0.9999... = 1 Discuss

1/3 = .333
1/3 + 1/3 + 1/3 = 3/3 = 1/1
kys

it doesn't equal it exactly but for most practical applications the difference is negligible.

>>738124891
if its an infinite series of 9s, then thats the idea.

It doesn't though. I don't care what anyone says. Being infinitely close to something implies that it still isn't that something. Therefore .999... will never equal 1.

It's true

Anyone who says otherwise is retarded

>>738124998
No, u dumb fuck. It is equal exactly

Every number has two ways of writing it.
1 can be written as 1, or 0.999....
1.5 can be written as 1.4999...
Etc.

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>>738125140
oh ok

>>738125183
That's the stupidest thing I've ever heard.

>>738125213
But it's try. If u want to, I can explain it to u

>>738125213
but is true

>>738125213
>That's the stupidest thing I've ever heard.
It is not.
For example I can write That's the stupidest thing I've ever heard or 1h4t'5 1h3 5tup1d35t 1h1ng 1'v3 3v3r h34rd.

everything can be numbered! :D:DD:

STOP BE SO UPSETTI
JUST WRITE YOUR NUMBERETTI

x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1

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shove that info up your ass and gtfo retards

>>738124891

Rolling for result

>>738125440

The answer is 40

>>738125297
you dont need to explain. you need to prove.

9+1=10
0.9+0.1=1
0.99+0.01=1
0.999+0.001=1
now imagine a countable infinity of nines after the .
if it is countable, we can just add a number that has as many digits after the . as the number of elements in our countable infinity. like 0.000000.....01
add them:
you get
0.99999....9+0.00000....1=1
this means to write one as 0.999999

you need an uncountable infinte number of nines after the 0.

q.e.d.

>>738125183
And whats with 1.00000...1?

i study math in my 4th semester and it is true that 1=0,9999...

0,999999 is getting closer and closer to 1 with every 9 you add. but technically it cant reach 1 ever. But infinity is something different.

x=0,99999...
10x=9,99999...
10x-x=9,99999...-0,99999...
9x=9
x=1

>>738125504
cool so what do you think of my interpretation ?
>>738125479
i study not math bur other natural sciences.

Stop freaking out, infinities are crazy like that

>>738125411
x = 0.9
10x = 9.9
10x = 9+0.9
10x = 9+x
9x = 9
X= 1

So 0.9 = 1

>>738125479
the concept of limits and infinity is the key here. it seems counter intuitive because 0.9 < 1 and 0.99 < 1 and 0.999 < 1 and so on no matter how many 9s we add, so long as there is a finite number of them.

0.999... represents an infinite number of 9s

>>738125750
>x = 0.9
>10x = 9.9
cant multiplication

dude .... whats 10*0.9 again ?

>>738125504
And what's with this?>>738125750

>>738125750
If x = 0.9 then 10x = 9

learn to multiply

>>738125829
11

>>738125776
yeah but an uncountable infinity right ? Have i shown it correctly senpai ?

i mean... there exist also countable infinities, but that wouldnt be sufficient to get 1=0.999999....

>>738125504
You cannot simply convert float to integer

>>738124891
1/9=.11111111
2/9=.22222222
:
8/9=.88888888
9/9=?

>>738125960
one cannot simply march into mordor. its dark.

>>738124891
Discuss if 1.0000...1 is 1 too.

math phd here

looks different so not the same

ur all welcome

>>738126062
/thread

>>738126025
its not,by definition. simple as that.

Only if it's rounded up..

>>738126086
So 0,000...1 to less is okay but 0,000...1 too much not. Why? I say 1.0000..1 is 1 too

>>738126081
Either you are really that dumb, in which case I will point out he is a painfully obvious troll, or you are a troll as well, in which case you can eat shit and die.

.999... (.9 repeating) is EQUAL to 1 regardless of what you asshats say.

>>738125750
>>738125829
I'm not the poster but I do know some Math.

The proof is ment for x=0.999999999...
0.9 takes lesser space than an infinite amount of nines though

>>738126086
this

>>738126025
>ends with 1
>ends
This implies a countable infinity, thus not equal to 1

>>738125364
This is now a cringe thread

>>738124891
0.99999... + 1 = 2
0.9999... + 1.0000...1 = 2

So 1.0000...1 = 1 too

>>738126187
Dude the proof yes. but the person i was referring to meant to "debunk" the proof.
And i said to him he cant multiply. simple as that.

I know what it was meant for.

>>738126220
This >>738126248

>>738126174
the problem is that at some point it ends with a 1. if it ends, you have a different kind of infinity

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>>738125501
First of all, it is 1.00000...0(9)
>>738125411
>>738125479
>>738125504
All of this is seems legit, but kinda wrong. Don't fuck with infinite numbers, its stupid.
There is 2 right ways to prove that 0.(9)=1.
First one:
What is a number 2, for example? It is a line which consistcs of two untits of length. So 3 consists 3 UL (I will call them that way cause don't know how to say eдиничный oтpeзoк in english) and so on. But there is also numbers like 2.5 . So it is 2 UL and something else that is less that UL but greater than 0. Then we divide our UL into 10 equal pieces and count, how much of them we got. Then if there is something else we will divide our UL in 100, 1000 and so on pieces.
So, what is a 1? Well, it is 1, obviously. But it is also 0 and 10*0.1.
So we got 0.something. But 10*0.1 is 9*0.1 and 0.1.
So we got 0.9 and someting and 0.1. But 0.1 is 10*0.01 and it is 9*0.01 and a 0.01.
So we got 0.99 and something. So, after an infinite numbers of steps we get that
1=0.99999999.....
Exactlu.
Second one:
picrelated. Cyммa бecкoнeчнoй yбывaющeй пocлeдoвaтeльнocти.

/thread

>>738126285
Look proof >>738126348

>>738124891
Close enough that I don't give a fuck

>>738126285
0.999... + 1.000...1 = 2.000...1

>>738124891
Please write me the number that is less than 1, but is as close to one as you can get.

That would be 0.999...

However, the LIMIT process of 0.999.... = 1.

It depends upon the context of those three little dots, ya know?

In the first example, it is a completed number (like pi can be), and in the second example, it is a classic limit process like you find in Calculus I.

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>>738125364

>>738126220
So is my proof that you need an uncountable infinity correct ?

Im still strugling here if its correct dude.

i need to know.

>>738126420
What is 9+1?

>>738126344
since when is basic math knowledge cringe worthy?
Seriously, read more or something.

>>738126446
0.999... = 1 regardless of context

>>738125092
your lack of knowledge in basic math is amazing

>>738126540
No.

Write me the number that is less than one, but as close to 1 as you can get.

Go.

>>738126535
You are just pissed because im right.

>>738126457
Huh?
No, it's not correct. By having ANYTHING after ... you are implying the infinite number of zeroes between the decimal place and the ending digit are a countable infinity. This means it is not the same as .999... being equal to 1

>>738124891
Is 1 - 0.000...1 = 1?

>>738126598
>"as close as you can get" on a continuous number line.
kek

Wrong because the function where x approaches 1 from the left does not include the point "1" in it's domain. nice try though. learn to limit.

>>738126684
And is then 1 - (Infinity) * 0.000...1 = 1?

>>738126611
right about what?
You didn't even make a statement that can be correct or incorrect.
You simply quoted a post out of context and frankly I don't even understand what you are implying.

I'm not mad, just disappointed the average intelligence of 4chan has plummeted so much over the years.

>>738126598
you're getting hung up on infinity my dude

Who cares there's no practical application for this because nothing is infinite in real life except OP's dick sucking capabilities

>>738126489
9 + 1 = 10
but that doesn't ever apply here

>>738126598
.999...8

Here you go.
a decimal place followed by a countable infinity of nines ending with an 8. That is the closest to 1 you can approach without equaling 1.

>>738126789
calculus much?

>>738126742
So 1 - 0...00001 = 0.999... = 1
So 0.00..1 has to be Zero because 1+0=1
So 1+0.000..1 = 1.000...1 = 1

>>738126906
* 1-0=1

>>738126789
>no practical application
sure, probably not in YOUR life. I'm assuming your day job is something humble like waste disposal or retail.

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>>738126334
>>738126187
Aaah my bad. I didn't have the energy to read the entire thread from the top.
Here, have a bird for the suffering and inconvenience

>>738126715
>>738126746
i'm just asking for someone to write down the simple, easy representation of the number that is less than one, but as close to one as you can get.

Just simply write it. Do it. doooo eeeeet.

>>738126647

Yeah. And this ive shown to lead to a contradiction which means that you need a UNCOUNTABLE infinity, just as you said. that was my proof that you need at least an uncountable infinity.

it was this...
>>738125479

is it wrong ?

>>738126906
Flaw in the first statement.

1 - 0.000...1 does not equal 0.999...

>>738126991
.9 with a line over it indicating it repeats endlessly.

_
.9

Just like 1/3 = .3 (repeating endlessly)

>>738126906
First line of the proof is incorrect.

>>738127053
So what is 1 - 0.00..1 then?

>>738126838
If you assume infinitesimal numbers to actually exist, 0.999... does not equal 1 and neither does 0.999...8

The thing is infinitesimal numbers do not exist in modern mathematics

>>738126991
someone already did
>>738126838

>>738126991
see >>738126838

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What you're thinking of OP is ℵₒ.
(Aleph-Null)
A transfinite set of natural numbers.
It does not equal 1. Because it will never reach, 1.

0.99999/3=0.33333
1/3=0.33333
case closed

>>738127103
a number slightly less than 1

>>738124891
https://www.wolframalpha.com/input/?i=.9+repeating+%3D+1

/THREAD

>>738127218
Which is what number? The number most slightly less than 1 is 0.999...

>>738124991

this means that 100% (3/3 or 1/1) is 0.999999...
not 1.
Idiot.

>>738127262
>https://www.wolframalpha.com/input/?i=.9+repeating+%3D+1

Incorrect. Wolfram Alpha is affected by a mathematical exploit of digital computing.

It will claim 'true' to avoid deadlocking a calculation. (Because it will never matter in practicality.)

>>738127317
either this is a troll or you are abso-fucking-lutely retarded

>>738125451
Don't panic, but it's 42.

>>738127317
the problem is if 1 is exactly 1 then 1/3 is not exactly 0.(3).
Is that what you mean ?

>>738127300
no, because 0.999... = 1

they are the same number written two different ways.

you can't just use mathematical notation to mean whatever you want. you have to follow the rules of the game

0.999... does not mean some really big but ultimately countable number of 9s

>>738127357
This. It would infinite calculate otherwise.

>>738126838
0.999...8
No I disagree.
Because then I can create a smaller number
0.999...89
See how that works?

I totally accept your formalization of an infinite string and then addition to the end of that string. But here's the kicker...

Once you accept that formalization, then what directly comes into play is infinitesimals.

Between it takes an infinite amount of infinitesimals to make the smallest real number. That's their definition, right? Right.

So by your own formal argument, I can create a number between 0.999... < x < 1, showing you that 0.999... does not = 1.

0.999... < (0.999... + dx ) < 1

That's 0.999... plus an infinitesimal.

And by using the infinitesimals (which is a core concept to your limit process), that is the only way we can get to the "end" of your infinite string without the ability to add another unit to the end of it.

This is why the Fundamental Theorem of Calculus, that wonderful epsilon-delta exercise we all know and love, is all about an interval around the limit number, and not about being equal to the number itself.

>>738127357
>mathematical exploit of digital computing
as a programmer and security analyst, this phrase makes me cringe.

What are you even saying, nigga?
I understand your point is that it would try to avoid infinite recursion, but that isn't why it says true. It says true because it's fucking wolfram alpha and it was asked to verify a simple equation; the answer to which is universally known.

>>738127357
If it's breaking from a "loop" it would say false, not true... but stuff like this is a mathematical axiom and it's programmed to know the equality. It's not calculating anything when you ask if 0.(9) = 1.

>>738127489
yes

If it's not 1, it's not 1.

Jeez, you kids.

>>738127490
So 1 - 0.0...1 = 0.99..8?
So 1-0.01 = 0.98?

>>738125501
It's not infinite, so it's 0.however many fucking zeros...01 bigger than 1

>>738127510
>date] [Auto] 6 new posts

Infinity isn't a number, but a set of numbers.

Your traditional set of cardinal, natural numbers.
1, 2, 3, 4, ... ect. Can be counted infinitely.
Labeled with ω (Omega)

However, a decimal between 0 and 1
Example being 0.9999 (repeating) would be labeled.
ℵₒ (Aleph-null)
They are two different SETS of infinity, and yes. One is larger than the other. Can you guess?

>>738127510
>Because then I can create a smaller number
>0.999...89

No shit, sherlock. That wasn't what I was asked to do though. I was asked to find the closest to 1. Going smaller doesn't prove anything.

>>738127555
trips are truth
>>738127560
also this is true as well. Unless specifically told to return a boolean true, it would return false on breaking the loop.

>>738127555
It says the answer because someone told him to say then to compute infinite. So its no argument at all. If 0.99... = 1 then where is the missing 0.0...1 ? And if the missing 0.000...1 = 0 then what is 0.00..1 * infinty?

>>738127645
Eh, sorta to the first. I don't think you could actually represent what 1 - 0.000...1 would actually equal but that would be the closest you could actually write.

The second line is just absurdly wrong though and a wild jump of logic

>>738127788
I wasn't going smaller, i was going bigger.
0.999...8 < 0.999...89
See how that works Sherlock?

>>738127874
there is nothing missing because 0.999... = 1 by definition
It is a different way of writing the exact same number

>>738127555
>>738127560

I am not a programmer.
However, I know the answer is not true.
0.999 repeating = ℵₒ
The larger transfinite set of cardinal, natural numbers.
However, 0.999 repeating cannot equal 1, if it can never reach 1. Thus the value is not the same.
(In practicality, it would make sense to ignore, let's say the 34th decimal. Because there is no use in using it. Unless you're calculating Pi.)

>>738127788
Idiot.

>>738126420
> ....1
How many math classes have you failed to think this?

>>738127965
How the hell do you figure?
Lets take the ending two digits of each

.999...8: 98
.999...89: 89

How exactly is yours bigger?
you even stated in your initial post you were making smaller numbers.

>>738127973
>Continuation

What I'm trying to say it. It was probably forced.
Thanks >>738127856
Is that it would make sense to say it equals 1, because its impractical otherwise for general use.

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>>738124891
depends on the context

>>738127971
So 1 = 0.(9) + 0.(0)1
0.(9) = 1.
So 0.(0)1 = 0?

>>738127874
There is NOTHING missing. They are equal.
Give it up with this .000...1 shit. It simply isn't equal to zero.

>>738126598
Find me a number between 0.9999... and 1

>>738128133
Flaw in the first line

1 = 0.(9)

adding anything to either side would make the equation false

>>738128133
Yes

Infinitesimal numbers do not exist, therefore 0.000...1 = 0

>>738128201
> Find me a number between 1 and 1

>>738128143
1-0.0000...1 = ? (In number)

>>738128219
>Infinitesimal numbers do not exist

Umm.. yeah they do.
They just aren't REAL; they're hyperreal

>>738127759
>Infinity isn't a number, but a set of numbers.
Then it is a category mistake to equate an infinite series of decimal expansions 0.999... with the number 1.

Which means 0.999... does not equal 1.

I will say again. When considering the limit process, which is Leibniz's definition, then absolutely, 0.999... =1.

But when considering it as a number, then 0.999.... does not equal 1.

My question has not been refuted.

write a number that is less than 1, but is as close to 1 as you can get.

And that answer is 0.999... < 1

It is all about which context "..." you are concerned with.

Does "bow" = "bow"

One may be a looped ribbon, or the front of a ship, or a tool that shoots an arrow. It may appear trivial that "bow" = "bow" until you realize that the context of one may not equal the context of the other.

This is the classical mistake that even learned professor of mathematics make because they simply parrot something they learned a long time ago, and are too arrogant to consider they may be wrong. It can happen, and does happen, even to great, smart, wonderful people.

Maybe another approach might help?

So if the two numbers weren't equal, their difference wouldn't be equal to zero.

Let d = 0.a_1a_2a_3... denote the difference 1-0.999..., where each a_j can take the values 0,1,2,3,4,5,6,7,8,9.

Then clearly a_1 has to be equal to zero (otherwise, added to 0.999..., it would exceed 1). Similarly for a_2, etc.

Therefore each a_j equals 0, thus showing that 1-0.999... = 0.

>>738128310
somewhere between .999...8 and .999... but you probably couldnt actually express the number

>>738128088
I am adding a smaller unit to increase the number.

1/3 < 1/3 +1/6, yes? Even though I added a smaller number than 1/3. That is how it works to creat a longer decimal expansion. It is time for you to retire from this argument, sir. Good day to you.

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>>738128351
its a tautology.

you say the difference is zero therefore they are the same.

>>738127645
>Infinite zeroes, end with 1
>Ends
You don't understand rules very well, do you?

>>738128219
infinitesimal numbers do exist.
See the classic seminal work by Abraham Robinson, "Non Standard Analysis". How do you not know this?

>>738124891

at least it's not one of the infinite "post pics you shouldn't share" threads, with the same pics over and over again. maybe you should post this 9.999999999....9 times a day.

>>738128361
If you cant express then it doesn't exist

>>738128442
I assumed you simply subtracted. It is impossible to tell what you did considering there is no way to represent exactly what size of infinity is between the decimal place and the ending digits.

Of course you could infinitely approach 1 by adding smaller and smaller amounts. This is the basis of Zeno's paradox.

However, no matter how many times you do that, you will be unable to express the changes as you will simply be increasing an already infinite number of decimals. Therefore, the closest to 1 that we can actually express is .999...8 as the number of 9s that ... represents could be any length countable infinity.

>>738124891

no. 0.9999.....9 is exactly 0.0000.....1 smaller than 1.

>>738128506
Yep, isn't that how equality between numbers is defined mathematically?

>>738128506
>its a tautology.
You have correctly identified his error.
If he has any integrity at all, he will recognize this, and admit it.

>>738128587
I can't fully express how much of a tard you are, but yet... here you are.

>>738124891

it approaches 1 for infinity while never achieving it

just like how OP approaches women.

>>738128629
I said Good day!

>>738128679
I'm willing to understand, but I haven't yet :(

>>738128318
It's so close it's equivalent.

>>738128679
A tautology isn't a logical error. 2=2 is a tautology but is nonetheless true.

>>738128679
People rarely recognize this. I see this mistake too often.

>>738128737
No it's literally equivalent.

>>738128743
Its an error in the sence that i doesnt prove shit. its an error in his thinking.
And you not understanding what was meant is strange to me.

>>738128705
> I say my assumptions are right but i Zell you the equation you gave me gives you an number that exists but nobody knows what it looks like

Thats not howto evidence. Go to church of you want to believe in things that you can't express or imagine

>>738128629
Your statement is not a proof.
It is another tautalogy.
"However, no matter now many times you do that, you will be unable to express ..." that 0.999... equals 1. EXACTLY.

You then go on to simply state your tautolgoy that it can't be true.

You have failed. Utterly.

>>738128766
Well, yeah.

>>738128813
So by your logic, geometry doesn't exist outside of 3 dimensional euclidean space? I mean, you can't truly express or imagine what a tesseract is..

>>738128737
>It's so close it's equivalent.
YES! exactly!
By the limit process, this is true!

But not if you are considering it in the context of an actual number. These are two different contexts, that is all I am saying, and one must understand which one is being used.

Can 0.999... = 1?
Yes if one is discussing limits, calculus, etc

Can 0.999... < 1
Yes, when is discussing numbers as complete.

Simple!

>>738128318
If you want 0.999... to mean "a decimal point followed by some very large but still countable number of 9s" then yes 0.999... < 1

But the rest of the math world will keep using 0.999... to mean "a decimal point followed by an uncountably infinite number of 9s" in which case 0.999... = 1

So yes, context does matter. In one context, you're using standard mathematical notation. In the other, you're using a notation system that you made up.

>>738128941
can you explain the calculus side of things to me ?

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>>738125183

>>738124891
For this conversation, I have one word that will trigger all the retards in this thread

L I M I T S

that is all

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time to shitpost this thread up

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>>738128816
>tautology
I'm not simply asserting what I said was true because I said it was.
I am simply stating that even within the realm of countable infinities, they can become increasingly larger yet still countable. We simply lack a way to express larger countable infinities in the form of Ellipses or any other method of repeating notation.

>>738125183
only if it's repeating which you did indicate with the ellipsis

>>738128766
>No it's literally equivalent.
... by DEFINITION via Leibniz
whose entire idea is to examine limits without actually having to evaluate the limit itself .

But when not discussing in terms of limits, then 0.999... takes on a different context.

>>738129103
I think this is the issue. Math means something very specific by ...

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>>738124891
Numbers are an abstraction and have no basis in reality.

>>738124891
I think OP is just the Captcha people trying to provoke more AI picture matching tests :(

>>738124891
Consider my epsilon-delta outline.
Show that a sequence
Sum from i = 1 to k of 9 x 10^(-i) approaches 1, and for every N > n there exists an epsilon < delta, where epsilon is the difference between the sequence evaluated at N and one, and delta is the difference between the sequence evaluated at n and one.
therefore the sequence converges to one.
QED.>>738124891

>>738129148
The only context where 0.999... does not equal 1 is when talking about non-real numbers, like hyperreals

>>738129223
I didn't know this became a YLYL thread, but I fucking lost.

>>738125411
Only if x is irrational like the .9999... that convieneintly became rational

>>738129190
Gotta love Caracalla

>>738127759
Go back to Vsauce

>>738124891
In the real number group these are two way to express the same number
Only in the surreal number group the concept of an infinitesimal number doesn t mean 0 ,but nobody care about the surreal number

>>738127759
Some infinities are larger than others.
This is true!
Mathematics is fascinating.

x = 0.999...
10x = 9.999...
10x-x = 9.999...-x
9x = 9
x = 1

>>738124891
I think it's funny how fags on /b/ think they're mathematicians from MIT. Debating like some 3rd rate community college philosophy majors. Kill Yourselves Already! Alt-right mother fucking cucks. Don't all of you have black cocks to suck and envy over?

>>738129249
>therefore the sequence converges to one.
Yes... a limit proces!!!
And it is correct that 0.999... = 1 in that context.

>>738129756
Do you mean larger in length or greater in number?

ITT uneducated anons probably from america try to discuss a calc day one concept of limits and infinities like were trying to invent a new math. Read a fucking book dipshits its not that hard.

>>738129959
larger in the sense that the elements of one infinite set cannot be placed into a one-to-one correspondence with the other infinite set.

i hate street signs, vehicles, bridges, apartment buildings, shorts ...

you guys are talking about liebniz but i think leibniz would use his "theory of indiscernibles" here

if 1 and 0.99... were the same, we wouldnt be able to tell them apart, but we treat them as separate entities, therefore they must not be the same thing

if they were the same thing we would never even attempt to compare the two

>>738125183
What is the 0.999 has an 8 on the end of it?

>>738131018
so larger in terms of the number of elements, ie "greater in number"

>>738125183
you're just as retarded as the deniers here.

Every rational number has an infinite number of representations.

Every irrational number has exactly one exact representation

.999~ = 1

i don't come here to learn math i come here to masturbate and hate myself

>>738131638
what would you say to >>738131034?

>>738124891
NO.

>>738131793
"We treat them as separate entities" is superficial and retarded. Algebraically, .999~ and 1 behave exactly the same, just like 1/2 and .5.

It's merely an alternate representation.

>>738131988
but we can tell them apart

>>738132017
how is that significant? What does "tell them apart" mean and what does it afford us in mathematics?

>>738124891

ITT we debate math that was solved in antiquity.

>>738132156
well leibniz said if two things were the same then we wouldnt be able to tell them apart

so if an infinite set was the same as a single instance we wouldn't be able to tell them apart, but we can

>>738132233

If this were true then we could not substitute one for another to produce an equivalent statement.

However, we can.

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>>738132156
If we have two apples and both appear to look the same on first sight, we would still be able to tell them apart on closer inspection.
If the question is: "Can we tell them apart?" Then the answer is: "Yes, but only upon closer inspection."
The question: "Why would we care to inspect them more closely? They're just a bunch of apples." isn't relevant to answer the first question.

>>738132376
only in mathematics can you use this though, everywhere else we can say "an infinite set is not the same as a single instance"

>>738132156
they may "mean" the same thing, but they are "referring" to two different objects

>>738124891
For the people who are dumb and hate algebra

1/9 = .11111......
(.111111......)*9 = .99999.....
(1/9)*9 = 1

>>738132376
it seems what you've really done is highlight a weakness of mathematics

because an obvious contradiction like "an infinite set is the same as a single instance" can be proven mathematically, while logically speaking, it is absurd

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Okay, let's take a look. First, we consider 1-0.999...= x. Which properties does this number x have? If 1 != 0.999..., then x>0 but x is smaller than any other positive real number, because it's an infinite series of zeros with a one at the end! So if 1 != 0.999..., then the consequence is, that x must be smaller than x/2 for example, which doesn't make any sense, because the smallest positive real number x doesn't exist for this reason. We can prove by contradiction that 1=0.999...

>>738125411
I remember that from my math's book from primary school.

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>>738128201
>>738124891
.999 does not = 1.

This logic is flawed because of our concept of "infinity".

There will always be a decimal number of .9 trailing 0.999... so there will always be a number between 0.999 and 1 so 0.999... does not equal 1 no matter how insignificant the difference.

I'm a physics professor.

>>738133766
This rest number can't exist if we are talking about an infinity of 9s, look at
>>738133631, in the real world, this is true of course, but in mathematics, the concept of a number that is infinitly small is contradicting itself.

0.999... = 9(1/10)+9(1/10)^2+9(1/10)^3... = (9(1/10))/(1-(1/10)) = 1.

Using the convergence of an infinite geometric series

>>738126991
1-ε, where ε is arbitrarily small, you silly retard.

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>>738134012
0.999.. is not infinitely small. It's a number that is nearly equal to 1 but it isnt. 0.99... isnt shrinking as a decimal is added to it.

Imagine a pie thats not quite whole. For practical reasons...it's 1 pie. This is life.

In mathematics though....it isnt 1 pie, it's slightly less.

>>738125092
U muricas are retards. There are 2 types of algebra and in first 0.(9) is prohibited and in another 0.(9) = 1. Also your beliefs on infinity sequences are comlete shit

>>738134565
I'm talking about how 1-0.999... is infinitely small if you assume that 1 != 0.999..., which results a logical paradox, not that 0.999 itself get's smaller, sorry for the misconception

>>738126838
IF you write a number with decimal point it means there is a bijection between N and all signs after point. It's impossible to define place of yours 8, so go kill yourself faggot

If you think that 0.999...=1 you are stupid

>>738127317
No, 3/3 is definitely 1 by the definition of how we came up with division. If you evenly split 3 items to 3 people, how many does each person get? The answer is exactly 1 with no left overs. Yet since we know through our definition of math and fractions that 1/3 = 0.333... and that 1/3 + 1/3 + 1/3 = 0.999... and yet that since 3/3 also equals 1, by definition 0.999... must equal 1 due to the transitive property of equality which states that if a = b and b = c then a = c, or if 3/3 = 0.999... and 3/3 = 1, then 0.999... = 1

It would be easier to discuss this problem by first discussing if Infinity = Infinity - 1 instead of if 0.990... = 1.

>>738135419
Typo. Meant "if 0.999... = 1".

The easiest way this was ever explained to me is that way:

Two real numbers are different if and only if there exists a third real number that lies strictly in between.

There exists no number that is both <1 and >0.(9). Therefore 1 = 0.(9)

<1 =/= 1

>>738135508
0. 9999 < 1
1 > 0. 999

>>738135508
There exists no such number that is <2 and >1 therefore 1=2

>>738126420
0.999...Pure Autism

>>738127510
You cant create a number that is smaller than 1 but larger than .999999...
.99999999... + x > 1 for any x, no matter how small
Thats why they are the same

>>738136300
0. 99(9)?

>>738136393
Not the same *

>>738136393
0. 999999+0. 000001=?

>>738136519
>>738136659
Im not saying .999999 with 6 nines, I mean an infinite number of 9s, Im just using shorthand.
Another way to think of it is if you have an infinite number of 9s, you would need an infinite number of 0s and then a 1 to get to 1 - but having a number that comes AFTER an infinite string is nonsense. The infinite never ends.

Another way to think of it is that if they arent the same number, then there must be a number half way between them. What number is half way between .999999... and 1?