@tvrocks

Yes, I had a mistake in my mapping which principians corrected. The point is, a mapping can be made and I believe you even made one. If a mapping can be made, then it must be of the same size, so that seems to be in direct contradiction to what you are propsing

&it horde: 1. I'm if this was not clear (looking back at it it wasn't) I was trying to say that in every situation there will be a standard infinity that may depend on the situation. I agree with this idea, and agree that it is situational.

2. 1/∞ is actually the definition of an infinitesimal, which is why I told you to look it up. Just look up the link I provided and then look up John wallis. Anyway though, if we are talking about using algebra to solve this, I donMt think you'd have a problem with this phrase: 1/∞=0 multiply both sides by ∞, and it ends up being 1=0 which is obviously not true. This would mean that the original equation is also not true though.

3. I actually dealt with that specific "proof" in my response to you. Tell me this, when you multiply anything by 10, will it move the decimal point to the right, and make it so that there will be 1 less sigdig? If so what would that mean when you subtracted them? I will also leave these here 2.2(10)=22 22-2.2=19.8 . .17(10)=1.7 1.7-.17=1.53
4.35(10)=43.5 43.5-4.35=39.15 what about this concept is new, and what would make it not apply?

@abge: I had actually never heard of the idea of making a mapping before, however, my intention was to make a mapping purely out of the 0-1 part of the 0-2 one and show that they were exactly the same on both sides. If you agree that the 0-1 part and 1-2 parts are the same size, then wouldnmt you be able to make 2 separate mappings soley from the 0-2 one after dividing it into different parts and realize that either of them are equal in size to the first, and that only one would be needed while the other would remain untouched?

Thorif: the &it horde should be @thorif, I'm sorry. Check out the link from lay page that explained what infinitesimals were.

Uh, infinity/infinity isn't 1. So (1/infinity)*infinity would not cancel out the infinities to leave 1.

@tvrocks

The main problem is that you still trying to treat infinite series as if they are finite. This is a common mistake, as we don't intuitively understand the concept of infinity. These ideas of doubling, halving, etc infinite series just doesn't work.

@abgemacht: if you believe that there are multiple infinities, then it does. Do you think that?

@tr: If 1 ∞ divided by the same ∞ were not 1, then Wtf is it?

Infinity

@thorfi: you're right. 0.9999... really is equal to 1. What I'm trying to say without using too much jargon) is that real numbers are defined as equivalence classes of Cauchy sequences of rational numbers, where two sequences are equivalent if and only if their difference tends to zero. 0.9999... represents the sequence 0, 0.9, 0.99, 0.999... and represents the sequence 1,1,1,1... which are equivalent so they define the same equivalence class.

By the way, the digit manipulation proof really isn't a proof because you first need to show that all the digit manipulation operations are valid for sequences of rational numbers, and that's actually much more work than simply observing that 0.9999... and1 define the same real number.

over infinity.

Any infinity divided by the same infinity or any other infinity is an indeterminate form. So is infinity times 0, so the other part of what you said in "multiply both sides by ∞, and it ends up being 1=0" is also wrong.

As I've said, there are two types of infinities: countable and uncountable.

The problem is that your arguments here lack rigor, which is critical for mathematics to work. If you truly believe your theories have merit, then you need to create a rigorous framework that works with established mathematics.

@tr: why is the top one automatically bigger than the bottom? (As if it weren't, it would be 1/something.)

@basv: Heh. Yeah, the wikipedia page gets into it in way more detail.

@tvrocks:

2. Well, yes, 1/∞ = 0 is "not true". Because it's simply *not defined*. It's not "false", either. It's just not a thing. It's like writing "adshy2134kjkjsd7" and asking if that's true or false. It's neither, it's simply undefined. You can't mix arithmetical operations with different classes of numbers. If you want to do arithmetic with infinities, transfinite cardinals is what you want to go looking at. That's a whole giant field all of its own.

3. It's an infinite series. You don't "lose" any digits of precision by moving the decimal point - there's still an infinite number of 9s to the right of the decimal point. That's the whole point. But if you really want more detail, just go read the wikipedia page on the topic - as @basv points out it's not like that particular proof is all there is to it.

@abge: my arguments are based off of applying the standard laws of mathematics and common sense to come to the conclusion that they are not the same number. Why, when both are divided by 3, would 1 yield .333... WITH a remainder of 1 no matter how far you went while .999... Would yield .333 WITHOUT that remainder? Also, tell me this, why would the standard rules of mathematics, that apply in literally every situation (except this one it seems you're arguing) not apply here, and why would new rules that do not work in every other situation apply here? Do you not think that |2x|>x in all situations? (Unless x is 0.)

@tvrocks: You *never get a remainder*, whether you're dividing 1/3 or 0.9(repeater...)/3. There is no remainder. You keep dividing "forever", resulting in 0.3(repeater...) in both cases.

And when you say "standard rules of mathematics", you mean "the rules of arithmetic". Which, several of us have pointed out several different ways - arithmetic doesn't apply when you're talking about infinities.

@thorfi: 2. You were saying it was defined as being 0 earlier... Check out the infinitesmal article i linked to earlier, that literally is the definition of infinitesmal. Here are some highlights "The English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections. The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal."

https://en.m.wikipedia.org/wiki/1/%E2%88%9E#History_of_the_infinitesimal


3. You agree that there are different sized infinities, correct? If so you would see that the 10 x would have an ∞-1 number of 9s behind it while x woukd have the same ∞ from the 10 x (10 x is based off of it) but would not be subtracted by anythig. To set it up with the number of 9s in the x compared to the number of 9s in the 10x (note, this is only talkinf about the number of 9s behind the decimal point, tje total number of 9s in each of them does not decrease) then it would be set up as ∞=∞-1 which, if you subtracted ∞ from both sides, it would yield 0=-1 which is also not true. You coukd substitiute in a variable x for ∞ if you wanted and still get the same result.

Also, Mathematics doesn't work using common sense. Common sense has nothing to do with it in the least.

@thorfi: set up 1/3 as long division and you will see that there will always be a remainder. Also, weren't you trying to use arithmetic to prove that .999...=1? Why does it only not apply when you don't like what it would mean if it did and also, why, if it truly didn't apply, would it not? Is it because the rules that work in every situation are not good enough to use with it?

@tvrocks

2. No. I said the same thing earlier - the thing is undefined and has no meaning.

3. No. The total number of 9s to the right of the decimal place is infinite and continues forever, regardless of which line you're on. But seriously, for far more rigorous proof, just go read the damn wikipedia page. I'm not copy pasting the whole thing here.

You can do it as a long division - you never actually stop, so you never *have* a remainder. Yes, you carry one, but that never actually finishes, so you never actually have a remainder.

@tvrocks

If this stuff is truly of interest to you, I recommend you see how far you can get through this:

http://www.math.brown.edu/~jhs/frintch1ch6.pdf

Feel free to ask questions.

Don't mistake the process for the result. The *result* of the long division is 0.3(repeater...).

Look one level up on the website for more info:

http://www.math.brown.edu/~jhs/frint.html

2/2=3/3=4/4=723/723=1 are all common sense things right? 13-13=0 is also common sense right? Math is based solely off of critical thinking and applying universal truths to find a result, which is what i am doing in this situation. Tell me this: if, no matter how far out you go with .99... Would it stop even if it theoretically did become 1? The only explanation for it becoming 1 is that as it became big enough the difference between it and 1 woukd become non-existent, however, even if that were true, it would still continue and would mean that .999... Were actually greater than 1. If a digit in which the difference between the 2 things becomes non-existant does not exist, then it will never become 1, however, if a digit in which it becomes 1 does exist, then it would become 1 but then keep going and become even bigger than 1. Unless it became 1 on a theoretical endpoint, which does not exist, it would continue and it would mean that .999... Is actually bigger than 1. I will now be going to bed and will not respond to any more statements.

OK, that's going in my profile.

Seriously, I think it's great you're interested in this stuff, but you need to be willing to learn about what has already been done as well as come up with your own ideas. Read those links I gave you and get a better handle on the foundations of math.

Oh dear. Absolutely none of those things are "common sense", as far as mathematics goes. Actually getting to grips with what arithmetic is, or even what integers are, let alone real or other numbers, in an actually properly defined way, is not common sense. It's a giant mathematical field of endeavour, that some of the greatest mathematical minds of history and today have struggled with. And if you want to really break your brain about number theory, try looking at Gödel's Incompleteness Theorems.

tvrocks I will give you one last reccomendation, forget common sense, it has no meaning at all, and it won't help you for anything in life except trolling

Mathematics is a language that we invented . As with all languages sometimes there are no available words to accurately describe situations. This is one of them .

EG there are lots of infinities count by ones or by 2's does that mean counting by 2's is half the size of the infinity described by counting ones ..

1.2.3.4.5 ect
or
2.4.6.8 ect

No we simply lack the language to describe this.