So...1/2 < 1, right?
So... 1/2 / &infin < 1 / &infin, right?

I've found a smaller positive number than tvrocks has found!

So...1/2 < 1, right?
So... 1/2 / &infin; < 1 / &infin;, right?

I've found a smaller positive number than tvrocks has found!

Well damnit. My &fu is weak

I noticed a general folly that it seems that almost all people who are for the idea of .999...=1 being that when confronted with clear evidence to prove the opposite they say either something like "you can't do math with infinity as it is not a number" or "the rules regarding infinity are way different than those regarding normal numbers" and, after thinking about it for a few minutes, i realized that both of these statements are complete bs and that "at least the people who say the first of the 2" are complete hypocrites. The idea of not being able to use math because it is infinity being used to disprove the problems with .999... equalling 1 are extremely stupid as the original "proofs" also tried to do math with a number that is infinite. Why woudl it only not apply when I was using it to disprove the theories that, according to that rule, should not have existed in the first place? This idea is outrageous, and any of the people who are gulilty of it are extremely hypocritical. It is basically the same situation with the people who said that the rules regarding doing math with infinity are different as, usually when confronted about it, they would respond because infinity is a concept, not a number. They then go on to break both the "rule" of not doing math with infinity, and the laws of arithmetic, and none of these people have actually given a good reason behind 1. why there would be different rules and 2. why those rules would lead to .999... being 1 instead of the result that basically all non-flawed math woud lead to. This is basically the only argument, besides the flawed x=.999... and 1/3=.333... "proofs " that supporters of the idea of 1=.999... have actually used, so i feel it a good idea to point out that the idea of "not being able to not use math to prove that 1 does not =.999... because .999... is infinite, and you can't do math with infinite numbers, but that it is ok when they do it to "prove" that it is equal to 1" is outrageous and, overall, idiotic. If you are going to continue on insisting that 1 is .999… then you either have to 1. Explain why the rules of mathematic don’t apply when trying to disprove the idea and 2. Why it is ok to do math in the standard way while trying to prove it is and 3. Why the rules would be different in the first place. (saying “it is infinite does not count.” If you are not willing to do these things then you should default to saying that they are not equal to each other. Currently this just seems like a fallback statement that people say just because they don’t like the idea that the math to support it is flawed. If you cannot actuallyally prove that the rules are different (you guys do have the b.o.p. after all as you are saying that the laws of arithmetic that are correct in literally every other situation) then you will lose as the math to support it, the conceptual reasoning, and the properties of the number are all flawed, and thus 1 does not equal .999…

@thorfi: things like "anything divided by itself is 1" are common sense. It may not be easy to come up with things like that without thinking about it, however, after thinking about it it makes complete sense and should be clear to all people. This debate doesn't matter very much though and i will admit that as in most of those cases, that it woudl have been better to say "careful thought" or logic instead.

@smokie: if there are “lots of infinities” then do you believe that some are greater in size than another? If so would you agree that the amount of rational numbers between 0 and 1 is greater than the amount of positive integers as 0 and 1 contains a lot of different infinities (for example, 0 and 0.1, 0.1 and 0.2 . Both of these are infinite.) I am mainly curious as the claim that there are lots of infinities would not make sense if you believed that all infinities were the same size as if that was the case there would only be 1 infinity.

@nescio: I am very happy to see that you agree with me on the concept that even though something is approaching something else that, even though it may get extremely close, it will never actually become that things.  do you believe that there is only 1 infinity? If so thehn wouldn’’t math with it be very easy as that infinity minus itself would obviously be 0 as they would be exactly the same? Wouldn’t that infinity minus itself be 1 as they are the same? Fi you believe that therye is 1 infinity then it would be very easy to do math with it. However, the idea that it would be undefined is flawed. If that infinity minus itself would be 0 then if you set infinity+5>intinity and subtracted by 5 it wou dlbe really easy to see that they are not equal as, if there is 1 infinithy atat when subtracted by itself is 0, than they would end up cancelling each other out and you would end up with 5>0 . If, on the other hand, you believe that there are multiple infinities, and that some are bigger than others, would you agree that it would make the most sense if there would be a standard infinity that depended on the situation as otherwise one infinite symbol could be less than, greater than, or equal to another infinity symbol? This would be the only way that math would make sense as otherwise it could result in something extremely weird, and the very concept of infinity would be flawed. If you do admit that there is a standard situational infinity then it would also be very easy to see that that standard infinity +5 is > the same standard infinity? I’m mainly curious on your response and, if you believe that math with infinite numbers truly is broken, then I would encourage you to either completely stay away from this subject, or to say that all math with infinite numbers is undefined universally, adnd say that .999… does not actually exist instead of saying that it is equal to 1. Looking forward to your response.

@jeff: congrats if you believe that, I actually mentioned the concept of 1/3infinity earlier when I was first presenting my theory that there is a standard infinity that depends on the situation, so did not feel like getting into it at the time. I personally believe that 1/i^i^i^i^I would be smaller than 1/I however, that does not seem to be a popular subject so I did not feel like getting into it at the time.

I have a life again now that school has started for me again, so I will probably not be extremely active, I will check in from time to time though. I do find this interesting. (although I don’t believe I’ll convince any of you>)

@tvrocks: If you believe there is some number smaller than 1/∞, then your postulate that .999... + 1/∞ = 1 is illogical.

Read Asimov's "The Road to Infinity". First, using infinity in algebra is a trick. Infinity isn't a number. Second, It's trivially provable that there is far more than 1 Infinity. There are an infinite number of Integers. That's a "countable" infinity. There's an infinite number of real numbers between ANY TWO integers. That's an "uncountable" infinity, and is clearly more.

BTW - @ghug - "imaginary" was an unfortunate popular name for the square root of -1. The correct terminology for any number including _i_ is "complex". And it's far from imaginary - (* WARNING: Nerd Sniping ahead *) http://acko.net/blog/how-to-fold-a-julia-fractal/

tvrocks: this entire thread consist of you, and people trying to teach you something. I'm pretty sure ☺, abgemacht and CSteinhardt all have degrees in math, physics, engineering, or similar fields. You could learn a lot from them.

Math is not like philosophy, in that you don't need to argue well to make a point. It's not a debate. I two people disagree about anything, either they have different assumptions/axioms, or one of them (or both!) made an error in their proofs. Right now it seems like you're using a different set of axioms than most of the other people in the thread. Please clarify what your axioms for defining real numbers are, and read up on the axioms that we use: https://en.wikipedia.org/wiki/Construction_of_the_real_numbers (I gotta say this wiki page is poorly written, so you might want to Google around for a better link. They'll all say the same thing).

Oh cool, just go ahead and insult all the people who tried to explain the non-obvious intricacies of modern math to you. All of them, because why the fuck not? It's not as if they've sacrificed time and effort to explain things someone they expected was in good faith trying to find out more about mathematics.

What is the difference between 0.999... and 1.000...? Give me the numerical difference. 1-0.999...=___. 1/i doesn't count, because it is not a numerical value, much like infinity itself is not one. Neither does 0.000...1, because it makes no sense. Stop treating infinity as a number, a lot of your problems will disappear.

Finally, have you noticed this thing I did here? I separated my thoughts into paragraphs based on content and tone. Perhaps you could do so, and make your walls of text at least vaguely readable. Also, you might end up summarizing your point in the meantime, so that the people trying to help you don't have to read the same thing several times.

You're wrong and that's okay. Everyone is wrong before they are right. But when people show you that you're wrong (by demonstrating why infinity can't be used arithmetically, and also referring you to all of math since like the 18th century and the assumptions we make in it), when they show why you're wrong (a reliance on common sense definitions of complex and abstract mathematical constructs), and when they show you how you can be right (adopt the standard definitions of things mathematicians use, such as infinity), at least listen. I don't really see you doing that.

This argument is somewhat like if someone decided that their definition of blue now contained the entire spectrum of visible light, and then was surprised at people describing things as red, since obviously everything is blue, because it reflects visible light, which is blue.

@tvrocks

If you spent as much time reading the sources people here have provided as you did insulting us, there would be nothing left to discuss because you'd realize you're wrong and why.

"I noticed a general folly that it seems that almost all people who are for the idea of .999...=1 "

For the idea? You mean people who understand these mathematical concepts? They are people who agree that, given certain axioms, this is the only consistant conclusion. (Ie it is a true statement)

You can be against consistancy - but you don't appear to be arguing that, it looks like everyone agrees to certain axioms and that we must be logically consistant; so the only problem with with logic.

I was pointing put that the theories they used were all very bad. I also never insulted them, i said rhat the ideas that they were bad, and i did say that they were bing hypocrites, however, that is not necessarily an insult and just has a bad connotation, while the denotatuon is not bad. I am grateful to them for trying to "correct me", and think they are all good people, i wrote the post like that solely to try showing how ridiculous some of the ideas were. Why is it ok to use flawed arithmetic to "prove"that .999... Equals 1 whike it is not ok to use actual arithmetic to prove that it doesn't? I am very curious on this concept.

@phil: i will admit i sounded more harsh than was my intention, i find it weird that you say that 1/i does not make sense while at least implying that .999... Does. What is the difference?

I also admit that i could probably use paragraphs to help, however, i am mainly making the cases on my ipad and it is way harder to format. I do listen to everything they say as, if i didn't, i would not have been abke to respond to it. I am not sure what you are teying to say with the blue as visible light spectrum thing and see no relation.

@basvan: aren't all conceptual things, by definition, at least starting off as philosophy? That is the thing that math is based on and, without it, math would not exist. I currently am about to leave my house so will look into the axiom thing asap.

@jbal: i understand the concept, thanks though.

*with your logic?

Either it is wrong, or it isn't obviously right (ie you are failing to communicate why it is right) - those who disagree all clearly see why they are obviously right, but are failing to communicate it back at you.

But abge ( who i have a measure of respect for ) seems to think you're not listening, so the fault on the communication side of things may not he theres.

Either way, not an interesting conversation, except the part i'm looking at, which is how to cimmunicate effectively :p

*using an ipad is a BS excuse. I'm posting from an iphone and rarely omit paragraphs.

@"@basvan: aren't all conceptual things, by definition, at least starting off as philosophy? That is the thing that math is based on and, without it, math would not exist. "

No, or depending on what you think 'counts' as maths. Most people consider arithmetic (sums) to he maths, an it is basic supermarket calculations (ie if one oragne costs 30 cent, and i want to buy 3 oranges and an apple, how much should i pay?, supermarkets tend to offer discounts from that amount for some purchases to make you feel like you're getting a good deal if you buy more than you need... Which wouldn't be possible if we didn't all have a basic level of arithmetic)

But it is a result of trade, not philosophy. Without trade we wouldn't have arithmetic.

Without philosophy we wouldn't have philosophy... We would still build mental tools to help deal with the world we are living in.

@abge: i personally do not consider hypocritical to be an insult as it is just a part of human nature. Being a hypocrite is not necessarily bad, it just means that people have either a. Not thought about it or b. Chose to ignore it. Being hypocritical does not harm anyone and just has a really bad connotation. (Although in retrospect i could have probably stated that better, and i do apologize.) i have researched the things they have given me in depth, and found that none of them would actually just be conceptually inacurrwte, or that i actually agree with them (and they didn't disprove the idea) i also spent maybe a minute insulting people while i spent 1 hour + researching.

@orthrauc: i have a problem with he inconsistencies and problems in the "logic" and that they donmt apply the "truths" that should be unjversal universally..

"@jbal: i understand the concept, thanks though."

It sure as hell seems like you don't.

Do you realise the 1/3 and 0.333 (repeating) are two representations of the same number? That you can have a decimal and rational representation but it will sill he tue same number? Is that something you are ok with?

Sure, and I'm sure many people will be very happy to discuss the philosophical status of math and philosophy, but for now let's do math. So you don't have to make your case or point out inconsistencies in other people's rationale, you just clearly state your axioms, demonstrate that they lead to the conclusion you want to proof, and qed.

So:
- axiom: real numbers are (equivalence classes of) infinite sequences of rational numbers, with two sequences considered equivalent if their difference converges to zero.
- notation: 0.9999... denotes the sequence 0, 0.9, 0.99, 0.999, 0.9999 and so on.
- more notation: 1 denotes the sequence 1, 1, 1, 1, and so on.
- To demonstrate: 0.9999... = 1.
- Proof: the difference between the sequence 0, 0.9, 0.99, 0.999, 0.9999 and so on and the sequence 1, 1, 1, 1, and so on is the sequence 1, 0.1, 0.01, 0.001, 0.0001 and so on. This sequence converges to 0. Qed.

Do you agree that this proves that, given my axioms, 0.999... = 1? Can you explain what your axioms are, and demonstrate that, given your axioms, 0.999... is not 1?

(Funny thing, i was going to suggest using a calculator, but mine seems to be smarter than usual... And not give me any errors)

I dunno, but I bought a street plan off a stall just off from Trafagar Square the other day. On the back of the plan £1.99 was printed. The stallholder said £2.
I think this is irrefutable evidence that .99p = £1

Y'all seem to be talking past each other quite a bit. Perhaps this would be a good way to help:

@tvrocks: The reason that 0.9999... is equal to 1 is because of the way that we define "equal" mathematically. One version of this is that we state that x and y are equal if and only if, for all choices of a specific z > 0 you pick, I can show you that both (x-y) < z and (y-x) < z. A consequence of this definition is that if x and y are *not* equal, you will be able to pick a z > 0 such that (x-y) > z or (y-x) > z. If you try to find such a z, you'll find it doesn't work.

There is a good reason that mathematics has adopted this definition: it turns out that choosing a different definition of equal makes a lot of things break that you really don't want to be broken.

So, here's my suggestion to help make progress: if you don't like the definition I just gave of equal, propose a different one in which 0.9999... is not equal to 1, and we'll show you some of the consequences that you probably won't like. [Alternatively, if you think that under my definition they are still unequal, give me a choice of z that proves it.] Sound like a fair plan?

" if there are “lots of infinities” then do you believe that some are greater in size than another?"

This is a nice question - since infinity is not a number (NaN),you can't say one is greater than or less than another NaN. The less than and greater than operators just don't work that way - unless you define them to have special (ie not universal) behaviour for dealing with infinities.

What we can do is show that two sets have a different cardinality.
The set {1,2,3} has a cardinality of 3.
The set {1,2,3,...} has an infinite cardinality.
The set {x: where x is a real number less than 1 and greater than 0} also has an infinity cardinality.

To test whether these two infinities are the same we need some sort of method to compare them. (Can't use =, as it can only be used to compare numbers and infinity is NaN)

So we can compare by mapping one element to another.
You could compare the set {1,2,3,...} to the set {2,4,6,...} and find that they have the same cardinality (even though you can clearly see that between any two integers, say 0 and 100, there is exactly twice as many number in {1,2,3,...}) we can map 1->2, and 2->4, and 3->6.

So for every one element in the first set there is exactly one element in tue second set aswell. (And you can do the both ways!) and we include every element of both sets. So we use this to say the sets have the same cardinality. (Both are countably infinite)

We can't do this with the sets {1,2,3,...} and {x: where x is a real number less than 1 and greater than 0}. There is always some number missing from the second set no matter how you map them. So there must be more numbers, so it must he a bigger type of infinity.

We call this uncountably infinite. (Note, i'm just defining things and picking labels, the standard names... And also note, i couldn't even write the set of real numbers between 0 and 1 using the same notation... Because there is no obvious series to write it down, i could go 0.1, 0.01, 0.001,... But it is not clear that this infinite series contains all the numbers between 0 and 1. Infact it doesn't even contain all the numbers between 0.1 and 0...)

**So in conclusion : what we mean when we say 'greater than' in english and in maths is different.

I might also ask what you think 'controversial' means, does it mean wrong??

Ok, after thinking about it I have realized the error of my ways: Terr is no hope of convincing you guys that I am correct. Sorry for the last message, it did come off more personally than I intended, (I did intend for it to be aggressive) so do feel guilty about that. Thank you to everyone if you were trying to "help" me "understand" the concept, your time was appreciated, and I did find a lot of the thugs talked about to be interesting. It seems that our views on math are fundamentally different though, so I was not convinced by your arguments. As I no longer feel like wasting any of your or my time, I will call this an end, and will no longer participate in this thread. (and I may also silence it for a while as I have little to no self control.) btw, this was not a troll post, sorry if it came across that way, I was legitimately curious and thought of some fun theories during it. (Such as the one that abge liked)

Did you just give up when challenged? CSteinhart asked for a really specific answer, either find z or choose a different starting point. But instead you said, no i can't do these things, but i'm still not conceding?

That is kinda crappy.

To play devil's advocate, I'll continue tvrocks' argument. Because in this case he may have an answer.

Let's give the "1/infinity" that tvrocks talked about a name, and call it epsilon. This denotes an infinitesimal number with the property that, for all positive, real-valued x, epsilon < x. So my answer to CSteinhardt is: z = epsilon. Any number in the sequence 0, 0.9, 0.999, 0.9999 is smaller than 1-epsilon, so its limit should be smaller than 1-epsilon/2.

tvrocks: "Terr is no hope of convincing you guys that I am correct". No, not without a solid mathematical argument, where you start with your axioms, and follow with a rigorous proof. If you're going to post a comtroversial idea, you should have something very solid to back it up with.

Well put Basvan, if you define 0.999... taht way and use infinitesimal numbers as they are defined in nonstantdard analysis, tvrocks argument is actually right. Whether he was unable to convince us, it's because he just hsa not enough mathematical formation to do so.

But I want also point that, maybe the big error of thsi thread (not only tvrocks) is that maths are not about "convincing" others. It's rather about getting conventions about definitions and axioms. Tvrocks wants to be against conventions because his "common sense" which, in spite of what I said in my previous comment, actually has a slight smell of "mathematical intuition", tells him to do so.

Now, "math intuition" is very dangerous (though can be very helpful too, since it's also usually the " genius insight") when it comes to facing real problems.

The point here is that we are not discussiong about a real problem here, but about axioms and definitions, and so it 's actually a pseudoproblem, where our "intuition" simply won't be "right" or "wrong". An instance of what I'm saying is the case of th riemannian geaometry which is equally valid than the usual, euclidian one. It's not that one is right and the other is wrong, both of them are valid and active and useful areas of mathematics.

And just to freak you all out...

did you know that using the *usual* axioms and defs o mathematics (including an axiom htat could take a thread much longer than this one called "election axiom") you can porve that you can take a boll the size of a bean, cut it into many slices, and then you can reassemble the slices so you get ball the size of the sun?

sorry for my dyslexia (or whatever), I meant
*porve -> prove
*boll -> ball