Good gawd! It is a religious thing!
I can call 'em.
I’m just keeping half an eye on this thread, so this has probably been tried, but on the off-chance that it hasn’t, and because given the length of this thread, one more superfluous post won’t matter greatly, what about this:
7777777, consider the axioms kindly presented by Indistinguishable in this post. You can view them as a kind of minimal set of properties that describe what we call ‘the real numbers’. In a similar sense, you could call the list ‘has red hair, is 1.75m tall, is female, is 37 years old,…’ a minimal list of properties describing a person we’ll call ‘Alice’. From such a list of properties, if it is complete, everything that can be said about whatever those properties pick out can be deduced—thus, the thing is characterized by these properties. When we talk about Alice, provided the name singles out a unique individual, we’re talking about precisely that thing that has Alice’s properties; and when we’re talking about the reals, we’re talking about precisely that thing that satisfies the axioms presented above.
How does that help us with the question of whether 0.9999~ = 1? Well, let’s first consider what the notation ‘0.9999~’ means. Basically, it’s a kind of shorthand: it stands for ‘91/10 + 91/100 + 91/1000 + 91/10000 + …’, where the ellipses denote infinitely many terms following the same structure. That structure is as follows: the n-th term in the sum is of the form ‘9*(1/10)[sup]n[/sup] = 910[sup]-n[/sup]'. Thus, the whole sum can be written as the instruction 'sum all terms of the form 910[sup]-n[/sup] for n = 1 to n = infinity’.
Now, remember the list of properties I talked about earlier? One property of the reals following from those in that list is that every infinite sequence of numbers of a given form has a unique value it reaches when the sequence is continued to infinity. The technicalities don’t matter for the moment, but basically, the criterion for this is that the gap between successive elements of the sequence gets smaller than any given value.
Another important criterion is that a sum of infinitely many terms has a finite value if the sequence of its ‘partial sums’, i.e. the sequence that we get if we take for its first element the ‘sum’ of the first term, i.e. just the first term, for its second element, the sum of the first and second term, for its third element, the sum of the first, second, and third term, and so on, if that sequence reaches a finite value. That is, for the problem at hand, if the sequence:
0.9
0.9 + 0.09
0.9 + 0.09 + 0.009
0.9 + 0.09 + 0.009 + 0.0009
.
.
.
Approaches a finite value, our sum will likewise have a finite value. Well, does it? Let’s say you give me the value 0.1, then I can tell you that the difference between the first and second elements of the sequence, i.e. 0.09, has a smaller gap than that. If you give me the value 0.01, then the difference between the second and third elements is smaller. If you give me the value 0.001, then… And so on. It’s clear that no matter how small a number you tell me, I can always find a gap smaller than that. Thus, the sequence converges, and hence, our sum has a finite value. This, just so you keep it in mind, follows strictly from the properties of the real numbers.
And we can do more than that—we can calculate that value. Let’s call that value v. So we have v = 91/10 + 91/100 + 91/1000 + … Let’s multiply that by 1/10. This yields v/10 = 91/100 + 91/1000 + 91/10000 + …, or in other words, the original value minus the first term. So we subtract this new expression from the original one, and get v - v/10 = 91/10, or in other words, 9**v = 9, thus proving that v = 1. If you don’t believe me, just ask a computer.
Now what’s important to realize here is that all of that follows necessarily from the list of properties we started out with, i.e. the axioms for the real numbers. These necessarily lead to the conclusion, via the above argument, that 0.9999~ = 1, because 0.9999~ is nothing but a shorthand notation for a certain sum, which we have shown has a certain value—and that value is 1.
So for that thing that is picked out by the properties, the axioms posted above, it’s unambiguously the case that 0.9999~ = 1—just like, for Alice, if red hair is percetly correlated with having freckles, the fact that Alice has red hair implies that she has freckles.
What you’re now doing is basically saying that no, Alice has in fact no freckles. But since this means that she can’t have red hair, it means that you’re not talking about the thing picked out by the set of properties described earlier—in particular, you’re not talking about something (or someone) that has red hair, since if Alice had red hair, she would have freckles. But this just means you’re not talking about Alice at all—you’re just talking about someone else.
This is what you’re doing if you’re insisting that 0.9999~ doesn’t equal 1: you’re not talking about the real numbers, since it follows directly from the properties of the real numbers that 0.9999~ = 1, as demonstrated above (and dozens of times at other points in the thread). Among the properties of the reals are certain rules of calculations as well as the fact that certain infinite sums have values, derived from the fact that certain series converge; and these properties lead to 0.9999~ = 1. If you’re insisting that it doesn’t, you’re insisting that these properties don’t (all) hold—but then, you’re just not talking about the real numbers everybody else talks about. You’re talking about your own ‘real*’-numbers, just as you’re not talking about Alice, but about some ‘Alice*’ when you’re insisting Alice doesn’t have freckles. There’s no debate on this, since Alice has red hair, and if you have red hair, you have freckles (at least in this hypothetical), so nobody that doesn’t have freckles has red hair, and thus, somebody without freckles can’t be Alice, since Alice has red hair. Similarly, if you say 0.9999~ does not equal 1, you’re not talking about the real numbers everybody else is talking about.
…999 = infinity
0.999… = 1
…999 + 0.999… = infinity + 1 = infinity
…999.999… = infinity
In normal math, at least.
In defence of 7s there are some mathematicians (well at least one) who doubt the existence of infinite decimals: NJ Wildberger on what he perceives to be the problem
Personally, NJB strikes me as being mad as a box of frogs.
You’re overstating the claim.
Certainly, one can find fault with the standard system of real numbers. And some people do as seen in the video.
A solution (and one presented in the video) is building an alternative system. And that’s totally fair and defensible.
None of that is actually any sort of defense for 7 7s, who is trying to fit his alternate definitions into the existing framework.
Great!
You fulfill the prophesy on my post #1247.
I told there that you will end up at the formula
∞+1=∞
and when you insert your infinity …99999.99999…into this formula you
will arrive at
∞=…99999.99999…=…99999+0.99999…=∞+1
and there you got it:
your infinities:
∞=…99999.99999…
∞=…99999
Next you should try to find errors in this kind of reasoning. Then you will be worthy
of receiving the truth concerning the number …99999.99999…
Now we seem to start talking the same language, as Exapno said " Every single person who understands math, regardless of their other personal beliefs, will agree on every single proof. That’s why everybody arguing with you in this thread - a group of otherwise totally individual personalities - all are saying the same thing as one another. That’s what real math is, a language spoken by all."
What happened to you? You seemed so eager to learn in your earlier posts, that I tried to help you learn what you didn’t know. Now you seem like a completely different person. I hope you feel better soon.
Perhaps you mistook me for someone else. I don’t know where did you got the idea
that I came here asking for help. I never said so. Or why did you think that you knew
something that I did not know. I am feeling alright all the time. I am just fighting
ignorance, it does not make me feel bad.Why did you think so?
You just sound confused. Sometimes you say that two numbers with different digits can’t be equal. Sometimes you insist that two numbers with different digits must be equal.
spittake
Uh, dude, I have a children’s book on my shelf here that explains the concept of infinity in a very accessible and rational manner. In fact, I have two - “The Number Devil”, and “Coincidences, Chaos, and All That Math Jazz”. These are both books aimed at either a child or young adult audience, which explain the concept in fun, easy ways. In fact, as you point out below, you’re the one who fundamentally lacks in understanding when it comes to the concept of infinity. In fact, let’s just skip right ahead to that, shall we?
You don’t understand Infinity. Infinity + 1 doesn’t actually make any sense, because “infinity” is not actually a real number. It doesn’t function the same way as natural, rational, real, or complex number, and is not technically a valid input for the “+” operator. It’s like saying “Zebra” + 1. Your input is nonsensical, so your result is as well. However, there are decent thought experiments which help explain how to apply things like plus, minus, multiplication, and division to infinity. “All That Math Jazz” devotes several chapters to it, but here’s a simple version.
Imagine you have a baseball team with infinite members. Let’s call them “Team Natural Number”. Their roster size is unbounded - quite literally an infinite number of players, each with a number on their jersey. They go to check into a hotel with an infinite number of rooms, each numbered. It’s easy to demonstrate that the team fits into the hotel - each player mathces their jersey number up to the room number, and all is well. But then the manager shows up, and he wants a room too. Where do you put him? Room 1? Taken by Player 1. Room 2? Taken by Player 2. Room 49582834? Taken by Player 49582834. There isn’t a room for him! You’d need a bigger hotel for him… except that the hotel is already infinite. So instead, what he does is, he pushes each player forward a room, and stays in room 1. Player 1 is now in room 2. Player 2 is now in room 3. Player 49582834 is now in room 49582835. So we have successfully added infinity + 1, and came out with the result… infinity. Adding to infinity does nothing. Same thing with subtraction - if a player is sick, simply bump all the players after him back a room - still infinite rooms. This is why what you’re saying is so… wrong. …9999 is infinity. …9999.999… is also infinity. Because they’re both absolutely asinine ways of writing infinity. Neither is bound - there is at no point a real or complex number which is higher than either. They are equal to each other, despite seeming different, because neither is an actual number in any real sense. You can’t treat them like they are numbers.
There is no error - you simply don’t understand infinity. You’re treating it like a number. It isn’t. And if anyone’s wondering why I’m bashing my head so hard against this, I failed every fucking exam this semester. I have to feel smart somehow. 
Yes, of course. Infinity has different mathematical properties than non-infinite numbers. For example:
2 * ∞ = ∞
This is all easily provable. I don’t know what’s confusing you.
Unfortunately, they’ll be hitting their heads on the ceiling, since due to the large horizontal expanse, Hilbert had to skimp on the vertical.
So can I:
See? I was wrong! 
I’m a social scientist and you’re a mathematician. Math may be your area of expertise but people are in mine. Especially people on the Dope. It’s an unbelievably awesome ongoing field study into psychopathologies.
Run, my little lab rats, run! 
Ha ha, it is you who are in trouble now.
You all fulfill the prophesy on my post #1247.
Let me repeat what is prophesied there: You will end up at a problem with infinity,
you will arrive at the formula ∞+1=∞ and you insert your infinity into it.
Remember what I said, it is *your infinity *, not mine.
It is you who are treating infinity as a number. Remember that, it is your mistake,
and it is all happening as was prophesied.
It is you who are confused with the number that was sent to you. You don’t
know what it is. You think that it represents infinity. You think that it equals 0.
But you are wrong. Do you consider yourself worthy of receiving the truth about
…9999.9999…?
Let me clearly show you your mistake:
…9999999 = -1 is true
now you think that
…9999999 = infinity
so that
infinity = -1
there it is, your mistake !
I told you it beforehand. You are treating infinity as a number. Can I say it more
clearly? Is infinity equal to -1 ? Tell me.
Let me repeat this a hundredth time: why do you think I don’t understand things
I am talking about?
It is you who are ignorant.
It is you who are treating infinity as a number.
It’s not just that, though. For some reason, these people all believe that they have a capitalized Revolutionary Truth that they, despite having minimal math or physics training (or because of it, since they aren’t brainwashed into accepting something obviously false), have discovered and must preach to the masses. It’s a lot like witnessing, with similar results.
Here’s how you prove you can’t add, subtract, multiply, or divide by infinity, using normal rules.
∞-∞=0
∞+1=∞
∞-∞+1=∞-∞
0+1=0
1=0
Since we know that 1 is not zero, we’ve proved that doing normal arithmetic with infinity leads to nonsensical results, which means we can’t do normal arithmetic with infinity, unless we’re willing to stipulate that 1=0 and explore the consequences of that stipulation.
Similarly, we can’t divide by a perfectly normal number, the humble zero, because what is 1 divided by zero? Is it the same as 2 divided by zero? If so, then 1=2. Oh, scary. Since we don’t want 1 to equal 2, we decide not to allow division by zero. Division by zero turns up in all sorts of real-world computation errors, we get the wrong result because we accidentally allowed dividing by zero in a place where it didn’t seem like we divided by zero.
As for the contention that 1 <> 0.9999… because none of the digits are the same, well, does 1 = 2/2? Two divided by two is one, so we’re perfectly happy to substitute 2/2 any place we write one, confident we’ll get the same result.
2/2 is just another way, using our notation where 2 stands for “one unit plus one unit”, of saying 1.
Just like 10 is the same number as 2, if we realize that 10 in binary notation means the same number as 2 in decimal notation.
So the fact that two numbers or two sides of an equation don’t look the same at first glance isn’t proof that they can’t be the same, otherwise we’d say that 6-2 <> 5-1, because one side has sixes and twos and the other has fives and ones.
That isn’t proof that 1=0.9999…, mind you. It’s just saying that just because we use different digits on both sides that doesn’t mean that the sides MUST be unequal.
Hey, if you want to invent a branch of math where we explore the consequences of allowing infinity to be manipulated using standard arithmetic, then go nuts. But some people have already tackled this sort of thing, so if you want to find out if you’ve discovered something really new you should read what they thought first.
Is it? I don’t think it is equal to -1. I mean, I don’t know what …99999 should equal, actually. It is a divergent series if it means what I think it means. Or it could mean something else. I assume it is a notation for expressing 9 + 90 + 900 + 9000 + 90000…, right? If it means that, then it doesn’t mean -1. If you think I should agree that it means -1, could you explain again why you think it does, and then I’ll see if I can follow your reasoning, since right now I don’t.
Actually, in most math we try not to actually use infinity. Usual math treats infinity like toxic nuclear waste that has to be carefully handled by remote control arms, carefully behind shields like limits and delta-epsilonics.
So…
Does it equal infinity? I’d say that the sum of 9 + 90 + 900 + 9000 + 90000… diverges without bound. We can call the summation “infinity” if you like. This means we absolutely don’t treat infinity as a number. Because if we treat it like a number, we find we get all sorts of results that we usually don’t want to accept. So infinity is not a number in most systems of math. It is a concept that we might use, but it isn’t a number even if we use it all the time, just like + and - and * and / aren’t numbers even though we use them all the time, they are operators.
To bring this back to that Tanton pdf, this is the gist of what he means at the end by “Welcome to Calculus”.
Further, there’s a nice sentence there at the end:
[QUOTE=Albert]
IF you believe that …9999 has a meaningful answer, then that answer has to be -1
[/quote]
That’s a might big IF. You have to get to that “meaningful” answer/system FIRST.