and 0.3333…=0.3333…?
Isn’t it a way too simple just to say that these two numbers are the same.
Why you think your simple way of thinking is the ultimate truth?
You have not reached infinity to be able to tell what is going on there.
It seems that there is a transformation when one reaches infinity.
If you did not achieve it yet, you should not behave as if you did.
The numbers transform into the symbol ∞
No-one knows how this happens, it is not taught anywhere, so it is also
impossible you had this knowledge, that would justify your behaviour.
Seven sevens, what do you want us to say? The limit as x approaches 1 of the function f(x) = x is indeed 1, using the definitions given in first year calculus. It’s not a real number or a vector, it’s a limit.
We’d like for you to take your derivation of why 0.999… ≠ 1 and show how this disproves the twenty or thirty other derivations that show 0.999… = 1. Please explicitly define the terms you use. Mathematics is concise, under any set of definitions, only one of these statements is true (assuming base 10).
I don’t want you to say anything if you don’t want to say anything, just waiting when your ignorance is over, unlike you who want me to force to admit that 1=0.99999…
What I wonder instead, why should not someone choose truth and knowledge instead of
lies and ignorance if he had a choice. It seems that many people have no such a choice.
You offer no choice. You want to force people to admit you are right. I have told many
times what I think about such a behavior: brainwashing.
Since you offer no choice, I will do it now.
Think that you have a once in a lifetime chance to choose something big.
A chance of a lifetime to achieve something big, it is hard for me to tell exactly what
because people already ridicule me too much.
I show you two options, and you must choose only one of them as your truth.
You cannot choose them both because as it is said “you cannot have your cake and eat it”.
Choose only one of these as your truth, and then the other one will be false for you:
…99999 = -1
1=0.99999…
and that will probably determine the rest of your life. It seems I alone have
chosen …99999=-1 as my truth and 1=0.99999…as false and that’s why I am fighting against the whole world.
But if I have the truth, and it not mine, how can you fight against the truth even
if you have the whole world on your side? I think you are wasting your time and energy in that case.
“My teaching is not Mine, but His who sent Me. If anyone is willing to do His will, he will know of the teaching, whether it is of God or whether I speak from Myself.…”
Yes, there are two paths you can go by, but in the long run
There’s still time to change the road you’re on.
And it makes me wonder.
I do believe that you are beginning to repeat yourself. Maybe you should try preaching from a different chapter, like telling us why the sons of Ham should be shunned or sodomites need to be stoned. Those are also golden oldies.
The Hamster King might at that point instead say “Alright, 9x = 2 + 3x. So x = 2/(9 - 3) = 1/3”.
A simpler example of this same phenomenon to consider in making sense of a system distinguishing .999… from 1 is whether .999… + .999… = 1.999… = 1 + .999…, as digit-by-digit calculation might suggest. [To which one can give various responses, in accordance with various interpretations, etc., etc., as I always say]
7777777, insults are not permitted in General Questions. This is an official warning.
These posts are also not appropriate for General Questions. This forum is for factual information, not for personal opinions. It’s clear at this point that your position is based on your personal beliefs rather than any actual mathematical theory. In addition, religious witnessing is only permitted in our Great Debates forum.
I am instructing you to drop this line of discussion. If you are unable to support your ideas mathematically (and it’s rather clear you cannot), take the discussion elsewhere.
Further posts along these lines will result in a warning, and you may find your posting privileges under discussion by the staff.
Colibri
General Questions Moderator
PS. While I’m not instructing other posters not to respond, I don’t see much point in engaging a poster who is not even attempting to argue factually.
People surely tire of me providing such defenses, but if I were to want to say that 9x = 2.999… and not 3, here’s one way to do so which is slightly different from previously discussed ways of doing so, but captures an important aspect of the intuitions expressed inchoately by some:
I’ll take this to mean x is a number at least as large as 0.3, 0.33, 0.333…, and strictly less than 0.4, 0.34, 0.334, … . We have no further information about x; there may be more than one number of this sort, but we’ve picked one out and called it x.
(The strictness of the upper bound here is, as we know, not the standard interpretation of decimal notation, but clearly such strictness is in mind for those who argue that 0.999… must be less than 1.000… because the first digit is smaller. So let’s understand how that would work [even though such objectors should also be made to understand how the standard use of decimal notation works])
I’ll take this to mean “10x is at least as large as 3, 3.3, 3.33, etc., and strictly less than 4, 3.4, 3.34, etc.”. And dividing through by 10, this is indeed equivalent to 1). No problem.
I’ll take this to mean “10x - x is equal to the difference of two numbers, the first of which is as described in 1), the second of which is as described in 2)”. No problem.
I’ll disagree at this point. The argument for the right-hand side is by subtraction, but just because A falls within the intervals [3, 4), [3.3, 3.4), [3.33, 3.34) and B falls within the intervals [0.3, 0.4), [0.33, 0.34), …, doesn’t mean A - B = 3. All we can conclude are things like that A - B falls within the intervals (3.3 - 0.4, 3.4 - 0.3), (3.33 - 0.34, 3.34 - 0.33), … = (2.9, 3.1), (2.99, 3.01), (2.99, 3.001), … . A - B could be equal to 3, or larger than it, or smaller than it.
The error, on this interpretation of the notation, is in presuming that knowing a decimal expansion of A and the decimal expansion of B lets one deduce the decimal expansion of A - B. It could be that more than one value falls within the same decimal expansion (so that 1/3 would not be the only value with decimal expansion 0.333…, but so could a value slightly less than 1/3, which could be our x).
On this account, the problem is never that one valued might have multiple decimal notations, but rather than one decimal notation might correspond to multiple values.
Glad to have been of help. Keep in mind, of course, that I’m only saying that’s one nonstandard way you could imagine interpreting decimal notation. You weren’t wrong to think that The Hamster King’s argument was perfectly legitimate, in the context of the standard interpretation of the notation. Please don’t think you were mistaken about the standard interpretation of things; all I hope to do is show how we can also understand other possibilities, formalizing the intuitions others of an admittedly mathematically unsophisticated sort struggle to express in their arguments.
(Also, I see now that while I intended to respond to The Hamster King’s illustration of the argument, I actually quoted The Great Unwashed’s. But it makes no difference.)