I don't care what anyone says. 0.999... isn't 1. That's retarded. It's like those math tricks they show in school that rely on bad logic. Because if it does repeat forever then at any given point you get 0.99999X where X is always a nine. So say you get somewhere to an "end". The digit would, for example be 0.99999999999999999. Now it may be fractional, it may basically be 1, but 0.9 and 1.0 are still distinct and discrete entities. And if it never ends then you continue to get 9s to an endless degree but never hit 1. This meme has gone way too far and people treat it way too seriously.

0/0 = 1 [math]btw[/math]

>>16779219
Agonizing over digits is the ultimate example of not seeing the forest for the trees. When you're writing 9s ad infinitum you're not doing anything related to any numbers, you're just drawing shapes.
"Infinitecimals" is actually the kind of thing that relies on "bad logic", a glitch in human thinking.
You should first realize that numbers don't exist to begin with, so it's enough a miracle already that we can think mathematically to all.

>>16779219
Limit of (X-1)/X = 0.999... and also 1 as X approaches infinity.
A function can't have 2 values for the same X so 0.999... = 1.

How many different ways can this be proven to you until you just accept that it's true.
It's not even that crazy. It just means 0.000...01 = 0.

>>16779219
If you're talking about analysis, equality of two real numbers [math] \{q, r\} [/math] means for any arbitrary [math] \epsilon > 0 [/math] you can choose, [math] d(q, r) < \epsilon [/math]. An "arbitrarily small" gap is expected and accepted. Real numbers as a purely mathematical model is weird and physically impractical. Blame the first mathematician who constructed it that way.

>>16779229
well said
>>16779202
the way I understood is
>what is infinitesimally close to 1?
>what is so close to 1 that nothing can ever be close to 1 than it?
>what is the closest thing to 1 that exists?
it's 1 itself of course

>>16779231
I mean logically, what even would be the length/volume/size of an infinitesimal besides 0?
The way the continuum is supposed to work is that you can always find further numbers between any two, right, but if we really want to shoehorn in the notion that there's this one number just before 1 for example, than that would essentail that the distance between then is 0. From this angle we can give the two a separate identity, but the properties at the end of the day are the same.

>>16779202
what is so close to 1 that nothing is between it and 1?
try to find another number except 1 to answer this question.

>>16779231
>has a 0.000...01
>no this is the same as just 0
So somehow you want me to ignore the clear little one value? I said in my post "it may as well be 1" but it literally isn't. There is some other value there. Sure we can round them off. But that 0.00000000000000whatever1 still has a little one in some place. There is some kind of value there as minute as it may be. It's one thing to say we don't need that level of precision and round it off. To claim it's some law and that it absolutely makes sense and should be this way is ridiculous beyond belief.

>>16779257
we just don't arbitrarily pick a "little" 1, when we got tired of counting
we pick the final 1
the 1 when nothing is between it and 0
only there is no final 1, because infinity never ends
only 0

what is the closest thing to 0 that exists?
0 itself.
that's what this means.

it's not equal to 0.
it IS 0.
it's just another way to write 0, a bit more elaborate.

>>16779257
If you divide 1 by 0.0...01 do you get any meaningful output?

>>16779202
this is like the breakfast test but for midwits instead of retards

>>16779257
>ignore the clear little one value
Not a value, that little one is a symbol.
There are always so many implicit assumptions that underlie this kind of reasoning that it's not even funny.
Alright if it's that self-evidently logical that there could be just a "number" that we can tell that it ends with a 1 but we can't ever get there then think about this: you have a the volume of a cube in space and you remove one of its bounding sides and the volume is projected that way into infinity. Now you have an object that is basically an infinite column in such a way that it has a beginning but not an end. As analogy what you're asserting is that it still makes sense to say that the end of that infinite length, there's still an other end and it's green. The notion that it doen't have and end by virtue of it being INFINITE (which literally means not finite, not ending) but we still describe that end is absolutely absurd. The defense you're still making is that "but I can imagine the end being green!", but the way you represent something in your mind implies nothing about the things itself, so don't confuse the two.
If according to you 0.00...1 makes sense then so should the following:
...1111... (a number that doesn't have a beginning and doesn't have an end, so it's infinite, but you can see FOUR ones omg it must be something)
1111 (it's actually just three 1s because the first and the last are the same, this number wraps around)
0.000...050...1 (there's both a 5 in there that you can never reach and a 1 at the very end there's two infinities in the middle of the number)
0.00...1...2...3...4... (and so on, all natural numbers occur as contiguous segments but you can never reach any of them starting from one of those)
You can try to find a rebuke to these notions but hinthint if you manage to find the reason you will also end up realizing why your idea about 0.00..1 being anything is stupid also.

>>16779219
Where in math did .999... ever arise from in the first place? It's not 1, so what? Goblteygook =/= 1. 1 actually exists as an adjective describing a quantity, like 1 apple(s). How exactly did you even arrive at this .999... to begin with to even claim that it exists?

>>16779219
all these "people" speaking of the """""""end""""""" of a fucking infinite (latin for motherfucking UNENDING) sequence of decimal digits.
protip, and the founding stone of the real numbers: if you can't find a number that goes between other two, then your other two numbers are one and the same
grow up
you learn about this at 15 tops
god

>>16779299
Absolutely based. All these people arguing about .999 and 1 are putting the cart before the horse. I can see why this thread is the ultimate troll.

>>16779231
another one
jesus fuck, there is no 0.000...001
where would you place the fucking one?
it is an infinite sequence of 0s
all zeroes
no one in sight
0.000... = 0
0.999... = 1
commit division by zero and divide yourselves by two

>>16779299
Nice dubs
If .999... exists, then it equals 1. If .999 exists, then it doesn't equal 1. Both are vacuously true!

>>16779304
>jesus fuck, there is no 0.000...001
And there is no .9999... Either!

I'm glad that the majority of people ITT are not retards.

>>16779304
>there is no 0.000...001
It exists insofar as I have defined it as existing just now.
It is what you get if you take 1 and subtract 0.999... from it. Because this value "doesn't exist," we can either say it's "0" or some other "null" value.
In any case, it acts in every way like 0 so it is 0.

>>16779304
>>16779325
Aren't you two saying the same?

>>16779325
>1-0.99999999 is 0 instead of 0.0000001 because... idc theres some 1 there bro just ignore it so i can say its just equivalent
No. Fuck you. And 1 is a prime too. This shit is so fucking retarded. Math needs to be corrected and made to work properly. I'm tired of this flimflam passing as legitimate.

>>16779336
Why are you so pressed when >>16779299
Just proves that .999... doesn't exist anyways, and >>16779307
Proves that you're both right

>>16779331
Yeah but he wants to be argumentative and so do I.

>>16779336
>1-0.99999999 is 0 instead of 0.0000001
That is incorrect. There's a difference between an infinitely repeating value and a terminating value.
0.333... != 0.333.

>>16779346
Just adding infinitely many 9s to something still will never be one. You just get a value closer and closer to 1. Because soon you'll have .9999bajillion quintillionths of 1 and basically have it. But approaching a value ≠ it is that value. Same for .333... and the rest. Mathematical woowoo with no real world value. It's nonsense and I refuse to believe it. If anything all that does is tell me mathematics needs reform and something is wrong somewhere. And maybe one day in the future someone will discover correct math. The same way singularities throw physics for a loop but ultimately one day can be solved with improved understandings.

>>16779355
>But approaching a value ≠ it is that value
>t. Doesn't understand limits.

>>16779367
>goes on forever
>wait no it has limits
This is what I mean. Complete woowoo gibberish. This is basically the singularity of math. Something breaks down and leads to absurdities. Except instead of going "maybe we should reevaluate our mathematical understandings" we have clowns who sit here saying why it's true and totally makes sense despite it flying in the face of even basic reason.

>>16779369
>>goes on forever
>>wait no it has limits
How many 9's can you tack on before 0.999... is greater than 1?

>>16779355
There was already a Foundational Crisis of Mathematics in the early 20th century (Hilbert vs Brouwer) right around the time the theory of relativity made rounds.
Even then two actual sides that people who actually understood what they were talking about mathematics took, was either that mathematics is nothing but manipulation of strings with manipulation rules (formalism) and the other is that mathematics is not an inherent property of objective reality but a mental activity so we must construct things in order to prove them (intuitionism).
The result of that foundational debate was that most mathematicians don't care and are midwits like you.

>>16779369
Basic reason is that there are no infinites.

>>16779374
None. You will never reach one. You either terminate at a value less than 1 or you just have an endless row of 9s all the way and it still won't make it to 1. Now if we use rounding we can say "its basically 1 because who cares about trillionths of a trillionth of 1". But to call them equivalent or the same thing? No.

>>16779382
>None. You will never reach one
Yup. That's what we call "a limit."
This thing you asserted was a contradiction is not a contradiction.

The value of a function as x approches infinity is exactly equal to the limit. There are even real world consequences of this being true (see: Zeno's Paradox and its solution).

0.999... = the limit of (x-1)/x as x approaches infinity which is also equal to 1.
These are all the same number. Cope.

>>16779382
How about we start from 1 because that should be a shorter route
How much should I step so that I don't step too much towards 0? Isn't any displacement basically already more than 0.00...1? Wouldn't stepping 0.00...1 imply I don't step at all?

>>16779388
No. There's a one there. We have the number system for a reason. You don't just ignore it because it's small. We can do rounding. But what is the logic here?
1 and 1.0 or 1.00 or 1.000... are equivalent for example. But to have any value like ...01 shows a 1 is there. And .999 either terminates at less than 1 or never ends at all so you still never reach 1.

Because otherwise how the fuck are you gonna say 0.999... is equivalent to 1 when 1 would be 1.0000000... with literally nothing after it except for null values.

>>16779202
>X equals not X
Into the trash.

>>16779391
Yes, there's a number system and dotdotdot is not part of it, forcing an "infinity" into it is abusing it in way it is not fit for, you're mistaking a direction for a completed totality.
Saying "there's a 1" is not meaninful after the point at which you lose the trail and jump somewhere arbitrarily.
The rule of this number system is that the value of a decimal digits is based on its position in the expansion: if you simply just write a digit saying it's "somewhere" then you're the one violated the foundational rule of the system and making empty meaningless statements. This is a very real and important retort against you that you should address somehow, just because you keep repeating your counterarguments that muddle this question make you not understand it or make sense to you, just because you keep doing that doesn't mean that what you're saying in any way counters or neutralizes the things that are said to you.

>>16779391
Look, dude, if you insist on this "problem" being solved then doing so is trivial. We'll do the same thing mathematicians always do and make some shit up.

We'll define some number (let's say q) to mean "the difference between 0.999... and 1." So 0.999... + q = 1. 1/3 then becomes 0.333... + (1/3)q. We can evaluate (1/3)q as 0.00...0333...+ (1/3)q.
You starting to see the problem yet? You really want math to be done this way?

>>16779414
That's what physicists do though, not mathematicians

>>16779417
Everything mathematicians do is made up. It's math.

>>16779421
That's not an argument. Mathematics still follows an innate logic and mathematicians don't employ handy little patches over gaps that they can't explain such as what physicists do and what you just advised just now.

>>16779202
The limits and analysis part doesn't matter at all for proving 0.9... = 1, which follows algorithmically from the definition of an integer in decimal notation. 0.9... = 1 is just a simple pedagogical tool to relate something clear and true to other, less clear, applications of limits and convergences.

>>16779428
What's the square root of -1?
Hell, even the concept of negative numbers was just pulled out of someone's ass as a "patch." My invention of some infinitesimal unit, "q," is almost exactly as justified as the imaginary unit.

>>16779435
No, you think that instead of saying "thing", just saying thing-but-one-letter magically makes any difference. It does not.
The examples you mentioned at least exteneded the capabilities of mathematics while still maintains the consistency of the system. Your idea, while maintaining the consistency, makes no functional difference whatsoever, the problem I had with you is that you can't make a distinction between superficiality and substance.

>>16779447
>Your idea, while maintaining the consistency, makes no functional difference
It makes the difference between 0.999... and 1.
Just FYI, there are number systems that do almost precisely what I did but with slightly more rigor. So you can take the problem up with them.

>>16779450
I meant functional difference as in synergizing with other elements of mathematics and opening up new possibilities, but alright then, if there are systems like that it means I'm ignorant.
But could you please tell me what those number systems are?

>>16779459
Surreal and hyperreal numbers.
They don't really go as faras I did in the direction I took them. Rather they just kinda define infinitesimals as existing and play around under the assumption that they do, indeed, exist.

When people say 0.999... = 1 it's assumed to be understood that we're talking about the real number line, which infinitesimals are not technically a part of.

>>16779466
Well, yeah, I really must admit I'm not at all well-versed in how surreals or hyperreals influence the discussion ITT, but based on a cursory overview it does seem like it's still the same intellectual pyramid scheme as with infinities just in an orthogonal direction. But at this point I'm knocked out.

>>16779478
Don't worry too much about it. It's a niche field with some topology use cases and that's about it AFAIK.

The "q" notation I invented just now is how we see that infinitesimals are either "not real numbers" or "equal to zero."

Take the (1/3)q example:
(1/3)q equals 0.00..333... + (1/3)q. We can simplify this to (2/3)q.
(2/3)q then equals (1/3)q which the only real number that could satisfy this condition is 0.

So either infinitesimals are zero and we're back to 0.999... = 1 or infinitesimals just don't exist on the real number line and we have to determine some other sets of rules they follow.
In any case, 0.999... = 1 as long as we stay within the domain of real numbers.

>>16779478
>>16779489
To add a couple clarifying tidbits:
We can use similar logic to show q/5 = 2q. Again, impossible with the real numbers that are not 0.
And (2/3)q would equal both 0.000...333.. and 0.000...666... at the same time. Not a real number unless these are all zero.

>>16779293
Its a symbol that represents a value. Something with no value is represented by a 0. Is a 1 a 0?

>>16779478
>>16779489
Hyperreals treat 0.9...9... as = 0.9... but 0.9...0 or whatever as < 0.9...
They claim to be an extension, a new infinite string after the first infinite string, but in practice they just break one string infinite with an unknown length of unknown digits.

>>16779497
Yes, according to Frege.
Unironically he defined 1 as the set containing 0.
And before that 0 is the empty set, according to him.

Besides that I sense some miscommuication here. 0 is a value, but if you interpret "value" as wortwhile desirability or degree of importance or esteem then you could view 0 as the lack thereof, but mathematically 0 is a quantity thus a value.
And since you're not engaging with my 99% of the arguments in the post you responded to, I won't try dig open your implicit (and faulty) reasoning that you yourself can't be arsed to explain either.

>>16779505
I represent the left side of the bell curve. A 0 is not a 1. Something that is 0.01 is not valueless, no matter how many 0s comes before the one. I'm sure a midwit can come up with a hundred paragraphs for why
>actually 1 isn't 1 its 0
And I wouldn't be surprised, but 1 is still not 0 is still not 1

>>16779507
NTA but following your exchange back a bit, you're choosing the hard way to look at it. If you add 0 + 0.9..., how do the nines all flip to zeros? The easy way is to just subtract 1 - 0.9..., there's never a 1 or a 9 at any place in the answer, only 0

>>16779507
Oh wait, you're actually making an important point accidentally.
For example if we say that we have 2 square inches, that means we have a measure of 2 multiplied by the length of an inch multiplied again by the length of an inch.
If we divide 2 square inches by the length of an inch, what do we get back? Just one side's length, 2 inches. We removed one factor of inchness.
What happens if we divide again by "inch", and remove another factor of inchness? We just get 2. And what does that mean at that point? Nothing.
Numbers by themselves have no value.
But the value things can take are quantified with numbers. As long as numbers are not multiplied with something, they are kinda sorta fictional.
The two tails of the bell curve say that numbers don't exist, it's midwits who think the world is made of pure maths.

>>16779503
Hmm,well... it sounds a lot like tetration then, in that it's reached the point where self-reference means it devolves into itself without actually constituting something unique.

>>16779202
1/7 = 142857/1E6 + 1/7E6
= 0.142857 + 1/7E6
= 0.142857 142857 + 1/7E12
= 0.142857 142857 142857 + 1/7E18
:
= 0.142857... + 1/inf
= 0.142857... + 0
= 0.142857...

>>16779231
now do it purely within the rationals

>>16779257
name me the place that 1 is on, for example in 0.1237, the 7 is at the 4th place after the decimal point

>>16779466
>Surreal and hyperreal numbers.
no, due to the transfer principle 1/3+1/3+1/3=9/9=0.999...=1

>>16779634
yeah

>>16779669
>9/9=0.999...
No it does not. That's retarded math equivalent to "9+9=99" and doesn't apply here just because you plug it in to the middle of something that is otherwise true

>>16779676
What's 1/9 in decimal?

>>16779676
1/9 = 0.111...
+
8/9 = 0.888...
=
9/9 = 0.999...

File: 999.jpg (46 KB, 540x499)

46 KB JPG

>>16779202

>>16779202
>compare the first digits between 1 and 0.999…
>theyre not equal
woooooooow who would have guessed these 2 numbers would be different.

>>16779835
>>16779836
>but steel is heavier than feathers

File: piaget-cons-test.jpg (88 KB, 1024x768)

88 KB JPG

>>16779836
>compare the first digits between FF and 255
>theyre not equal
Yes, perfect sense, the same content gets diferent depending on the form that contains it, AND it doesn't mean you're at the level of a 6 year old to say that.

>>16779687
>>16779761
This is some 2+2=5 bs

>>16779927
Not really because integer arithmetic is natural and intuitive. You can actually put two pebble and two pebbles together and count it and see that it's four pebbles.
But "real" numbers are actually artificial, contrived and ultimately imperfect if not straight up faulty.
People who want to assert that 0.999... is not 1, are doing so because they don't actually want to take the effort to think about it, rather just look at it and stubbornly defend the hunch they had at first sight. That's not a valid form of thinking to be respected.
The reality is that 0.999... is an informal notation, and it is not a well-formed expression, it is a corner case of the decimal system.
The incorrect assumption that people who don't understand this ITT make is that "if I see digits then they must be meaningful", then proceed to invest themselves into their prima facie impression and make excuses.

>>16779953
On the flip side what I see ITT are people trying to make fake numbers meaningful by equating then to real ones. 0.9..... is meaningless, but its also not 1. 9/9 is nothing less than whole aka 1.
To declare that 1 is not 1 is you trying to create meaning

>>16779953
>corner case of the decimal system
It's more like a main theme of the decimal system. Every rational number with a denominator that divides the base has two decimal forms. Only the sketchy numbers like irrationals and rationals that don't divide the base don't have two.

>>16780014
You're right, most of us did fall into that trap.
Thought to describe what the critcism has been withotu buying into the false premise: when people try to make 0.999... a thing they can't define it in a way that produces something that is distinct from 1.

[math] \displaystyle
1= \dfrac{3}{3}=3 \cdot \dfrac{1}{3}=3 \cdot 0. \bar{3}=0. \bar{9}
[/math]

>>16780021
Yeah, I didn't phrase that quite right, but realistically how often do people write 7.4999... to mean 7.5? Almost never, so clearly one of the decimal forms is canonical.
Besides that, the infinite expansion is a quirk of reals, not necessarily idiomatic to rationals which aren't simply just regular old reals among the irrational reals, they are a different concept that even though can be extended into reals, rationals and reals are not bijective.

>>16780028
3/3 does not equal not 3/3. This is creating decimals out of nowhere and betrays conservation of numerical mass. When is this applicable in the real world?

>>16780031
[math]
0.5_{10} = 0.333..._7
[/math]

>>16780028
Thanks [math]\LaTeX[/math] bro.

>>16780032
>3/3 does not equal not 3/3
your Fields medal is in the mail

>>16780037
Thank you now when is this kind of 'math' applicable in reality?

>>16779219
>So say you get somewhere to an "end".
He didn't.

>>16780033
The decimal points should give it away that you're still using reals in your answer.
When I said rationals I meant rationals.
[math]\frac{1_{1}}{11_{1}}[/math]

>>16780031
I'm just saying if you use decimal notation to compare two numbers, such as 0.5 and 1, you're implicitly accepting the decimal definition of an integer, which is an integer followed by arbitrarily many zeros. Otherwise you can't make the comparison to begin with.

>>16780043
[math]
11_1=1 \cdot 1^1 +1 \cdot 1^0 = 2_{10}
[/math]

>>16780043
[math]
0.333..._7=\frac{3}{7}_{10}+\frac{3}{49}_{10}+\frac{3}{343}_{10}+\frac{3}{2401}_{10}+...
[/math]
call back when you find the moment it isn't rational

>>16779891
>retard thinks he can change bases to prove me wrong
nice try einstein

>>16780252
That guy's been doing that shit in every vaguely math related thread since late August.
Most likely his coursework introduced him to modular arithmetic and he thinks the fact that "10" can represent basically any number you want it to represent is profound, esoteric knowledge that 90% of this board haven't already been introduced to.

>>16779202
>0.999... = 1
The problem with this equation is that the RHS is defined but the LHS is a fantasy based on incoherent fairy tales about "infinite sets" and "limits" of "infinite sequences". If you could formulate the equation without appealing to such fairy tales, I will accept it.

>>16780381
You don't need any of that. In decimal notation, an integer X is defined as X.000... The equality follows algorithmically, even if you're working in a system bounded by the physical constraints of the universe, even the life or the attention span of a human.

>>16780381
Well said Brother!
The time fast approaches when we will be strong enough to storm the citadels of the infinity loving Heretics and retake the HOLY LAND of MATHEMATICS for the GLORY of GOD!
Blessed are we of THE ONE TRUE FINITE FAITH!
For we do GOD's will!

>>16780100
That's not what I'm saying, I'm trying to make a distinction:
[eqn]
(i)\;\;\;\;\;\;\;\;\frac{1}{3} = 0.333... = x\\
(ii)\;\;\;\;\ulcorner\frac{1}{3}\urcorner \not = \ulcorner0.333...\urcorner\\
(iii)\;\lvert\ulcorner\frac{1}{3}\urcorner\rvert = \lvert\ulcorner0.333...\urcorner\rvert = x \\
(1)\; \Gamma_{\mathbb{Q}}\vdash \ulcorner\phi\urcorner{} \iff \exists{}n:\mathbb{Z}\;\exists{}d:\mathbb{Z} \; [d \not = 0 \wedge \ulcorner\frac{n}{d}\urcorner \equiv \ulcorner\phi\urcorner ] \\
(2)\; \Gamma_{\mathbb{Q}}\vdash [\ulcorner\phi\urcorner{} \implies \lvert\ulcorner\phi\urcorner\rvert \in \mathbb{Q}] \\
(3)\; \Gamma_{\mathbb{Q}}\nvdash [\lvert\ulcorner\phi\urcorner\rvert \in \mathbb{Q} \implies \ulcorner\phi\urcorner{}] \\
(4)\; \Gamma_{\mathbb{Q}}\vdash [[\ulcorner\phi\urcorner\ \wedge \lvert\ulcorner\phi\urcorner\rvert = \lvert\ulcorner\alpha\urcorner\rvert] \implies \lvert\ulcorner\alpha\urcorner\rvert \in \mathbb{Q} ] \\
(5)\; \Gamma_{\mathbb{Q}}\vdash \ulcorner\frac{1}{3}\urcorner \\
(6)\; \Gamma_{\mathbb{Q}}\nvdash \ulcorner0.333...\urcorner \\
\therefore \\
(7)\; \Gamma_{\mathbb{Q}}\vdash \ulcorner\frac{1}{3}\urcorner{} \implies \lvert\ulcorner\frac{1}{3}\urcorner\rvert \in \mathbb{Q} \\
(8)\; \Gamma_{\mathbb{Q}}\vdash \ulcorner\frac{1}{3}\urcorner{} \implies \lvert\ulcorner0.333...\urcorner\rvert \in \mathbb{Q} \\
(9)\; \Gamma_{\mathbb{Q}}\nvdash \ulcorner{}0.333...\urcorner{} \implies \lvert\ulcorner{}0.333...\urcorner{}\rvert \in \mathbb{Q} \\
(10)\; \Gamma_{\mathbb{Q}}\nvdash \ulcorner{}0.333...\urcorner{} \implies \lvert\ulcorner\frac{1}{3}\urcorner\rvert \in \mathbb{Q} \\

[/eqn]

I hope you don't mind that I made my argument painfully explicit. There is a difference between the formula/syntax/representation and the thing denoted, which you also implicitly agree with since you're trying to make your point by varying bases.

>>16780100
>>16780484
We absolutely can attribute a type to a syntactic form, and one type can be a more immediate candidate logically than others, so that's why I'm saying that the formula [math]\frac{1}{2}[/math] is more rational than 0.5 which is only rational by virtue of its denoted value, we only ascertain that after multiple potential steps.
It doesn't matter what base you choose, semantic and syntactic integers maintain their rules, so you'll only be able to form "proper" syntactic rationals with those, the point I'm making still stand, there is an important meaningful distinction between form and content, and most people such as you aren't really aware and conflate those notions when reading what I or others say and start empty arguments.

>>16780252
Ahah! So you see, numbers of different categories cannot be trivially compared, be that one being finite and the other infinite or being in different bases. So you agree that we can't compare neither FF and 255 nor 0.999... and 1, thank you, that is exactly that point I was trying to make.
>>16780372
>Most likely his coursework
Lol you're probably confusing me with another anon, I haven't formally studied maths since high school.

>>16780401
Not true. X=X.000... arises from manually performing the long division algorithm for X/1 to any precision. You will not find a similar result showing that X=(X-1).999..., because .999... doesn't exist.

>>16780514
Division has nothing to do with it. The decimal notation of an integer is defined as having arbitrarily many zeros in every place after the decimal. Otherwise you can't subtract 0.5 from 1 because 1 wouldn't have a tenths place.

>>16780531
So what's 0.999... in fractional notation?

>>16780533
Who cares?

>>16780534
It directly relates to whether it is equal to 1.

>>16780531
>The decimal notation of an integer is defined as having arbitrarily many zeros in every place after the decimal
That's incorrect. You can't just define things to be true. Your attempted definition would lead to the following conclusion: 1=0.999...000...

>>16780533
9/9

>>16780546
Perform the long division. You'll get 1.0000...., not .999.... No division will result in the latter.

>>16780544
>You can't just define things to be true
Yes, you can, it's called axioms
>Your attempted definition would lead to the following conclusion: 1=0.999...000...
No, because 0.999... is defined as going into infintiy and beyond, there's no end to put zeros to, 0.999... doesn't have an end.

>>16780548
>it's called axioms
Your axiom is false and to be rejected.

>>16780547
You get both 1.000... and 0.999..., but you have to look at it from a different angle to see each.

>>16780536
What does an alternate notation have to do with anything? If a - b = 0, then a = b. This is simply the additive identity. Otherwise addition fails.
In decimal notation,
0 is defined as 0 with arbitrarily many zeros in every place after the decimal.
1 is defined as 1 arbitrarily many zeros in every place after the decimal.
0.999... is defined as 0 with arbitrarily many zeros in every place after the decimal.
1 - 0.999... gives 0 with arbitrarily many zeros in every place after the decimal. This goes on until you stop paying attention, die, or the universe runs out of storage space. That's the definition of 0.

>>16780553
We agree.

>>16780544
It's literally how an integer is defined in decimal notation.

>>16780553
>0.999... is defined as 0 with arbitrarily many zeros in every place after the decimal.
Lol I mean *arbitrarily many nines

>>16780550
That's not how axioms work. You can arbitarily pick anything as axioms for a system, see where that takes you and ideally check whether it procudes a consistent and complete system.
One famous example for a system that isn't consistent is mathematics.

>>16780561
So what you're advocating for is the invention of an entirely new system in which 0.999... != 1.
I suppose you're allowed to do that but that's outside the system of real numbers.

>>16780563
>So what you're advocating for is the invention of an entirely new system in which 0.999... != 1.
No. 0.999... is 1.

>>16780564
Then what's this whole business about integers having infinitely many zeros about?

>>16780567
I think there are three of us talking currently.

>>16780567
NTA but you don't need infinitely many zeros. The equality holds no matter how severely finitist your perspective is. In decimal notation, subtracting 0.999... from 1 gives you the definition of 0.

>>16780568
>>16780577
Well the "axiom" I was disregarding was that integers have infinitely many zeros.

>>16780578
Integers are defined, in decimal notation, as having arbitrarily many zeros in every place after the decimal. If you want to believe in infinity, that's on you.

>>16780584
No they're not. They are shown to have arbitrarily many 0s as a result of the division process. You should know not to add unnecessary axioms when they can be derived from calculation.

>>16780591
Division has nothing to do with anything.

>>16780595
>Division has nothing to do with anything
Right, might as well do away with decimals entirely.

>>16780597
Who cares? The topic is whether 1 = 0.999... in decimal notation.

>>16780595
>>16780597
Anyways we're getting away from the original discussion. Whether you want to take as an axiom or not, we can't do the same with .999... It is a complete nonsense scribble with no calculations justifying its existence.

>>16780606
It's the integer 0 with arbitrarily many nines in every place after the decimal.

>>16780607
No, for example 0.9 doesn't fulfill the rule that 0.999... implies. Saying it's just simply an "arbitrary" amount is inaccurate.

>>16780607
>Arbitrarily many 9s
Incorrect . You can have arbitrarily many 0s with 0.000... because 0=0.0=0.00=...
But .9=\=.99, and so on
You

>>16780611
It does if everyone dies after the first 9 and the universe collapses. Otherwise, you're lying and is wasn't "arbitrarily many," it was fewer than that.

>>16780614
The inequality 0.9 ≠0.99 is completely irrelevant.

>>16780619
Do you know what arbitrary means?

>>16780625
Do you know what arbitrarily many means?

>>16780629
Yes, it means it's up to anyone's individual judgement or whim.

>>16780634
You can go on writing the known digits of 1 - 0.999... up to your whim or judgement. None of those digits can ever not be 0. That means the answer to 1 - 0.999... is 0 followed by arbitrarily many zeros (you decided when to stop on your whim) which is the definition of 0, in decimal notation.

>>16780629
>>16780634
>>16780625
this would never happen if we were using base 15

>>16780644
You can't even show that .999... exists, and now you expect me to use it in a calculation? Stop being ridiculous. Even if we assume that this can be done, the subtraction process requires us to start at the right of both numbers, which is impossible.

>>16780644
in real math the nines wouldnt stop just so you are aware boys

>>16780606
>.999..
>no calculations justifying its existence.
[eqn]
\frac{10^x-1}{10^x}
[/eqn]
Where x is any natural number.

The output is 0.9...9 where x is the number of 9's. As x approaches infinity, the limit is exactly 0.999...

>>16780658
Be that as it may, it's easier and more fun to prove it from the perspective that they might.

>>16780665
>Limits
>Infinity
Heh, sorry kid, but the most math I do is algebra 2. That's all I need for my high 6 figure finance job. Don't you think if that stuff was real, that it'd be required for my job?

>>16780671
>Don't you think if that stuff was real, that it'd be required for my job?
Literally anyone could say this about any job for any level of math.

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>0.9 = 1
fukkin r*tard lmfao

>>16780665
i said wouldnt stop i meant forever not infinity anon you fucking clown anonymous

>>16780684
Fine. If you keep increasing x "forever" you get 9's that go on "forever."

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>>16780677
Any job, sure, but not a high six figure job like mine. Money talks, and BS walks. You'll learn that eventually, kiddo.

>>16780686
if you divide one by three you get 3 which doesnt stop
but if you used base 15 and divided something like number one by three you get a nice and round 0.5

>>16780687
>not a high six figure job like mine
Yeah. Your Onlyfans obviously didn't teach you how to proofread your post.
You intended to say the opposite of this.

>>16780695
Heh, like I said, money talks. Just the mere mention of my income has complete strangers scrambling to correct me for free. All because they think I might hire them.

>>16780701
Hire me to do what? Hold the camera for you? Cuz I'm sure as hell not holding the cucumber.

>>16780701
i dont care look at this post >>16780689

>>16780705
Hire you to get paid. That's all anyone gets hired for. You'll learn one day kiddo

>>16780719
Heh, there's no base 15 in my finance job. There's base 10 and base 2 occasionally.

0.99999... is a merely a decimal representation of 1/3 x 3, or 0.33333... x 3.000000...

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>0.9 =\= 1 (100% true)
>0.99 =\= 1 (100% true)
>x2, 200% true
>2/2 truthes confirming absence of equality to 1
>0.999 =\= 1 (100% true)
>x3, 300% true
>3/3 truthes confirming absence of equality to 1
>0.9999 =\= 1 (100% true)
>x4, 400% true
>4/4 truthes confirming absence of equality to 1
>0.99999 =\= 1 (100% true)
>x5, 500% true
>5/5 truthes confirming absence of equality to 1
>0.999999 =\= 1 (100% true)
>x6, 600% true
>6/6 truthes confirming absence of equality to 1
>0.9999999 =\= 1 (100% true)
>x7, 700% true
>7/7 truthes confirming absence of equality to 1
>0.99999999 =\= 1 (100% true)
>x8, 800% true
>8/8 truthes confirming absence of equality to 1
>0.999999999 =\= 1 (100% true)
>x9, 900% true
>9/9 truthes confirming absence of equality to 1
>0.9999999999 =\= 1 (100% true)
>x10, 1000% true
>10/10 truthes confirming absence of equality to 1
>...
>0.999... =/= 1 (...should we question?)
>x∞, ∞% true
>∞/∞ truthes confirming absence of equality to 1
>"NUH UH, LOOK AT THIS! DO THIS ONE!"
>>0.999... = 1
>(ambiguous, 50% maybe true)
>x∞-1, ∞-1% true
>∞-1/∞ truthes confirming absence of equality to 1
>1/∞ potential ambiguities confirming presence of equality to 1
>"LOOK IT FINALLY MAYBE HAPPENS ONE TIME AFTER THE COUNTLESSLY TESTABLE MASS OF ALL OTHER TIMES IT NEVER HAPPENS! THAT MEANS 0.999...=1"

>>16780731
In order to perform multiplication, you have to start at the right of the numbers, which is impossible for .333...

>>16779676
5/9=0.(5)...
67/99=0.(67)...
137687478724747/999999999999999=0.(137687478724747)...
1/3=3/9=0.(3)...

>>16779863
don't even get me started on tungsten!

>>16779953
>But "real" numbers are actually artificial, contrived and ultimately imperfect if not straight up faulty.
good thing then that 0.999... is a rational number

>>16780735
???
No, that's only a problem for transcendental numbers.

>>16780735
>xy = z
NTA but that's not true if the digits of x and y repeat. Your objection only comes in to play if they don't, which leaves open the possibility that you'll never be able to write down the next digit of z.

>>16780031
>so clearly one of the decimal forms is canonical.
i suppose, but that doesn't suddenly make the other one apocryphal

>>16780745
>apocryphal
It's quite literally a party trick for all intensive purposes.

>>16780552
would you be so kind so as to write both in latex for is?

>>16780743
>>16780744
Oh really? Try multiplying the following two numbers by the long multiplication procedure:
>.(67589)....
>.(986786556)...
You literally can't. You have to switch them to fractions, then multiply numerator and denominator separately, and then perform long division. You can't go straight to multiplication with them, just like you can't go straight to multiplication with .333... and 3.
>.333.... X 3
>=1/3 x 3
>=3/3
>=1
No .999.... Arising from the calculation.

>>16780750
>all intensive purposes
Moron detected.

>>16780484
>didn't find the moment
shocker

>>16780742
Please refer to >>16780484 and >>16780503 to clear up the unintended type error in your polemic.

>if 0.9 = 1, requires at least a mistake of 0.1
>if 0.99 = 1, requires at least a mistake of 0.01
>if 0.999 = 1, requires at least a mistake of 0.001
>if x = y, requires at least a mistake
>inb4 "if 0.999...=1 requires at least a miniscule negligible infinitisemal mistake"
>accepting mistakes
ngmi

>0.9 = 100% certainly only 0.9
>0.99 = 100% certainly only 0.99
>0.999 = 100% certainly only 0.999
>0.999... = 50% maybe 1.0, or 50% maybe 0.999...
seems like the probability of "0.999...=1" is equatable to the problem itself.
a thousand, a million, a billion, quintillion, googol, grahams number, tree amounts of certain case scenarios where n=n is also n=absolutely not 1, juxtapositioned besides a single ambiguous case scenario in which n=∞ where it maybe might equal 1, sum culminates that the probability of it equalling 1 is the same as the singular infinitesimal left between the two different numbers confirmed against all and every multitude and multiples of known n, otherwise guaranteeing that it is not equal to 1 unless by mistaking the context for allowing the existence of unknowable unknowns, and if so, the further contradiction against that very allowance to make a determination of knowing via unknowable unknowns that 0.999... = 1

i suppose, by not-epic ramanujan math, the sum of all wrongs is a right, and the sum of all rights is a wrong.

otherwise, 0.999... =\= 1 because 0.999... = 0.999...

>>16780752
0.999... doesn't arise from a calculation, it's the integer 0 with as many nines as you need to keep comparing it to another decimal.
>.(67589)....
>.(986786556)...
You can do that by taking the 9 of the first and 5 of the second.

>>16779202
base fifteen yes or no 1/2 is o.77 think abot genociding britain yes or yes

>>16780734
>>1/∞ potential ambiguities confirming presence of equality to 1
>1/∞=0
>0 potential ambiguities confirming presence of equality to 1
yep

>>16780752
>>16779761

>>16780759
i see no error

>>16780760
>he doesn't know that ...999=-1

>>16780756
It was never about [math]0.333_{7}[/math] not being a rational number and I explained clearly where the misunderstanding was, at this point there's nothing more I can do to help you understand.

>>16780762
>0.999... doesn't arise from a calculation
it actually can.
[math]\sum_{n=1}^{\infty} \frac{9}{10^n} = 0.999...[/math]
if n=3, the result is 0.999
if n=7, the result is 0.9999999
so, this simple calculus equation explicitly allots for a specific unique number which is unambiguously not another number and only equal to a decimal followed by bunch of repeating 9's

>>16779761
>>16780766
Long addition starts at the right, which is impossible with 1/9 and 8/9.
>>16780762
>I call this thing a number
>But it won't arise from a calculation
>I just use it to say it's not the same as other numbers
Just admit that it doesn't exist
>You can do that calculation
Show me on paper or latex

>>16780775
>"i am king of math"
>chokes at 1/9
lol

>>16780769
The error is that you made a redundant statement with saying "0.999... is rational number" because it doesn't interact in any way with my claim you responded to.
1/9 is a rational number and not contrived. 0.999... is rational number and contrived. They are the same thing but not the same somehow, so as you can see something in here is overloaded semantically and I explained how in >>16780484.
I'd be more than happy to explain it to you niggers if you could just tell me what you don't understand but alas you're dumb monkey niggers.

>>16780774
It doesn't arise from division, which is what the other anon is obsessed about. In decimal notation, ()... just means you keep cycling through whatever digits are in () until you can't or don't need to keep comparing them to the digits of some other number.

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Same retards who post the wrong logic and answer in this thread also post the wrong logic and answer in picrel

>>16780775
If you can define a number as the integer 1 with arbitrarily many zeros in every place after the decimal, then you can also define a number as the integer 0 with arbitrarily many nines in every place after the decimal.

>>16780777
Nice trips. I'm the king of math involved in finance. We don't add .111...+.888... Like we do with terminating decimals, because we can't, because the process begins at the right of the number, and these numbers have no "right". We convert them to fractions, and the numerators to get 9/9, and perform long division on 9/9 to get 1.

>>16780753
>fell for the bait

>>16780792
50%

>>16780794
>This guy again that doesn't even know what the word arbitrary means
Numbers arise from counting. Decimals arise from splitting into parts. Nobody makes up nonsense to define.

>>16780801
It's actually 2/3 if you read carefully and passed kindergarten

>>16780803
In decimal notation, an integer is defined as having arbitrarily many zeros in every place after the decimal. It has nothing whatsoever to do with splitting into parts or division.

>>16780805
the same box as the box with the gold ball you took out of the box
there are two (2) boxes with gold balls here
which means after taking one gold ball (1)
it means there is 2/1 to find more gold balls in other words it is certain that there are more gold balls

>>16780811
There are three possible gold balls to choose from. Two out of those three are in box 1. Box 1 is the only box that has two gold balls. Therefore, it is 2/3

>>16780809
And we only agree on that definition because it's valid. There's no valid justification for .999...

>>16780796
>because we can't,
you can't, anyone else can

>>16780826
What's the difference?

>>16780845
When you do that, you're not performing the algorithm correctly, making your argument invalid.
>>16780847
Think about it all little. maybe you'll figure it out.

>>16780851
Just say you can't think of one. Because you can't. There's no difference between using as many zeros as you need to compare two decimals and using as many nines as you need to compare two decimals. Neither of those things follow from splitting a larger number.

>>16780855
>I can't think of one
Fine, I'll do it for you. Adding arbitrarily many 0s to 1. Is the same as 1+0/10+0/100... and is always equal to 1. The same doesn't apply to 0+9/10+9/100.... it is never equal to 1, no matter where you terminate the calculation.

>>16780860
>0+9/10+9/100.... it is never equal to 1
Sure, I can't prove that 0 + 0.999... = 1. But I don't need to because I can prove that 1 - 0.999... = 0. And if a - b = 0, then a = b, otherwise every major property of addition fails.

>>16780870
Again, you haven't proven .999... exists, and furthermore, you can't perform the subtraction 1- .999..., because subtraction starts on the right. Am I seriously the only person in this thread that learned the 4 basic arithmetic operations in elementary school?

>>16780872
>.999... exists
You cant write the integer 0 then keep writing nines until whenever you want or need to stop?
>subtraction starts on the right
No it doesn't. Subtraction is the result of a number minus a number. The first digit of 1 - 0.999... is 0. The second digit is 0. The third digit is 0. You can keep doing this until you pass out or die, you'll never get a digit that isn't 0. Whenever you stop or die, you'll always get the integer 0 followed by arbitrarily many zeros. Which is the definition of 0 in decimal notation.

>>16780876
>You cant write the integer 0 then keep writing nines until whenever you want or need to stop?
I can, and no matter when I stop, it is not equal to 1, and certainly not equal to the non-existent .9....
>>16780876
Mechanical subtraction, used to actually perform calculations and prove results, starts on the right. When you claim to perform 1-.99... , you are performing the algorithm incorrectly, which invalidates your result.

>>16780880
>no matter when I stop, it is not equal to 1
Yes, again, I can't prove that 0 + 0.999... = 1. But I don't need to because a - b = 0 works just as well.
>someone else's algorithm starts on the right
Who cares? My algorithm writes every provable digit of 1 - 0.999... from left to right. Until my hand falls off or the computer breaks. Every digit I wrote is true and proven. No imaginary digits exist, let alone matter in any way to the definition of 0.

>>16780885
>Can prove with subtraction but not addition
>Making up you own algorithms with no justification
I now know you have no idea what you're talking about.

>>16780891
>Can prove with an elliptic curve but not a power series
>Greentext greentext greentext, greentext greentext
Good job lol.

>>16780893
Good job retard

>>16780896
Sorry your H1B didn't work out. On the bright side, now you can stay home and Make India Great Again.

>>16780898
Lol projection. I wasn't going to say anything, but all your posts reel of ESL

>>16780901
>reel of ESL
Lol.

>>16780902
The l and k keys are right next to each other. The fact that you have no idea what I was trying to say proves that you're a stinky Indian.

>>16780905
>no u
It must have really hurt when I pointed it out.

>>16780911
>Can't prove his ridiculous theory
>Resorts to name calling
>Is probably brown
It must suck to be you

>>16780915
You should channel your anger into shooting up your own call center. You'll be famous.

>>16780917
Why do you continue to project? The entire thread can tell that you're the one getting worked up.

>>16780925
Do you still appeal to muh lurkers in India? Is that a caste system thing?

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>>16779376
the physiognomy comparison here checks out

>>16780936
I'm just wondering what you're trying to prove, when you're the only buttmad person here. Just admit you took the bait and got trolled.

>>16780942
If you weren't an Indian, you wouldn't post a reply to this. If you were, you would.

>>16780947
Watch the sam Hyde video on flat earth and then you'll understand how I know that you're ass blasted

>>16780953
No thanks but thanks for admitting you're an Indian lol.

>>16780954
I accept your concession

>>16780957
>India 101
Yep.

>>16779219
Maybe I'm biased because it's literally on the first page of my first math textbook.
1/3 = 0.3333...
2/3 = 0.6666...
3/3 = 0.9999......which equals 1.
obvi

>>16780784
>The error is that you made a redundant statement with saying "0.999... is rational number" because it doesn't interact in any way with my claim you responded to.
>>16780736

>>16780958
I accept your concession

>>16780984
Yes, that is a redundant statement and it's evident that you have no idea what is even being discussed here by the fact that you referenced a post that is exactly an example of what I formalized in >>16780484 as (1), (2) and (4), as if you voiced any contention against what I said. Stop shadowboxing already.
You can't expect my arguments to magically possess your eyes and make you read all the way, your attention span is not a measure truth.
A rational number can be coerced into the form of a real representation, but for the purposes of rigorous and exact constructive reasoning a number like 0.5 does not witness that its denotation is rational number, which you conciede with your reasoning in >>16780736 , one must present the rational construction, and all this demonstrates that there is a preceptible difference between form and denoted value.
Our minds don't penetrate into a platonic world of pure ideas when we work with numbers, so we can't just comfortably pretend representation it's not a significant aspect.
A real form is not and can never be a proof that its denotation is a rational number; we can only meta-matemathically decide value-level equivalence and make a connection, but it's not the same as giving the canonical construction in the target domain. And this is a non-trivial distinction, I am demonstrating to you that these mathematical objects we're analysing have a set of related but logically orthogonal factors.

>>16780984
>>16781347
The real numbers are indeed a field extension of rationals not because the invention of real numbers prerequisited the existance of rationals in order for them to be molded into the reals, the real numbers would be a field extension of rationals even if the reals had been invented first. But if someone is not able to comprehend that they are independantly valid concepts and he is not able to understand the idea of a term being considered as type of one and not the other through syntactic means, I think it's fair to assume that that person's perspective on mathematics is not too solid (even if evidently most people are like that).
However paradoxically enough when I make the claim "0.5 is not rational" everybody intuitively reaches for the "short proof" that 0.5 = 1/2. This has two unintended consquences: one is that it implicitly conciedes that 0.5 is itself not a proof of the denoted value being a rational, or else you could just claim with just as much indignation that the form 0.5 by itself is self-evidently rational. But people don't do that, they correctly intuitively sense the distinction and provide the needed transformation without being able to consciously distinguish between representation and representee. The second consequence is that it is someting inherent in mathematics because everyone intuitively reaches for this "short proof".
So what I formalized in >>16780484 is nothing new, nothing revolutionary, I'm quite literally just formalizig the intuitive attitude all people have. And by virtue of it being a unversal reaction it's arguably revealing something important mathematically. It's not prescriptive, I was describing something people already abide by, they should just make the realization.
Regardless whether its denotation is rational, a real construction is a real construction, not a rational construction.

>>16781143
Power outage in Mumbai?

Gentlemen please.
Lets all just calm down.
Cease this unseemly squabbling.
Its clear for all to see.
The Universe is finite and discrete
Infinity does not exist.
Therefore neither do infinite decimal expansions.
Its not a case of arguing about what 0.999... equals.
Its a case of BURNING AT THE STAKE EVERY LAST ONE OF THOSE GOD CURSED INFINITY LOVING SODOMITES!
We shall not rest until these abominations before GOD are CLEANSED from this EARTH with HOLY FIRE!
We are THE ONE TRUE FINITE FAITH!
Sent to reclaim Mathematics for the GLORY of GOD!
DEUS VULT!
DEUS VULT!
DEUS VULT!

>>16781347
>constructive reasoning
ah, i see the issue now, you are not even good at your subset doctrine

>>16781552
How could you see anything when your head is so far up your ass?

>>16781446
God is infinite you heretic.

>>16781686
>heretic
I'm a heretic and an anarchist.

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>>16779202

>>16781686
SPLUTTER!
Those are fighting words, YOU BLASPHEMER!
Our HOLY Inquisitors are on their way. They have a fine assortment of sharp and pointy things which will help wring a confession out of you.
Then at least your soul can be saved before we burn you.
Praise GOD!

[math]
\boxed{0 < p < 1} \\
1 = p + (1-p) ~~~~~~ \overset{1}{ \overbrace{[=====p=====|==(1-p)==]}} \\
\text{divide p using x} ~~~~~~ \overset{1}{ \overbrace{ \underset{p}{[ \underbrace{=====x=====|==(p-x)==}]} ~~ + ~~ (1-p)}} \\
\\
\text{solve x and (p-x), when length ratios must be the same} \\
\dfrac{x}{p-x}= \dfrac{p}{1-p} \Rightarrow x- xp = p^2 - xp \Rightarrow \underline{x=p^2} \Rightarrow \underline{(p-x)=p(1-p)} \\
\overset{1}{ \overbrace{ \underset{p}{[ \underbrace{=====p^2=====|==p(1-p)==}]} ~~ + ~~ (1-p)}} \\
\\
\overset{1}{ \overbrace{ \underset{p^2}{[ \underbrace{=====p^3=====|==p^2(1-p)==}]} ~~+ p(1-p)+(1-p)}} \\
\overset{1}{ \overbrace{ \underset{p^3}{[ \underbrace{=====p^4=====|==p^3(1-p)==}]} ~~+ p^2(1-p)+p(1-p)+(1-p)}} \\
(1-p)+p(1-p)+p^2(1-p)+p^3(1-p)+ \cdots =1 ~~~~ \left | ~ \times \frac{1}{1-p} \right . \\
1+p+p^2+p^3+ \cdots = \dfrac{1}{1-p}
[/math]

0=∞=-∞=0=∞=-∞=0...

All numbers equal all other numbers, including 0. We are the limitation because of our limitations, but limitations are not such, so we are not such. 0 can add and take away value as it holds place and denotes space in kind. Such as any number, lest ye be blind. But awareness is the braille where any sense fail. Contemplation the hand that makes truth so grand. You are sum of all faults and victories, and the dividend of what you determine.

>>16781626
but enough about yourself