/sci/ - Science & Math

>>12349297
0.33... = 1/3
1/3 + 1/3 + 1/3 = 1
0.33... + 0.33... + 0.33... = 1
0.99... = 1

>>12349297
10 x 0.99... = 9.99999...
1 x 0.99... = 0.99999...
Subtract these two and you get
9 x 0.99... = 9.00000...
Now divide by 9 and you get
1 x 0.99... = 1.00000...

>>12349297
1/inf=0

>>12349297
There is no decimal number between 0.9999... and 1. Think about it. If there was, how do you express it? Infinitely many zeroes followed by a 1 at the end? How can such a number exist though when there is no end? It can't.
The real answer is that decimals are just shorthand for repeated additions of fractions and so this is just the summation from i=1 to infinity of 9/10^n. A basic knowledge of infinite series confirms that this series is equal to 1. This is always true because in order to express an infinite decimal representation you have to assume that an infinite series can exist.

>>12349408

You just add another period to the ellipsis expression, viz.

0.999... < 0.999.... < 1

What calculus teaches us is that you can always do just a little bit more, but you can never really, truly have the answer. This is why notational conventions such as the above are adopted. Paul Cohen showed the validity of this type of thing using a technique known as "forcing".

>>12349372
This is how I grasped the concept myself long ago. OP is just a br*inlet.

>>12349297
>The way 0.999... is written clearly shows that it is a decimal number between 0 and 1, exclusive
try to actually prove that if a decimal starts with 0. then it lies in 0 and 1, exclusive. you will not succeed, because it's not true.

>>12349297
what's 1.000...-0.999...? what's 1 divided by that number? can you do some finite number of operations that would make 1.000... and 0.999... differ by, say, 1? think about those questions for a while

>>12349372
Is this like, the translation b/w fraction and decimal has issues thus 0.9999999999... = 1 seems to not appear intuitive initially, but the fractional form encoding makes it easier to conceptualise?

>>12349429
>you can always do just a little bit more, but you can never really, truly have the answer
true, with every new 9 added, the result gets closer and closer to 1, but never truly reaches it
however, as the number of 9s approaches infinity, the number gets so close to 1 that for all intents and purposes, it might as well be 1, that's when it becomes "equal" to 1

>>12349565
wrong. 0.999... is not approaching anything, it's the number which is being approached. it's not so close to 1 that it might as well be 1. it literally is 1.

>>12349297
It represents a geometric series that sums to 1

>>12349297
If 0.999... is greater than any finite amount of 9s then there is no number between 0.999... and 1 since any number you propose is less than some finite amount of 9s. That is, 0.999... is as close to 1 as you can get, infinitely close. Thus they are equal.

>>12349297
it's bullshit
>you have one apple (1)
>you bite it a little bit (0.99)
>1 != 0.99

>>12349297
>So 0.999... < 1 by definition, right?
Yes.

It's just a way to write the number in the decimal system.

>>12351636
> 1! = 0.99...

This is fact, you fucking retard

>>12351694
how does an entire apple equal to a bitten apple?

>>12351699
Who the fuck is talking about apples here.
1 factorial equals 1, which is the same as writing 0.9...

>>12349429
Nice trolling.

>>12351703
the value of an apple is 1, dumbass

>>12349773
There is no integer between 0 and 1, they are infinitely close, thus they are the same number!

Because that's how you construct R from Q. Take the set of all sequences (q1,q2,...) in Q. Take the subset of all sequences that "converge" (actually Cauchy sequence). Identify sequences if their difference converges to zero. Define operations = real numbers.
So 0.99999 is just the sequence (0.9, 0.99, 0.999, ...) and it is equivalent to 1=(1,1,1...) since their difference converges to 0.

In other words, 0.999... is in reality the notation for a limit.

>>12349372
actually it can be 1/3 != 1/3 != 1/3
e.g. 0.14 + 0.25 + 0.61 = 1
also pic related

>>12349297
It's easier when you imagine 1 to represent the fractional range from 0.000... to 0.999... 0.333... is a third, 0.999... is 1, 0.000... is 0.

>>12351908
This reasoning doesn't work in the integers

>>12353685
Why should the reasoning work with numbers then? There being a number in between every two numbers can just as well be a false assumption as shown by 0.9... and 1.

if you step away from the computer for two seconds, and look into your brain and think about looking at the number 1, and imagining .999, they would be so closely related that they would appear as one.

>>12351699
.999... isn't a bitten apple, the comparison would be if you cut up an apple into a .9 piece and add a .09 piece and add a .009 piece ect ect and continue to infinity then what you have done is cut up an apple into pieces and put them all back together again, as in EVERY SINGLE PIECE of the apple. NO BITES ONE APPLE.

>>12353839
integers are not densely ordered, real numbers are. The two structures are completely different and have different properties.

>>12349297
Wouldn't this break binary by implying that 0.0000..... = 1?

>>12349401
1/inf = undef

>>12349297
There's been a million proofs for this, and if you don't get it then you're not cut out to be a mathematician. Why bother concerning yourself with questions like these when they only appear in the context of math?
Face it, you're wasting our time and your own.

>>12353899
it's literally defined, right there.

>>12353896
no the equivalent is .1111.... =1

>>12349372
Prove that 1/3 = 0.333...
>>12349380
Prove that you can multiply and subtract from infinite decimals
>>12349472
You’re just a retard since it isn’t a proof, it’s just an aritmetic trick to deceive brainlets
>>12349760
Even by the geometric series you have to prove that the sum of infinite series is the exact value instead of an almost perfect approximation
>>12352175
This is the best answer given here. 0.999...=1 sounds retarded at first but it’s a consequence of how the set of reals is defined. Some mathematical statements derive from the axioms tat we choose to work with, and so they are only useful within that logical system we’ve built. The fact about 0.(9)=1 doesn’t change under Q, N, Z since 0.(9) doesn’t even exist in those sets. I’m assuming higher order sets such as complex and quartenions would follow the same logic since they’re a superset of the reals, but idk much about them.
As >>12353874 says, there’s no such thing as the “nearest” number in the reals (only in N and Z, in Q there isn’t but you can’t get infinitely repeating decimals there however) therefore it should follow that 0.999...=1.
Talking about 0.333... or infinity when it comes to physical reality is nonsensical because our physics knowledge dictates that there’s a limit to things, and you can’t (yet) get infinitely big or small

my favourite /sci/ meme

>>12351699
They are still one apple.

There can never be any difference.
1 - 0.999... = 0.000...
The only way to get anything other than zero is if the nines ends.

>>12353899
1/inf = 0
1 + inf = inf
1 - inf = -inf
inf + inf = inf
inf/inf undefined
inf - inf undefined
1^inf undefined

you can't do everything with inf as with a number, doesn't mean you can do nothing tho

>>12349565
>with every new 9 added
No nines are ever added.

>>12353924
>Prove that 1/3 = 0.333...
1/3 = 3/10 + 1/30
= 0.3 + 1/30
= 0.33 + 1/300
= 0.333 + 1/3000
:
= 0.3... + 1/inf
= 0.3... + 0
= 0.3...

>>12353924
Prove that you can't multiply and subtract from infinite decimals

>>12351636
how much can you eat until you get to 0.99...?
or what is 1-0.99...?

>>12353924
>Prove that you can multiply and subtract from infinite decimals
1 = 1.000...
2 = 2.000...
3 = 3.000...
3-2 = 1
QED

>>12351636
>you have one apple (1)
>you don't bite it at all (0.999...)
>1 = 0.999...
ffy

>>12349297
It is not.
0.999... = 1 losers are wrong.
Only if you define 0.999... as the LIMIT of the infinite series would that equal one.
0.999... will never equal one.
It's LIMIT equals 1.
You must treat 0.999... as the limit of the infinite series to keep your sanity.
RAAAAAAAAAAAAAAAAAAAAAAAAAA!!!!!

>>12353839
>Why should the reasoning work with numbers then?
because you can easily prove it:
if a < b, then (a+b)/2 satisfies a < (a+b)/2 < b
your homework is to figure out what goes wrong when you work with integers only

>>12354112
low iq post

>>12354112
0.999... doesn't have a limit, it's simply 1. It's a number, not a sum.
You're thinking of 0.9 + 0.09 + 0.009 + ..., which is a sum that has a limit defined as 1, not a number.

>>12354018
Retard
>>12354017
Good proof

>>12354112
>Only if you define 0.999... as the LIMIT of the infinite series would that equal one.
that's exactly how 0.999... is defined

>>12354161
This is the correct answer and will dispel the angst of 0.999... != 1 correct believers.
THE LIMIT EQUALS ONE. NOT THE THING ITSELF.

>>12354190
>THE LIMIT EQUALS ONE
limit of what?

>>12354149
Wrong brainlet.
0.999... is interpreted to be the limit of an infinite series. That is what THE THREE DOTS MEAN.
0.999... = Lim x->inf, Sum(i,1,x,9*(10^-i)) = 1.
If you remove the limit and simply have the summation, it will never equal one. The limit is implied and is the only way it is coherent.

>>12354190
no dude, the number 0.9999.... IS the limit of the sequence 0.9, 0.99, 0.999, ...
the "thing itself" is the limit of another sequence which converged to 1.

showing that 0.999... and 1 are both the limit of the same sequence doesn't show that they are the same, it just shows that they are topologically inseparable. I don't see any proofs in this thread that the reals are Hausdorff, so there's no reason for me to believe 0.999... and 1 are the same just because they are both the limit of a sequence.

>>12354114
>if a < b, then (a+b)/2 satisfies a < (a+b)/2 < b
it doesn't have to, as is shown in the case of 0.999... and 1. by all means prove that they are the same number without assuming your condition holds.

>>12354233
a < b
a + b < b + b
a + b < 2b
(a+b)/2 < b

a < b
a + a < a + b
2a < a + b
a < (a+b)/2

at least one step must be wrong. which one?

>>12351908
>There is no integer between 0 and 1
But there is a number. Try again.

>>12354224
Based anon. The limit of a thing, and the thing itself are not the same.
Obviously people first learning this don't realize 0.999... is notationally defined with the limit of an infinite series implied.
And others that have learned some math forget that the limit is implied, and without a limit the infinite series is something different.
Brainlets right in their intuition, Pompous asses in their knowledge of operationally defined notation.

>>12354114
You are dealing with infinite operations, and cannot assume some basic arithmetic proof can be satisfied in the same way it can with finite integers.

>>12354245
the thing = the infinite sequence (9/10, 99/100, 999/1000,...)
limit of the thing = 1

0.999... is defined as limit of the thing
ergo 0.999... = 1

>>12354242
How do you perform those arithmetic steps on an infinite process that will never finish (i.e. the infinite series 0.999...) and not simply finite integers? The steps in your reasoning implies a static finished thing that can then be subject to arithmetic operations. Think about that anon.
Think about some infinite churning machine, and where in the steps of your reasoning you fail...
0.999... without the implied limit != 1.
JOIN US ANON. TAKE THE INIFINITEPILL!

>>12354253
Correct, as an operational definition...
But don't be a pompous ass and a brainlet yourself, when a brainlet first learning this stuff interprets 0.999... to be the infinite series, and not the LIMIT of the infinite series.
THE THING != 1.

>>12354262
who's talking about an infinite process that will never finish? I'm talking about real numbers, anon
0.999... is a real number, anon

>>12354268
0.999... is not the thing
0.999... is the limit of the thing

>>12354271
That's the point of the whole thread.

>>12354205
>brainlet
You lose.

0.999... never changes.
It's not a sum.
It's not a process.
It's literally the number 1.

>>12354271
Yes, 0.999... is a real number.
But you will never be able to make your number
>>12353839
This anon is right.
Just because you cannot construct a number doesn't mean that 0.999..., NOT THE IMPLIED LIMIT, is equal to one.
It is an infinite process, will get forever close to one. Will never be one.

If we remove the implied notion of limit for infinite series...

0.999... < 1.
1 + 0.999... < 1 + 1.
1 + 0.999... < 2.
(1 + 0.999...) / 2 < 1. *Your failure is related to this step if you think....

>>12354285
point of the thread is discussion about the real numbers 1 and 0.999..., yes

>>12354306
>It is an infinite process
No, it's not a process.

>>12354302
Wrong. 100% wrong. It is of a completely different character.

>>12349297
most answers are retarded. Its because infinity does not exist.
1/3 *3 = 1.
you can pretend 1/3 is "0.33333...." and the 3s run off the page and out the window i guess?

>>12354311
Wrong. 100% wrong.

>>12354306
so you're saying that x < 2y doesn't imply x/2 < y
wow you sound like a brainlet

>>12354311
whats 0.99999... times 2?

>>12354328
2

>>12354306
>0.999... is a real number.
>It is an infinite process, will get forever close to one. Will never be one.
real numbers aren't processes and they're not getting closer to anything. they're static.

>>12354302
>>12354309
You pseudointellectuals latch onto the most minimal amount of formalism and become tyrants in it, when supposed brainlets have a better intuition.
I can see how communists indoctrinate college students now with your psychology.

0.999... is not 8 symbols. It does not equal eight. It implies the limit of an infinite series, which does equal one.
But the infinite series and the limit are not the same thing, and when an anon first taking calculus or whatever does not imply the limit in their questioning they are both right and wrong.
Wrong in their understanding of operational definitions; Correct in their intuition of the interpretation they convey.
Basic proofs in integers do not always work for infinite series. The infinite series itself, not its limit, does not equal one. The proofs used asserting the infinite series equal to one are not valid, only that the limit of it equals one.
It is a very subtle point and usually a confusion of language.

>>12354343
0.999... isn't a limit or a process, it's an alternate spelling of the number 1.

No amount of deconstructionist verbiage you post will change that.

>>12354326
You can never generate the number in your proof... the infinite series will obliterate it.

>>12354326
The infinite series will end up equaling the constructed number, invalidating such a proof.
Maybe topology anons are the only true mathematicians...

>>12354356
>generate the number
how about you write something which has an actual mathematical meaning

>>12354352
Division by zero fucks up things.
The same goes for infinite series....
They do not admit division in the same way as integers!
IT IS AN EVER GROWING PIZZA! NO FAIR SLICE CUTTING!

>>12354369
Not your personal thesis army, I have better stuff to do...

>>12354363
infinite series have nothing to do with it. this >>12354242 is true for all real numbers a,b.

>>12354374
"Division by zero" and "infinite series" have nothing to do with the number 0.999...

>>12354352
Nice pilpul communist, but the limit is implied.
I am not the postmodern, neomarxist deconstructionist. I am simply helping out people first learning that their intuition is correct, only their formalism is wrong.

>>12354387
Everything you've written is literally pilpul.

>>12354380
It has everything to do with it.

>>12354391
No, it has absolutely nothing to do with it.

>>12354390
No. I state the truth.
Limits of infinite series, and the series themselves are not the same.
Your notion of the reals is flawed.
The best definition of 0.999... is Summation(i,1,inf,9*10^-i). Most anons think this, even when the limit is implied.
If this is the definition of 0.999... , then it definitely != 1.

>>12349408
Infinite series can exist, but not certain operations on them, i.e. division, which is used in false proofs.

>>12354403
No, the best definition of 0.999... is a string of all nines that never changes.
It's impossible for 0.999... to be any different than 1.

THE BEST DEFINITION OF 0.999.... IS YOUR MOM, AND HOW MANY TIMES SHE HAS TAKEN MY BIG CHUNGUS IN HER SLANT ASS.

>>12354419
YOU WERE BORN OF THE ASS.
YOU ARE A BUTTHOLE BABY.

>>12354403
the ONLY definition of 0.999... is the sum of the infinite series 9/10 + 9/100 + ... which is by definition the limit of the infinite sequence (9/10, 9/10 + 9/100,...)
there's no ambiguity and there's no place for any discussion. this is the literal meaning of the symbol 0.999...
yes, "if we define 0.999... differently, then it's not equal to 1". guess what, if we define 2 differently, then it stops being equal to 1+1.

>>12354328
UNDEF!

>>12349372
Thats wrong.
Math is a tool for geting work done.
Its like a saw blade the "errors" the dust left behind.
Math isn't reality, its not true and its only use is as a tool.
There is no 2 in nature, no copies, no duplicates and no randomness.
Reality is hard, fact.
That fact is God, the failure to understand this is a failure of education.

>>12354433
Limit(x,inf,(Summation(i,1,x,(9*10^-i)))) > Summation(i,1,inf,(9*10^-i)).
I WIN.

>>12354433
Infinty fuckin all ya'll in the ass.
Get on my level.
P != NP.

>>12354419
Limit implies motion.
Math is alive.

>>12354474
Summation(i,1,inf,(9*10^-i)) = Limit(x,inf,(Summation(i,1,x,(9*10^-i)))) by definition, lol

>>12353529

>>12354493
0.999... has nothing to do with limit or motion.

What about 0.1111111111111111111111111111111111111111111111111111111111.....22222222222222222222222222222.....................333444555666777888999111222333444555666777888.....

>>12354497
Correct.
1 > Summation(i,1,inf,(9*10^-i)) = Limit(x,inf,(Summation(i,1,x,(9*10^-i)))).
1 > thing getting ever close to one = limit of something that gets infinitely close to the thing getting ever close to one.
What is so hard to understand?

>>12354501
Fake and gay.

>>12354501
Once again, not your personal thesis army.

>>12354554
Summation(i,1,inf,(9*10^-i)) is not "getting ever close to one". it is one.
Summation(i,1,inf,(9*10^-i)) = Limit(x,inf,(Summation(i,1,x,(9*10^-i)))) = [math]\lim_{x\to\infty}\sum_{i=1}^x\frac{9}{10^i} = \lim_{x\to\infty}\left(1-\frac{1}{10^x} \right) = 1 - \lim_{x\to\infty}\frac{1}{10^x} = 1 - 0 = 1[/math]

>>12354501
I have the part where I set those two terms equal to zero "by iteration" which is a a normal thing. However, if you don't buy that then you can just repeat until the remainder is [math]\frac{\varepsilon}{100000}[/math] and the final equality still works out the same.

>>12354501
Operation on infinite series are not the same as integers anon....
If you properly expanded your proofs instead of getting confused in the symbolism you would see....

>>12354579
>Operation on infinite series are not the same as integers anon....
they are if the series converges absolutely anon......

>>12354501
This is actually laughably false.
Brainlets cannot see for whatever reason,
too tied up in symbols...

>>12354596
You fail to understand basic order of operations when dealing with operation on infinite series anon. You are so coached in symbolic pilpul that you cannot see.

>>12354676
firstly, I'm not Tooker. secondly, operations with infinite series work the same as operations with finite series if the series converge absolutely.

>>12354714
A bit circular don't you think, given that that is where the flaw is??? lol

>>12354714
Absolutely wrong. Dig into this anon and you will see.

>>12354756
>Absolutely wrong
nice argument

>>12354801
multiplication and division of infinite series is not always defined. must be harmonic.

>>12354756
Suppose
[eqn]\sum_{k=1}^{\infty}a_k=5.[/eqn]
Operations with this series work exactly like operations with 5. This is true for all absolutely convergent series.

>>12354825
Also, what the fuck is the problem with my TeX never rendering?

>>12354827
TeX does not render bullshit.

>>12354825
You are already assuming it equals 5, which it does not, in the case explained here. You use circular reasoning.

>>12354832
Yes, but I was asking why my string
>[eqn]\sum_{k=1}^\inftya_k=5[/eqn]
didn't render.

>>12354825
Infinite series implies infinite operations, so you cannot multiply and divide without fucking wrecking yourself. only harmonics allowed, addition subtraction. Totally different algebra for inifnities.

>>12354840
I didn't assume it, I supposed it.

>>12354823
>multiplication and division of infinite series is not always defined
it is when the series is/are absolutely convergent

>>12354843
TeX can smell your communist bullshit from a mile away.
lim 0.999.... = 1.
0.999... != 1.

>>12354846
If two expressions, five and sum infinite sum, or such a sum and any real number, are equal then their operations are equal.

>>12354847
You mother assumed the suppository.

>>12354850
No it follows from the Lemma that f(x) is continuous at x=9 which means the value and the limit are the same.

>>12354853
Yes, but if they are not, operations are not as well.

>>12354854
You know it's the one who you call my mother that cuts people's guts up to turn them into one-man human centipedes right?

>>12354866
You mother is subservient to me.
lim 0.999... = 1.
0.999... != 1.

>>12354857
This is wrong. Get with the times.

>>12354880
>lim 0.999... = 1.
true
>0.999... != 1.
false

0.999... = lim 0.999... = lim 1 = 1

or did you forget that limit of a constant is the constant itself?

>>12354897
\lim_{x\to\infty}\sum_{i=1}^x\frac{9}{10^i} = 1.
\sum_{i=1}^x\frac{9}{10^i} != 1

>>12354880
>subservient
If that's true, then all of the people you were trying to keep clean are going to end up as filthy as can be. All those fathers that trusted you with the well being of their beloved children are going to be so angry with you. Do you care?

>>12354897
It follows from the Lemma that f(x) is continuous at nine. Continuous means that the value of the function at a number is equal the limit of the function as its argument approaches that number.

>>12354910
same thing
see https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/#Sequence,_Series_and_Limit 8th item

>>12354921
But 0.999... != 1, so it is not continuous. Only its limit equals one.

>>12354931
0.999... is not a function you retard

>>12354925
limit of a series, yes.
Series itself, no.

>>12354939
0.999... is the limit

>>12354935
Yes it is. 100%.

>>12354931
It is continuous. You can look where I have used L=1 in the proof of the theorem and use L=0.999... to produce the exact same result.

>>12354935
It is a limit of an infinite series....
Or could sometimes mean the infinite series itself, no limit suppositoried.

>>12349297
The trick lies in ... Its 0.999 repeated to infinity. It never stops.

>>12354955
You can prove 1=2 using undefined operation, division by zero.
You are still wrong... infinitely wrong!

>>12354961
0.9990.9990.999... = 1?
0.999... = 1?
How are you all so wrong!?
0.9990.9990.999... !=1.
0.999... !=1.

>>12354957
>Or could sometimes mean the infinite series itself
never in this discussion though, because equating a sequence to a number doesn't make sense. or rather it's always trivially false.

>>12354961
It will never reach...
Like you mom never getting paid for all the dirty things she does.

>>12354961
>It never stops.
you're right. it never stops because it never started. it doesn't do anything, it's a constant value.

>>12354975
The limit ONLY is equal to 1.
Not the infinite series itself...

>>12349408
... instead of overline
0.999... + 0.000...1 = 1

>>12354880
>>12354897
>>12354910
>>12354931
The function is continuous. f(9)=0.999... and lim_{x to 9}f(x)=0.999... Furthermore, limits are unique so 0.999...=1.

>>12354979
It is constant, infinite accumulation. Infinite series. It is improper to multiply or divide an unharmonious amount to it.

>>12354985
0.999... is the limit
do you want citations from real analysis textbooks or what?

>>12354988
This is the right answer.
0.999... + 0.000...1 = 1.
1 - 0.000...1 != 1.

>>12355007
>0.000...1
there's no such number

>>12355005
We are talking about the distinction between the infinite series, and its limit which is equal to 1.
If your talmudic legalese means 0.999... = limit of the infinite series then yes, it equals one.
If some anon interprets 0.999... to mean the infinite series itself, they are correct in saying it is not equal to one.

>>12355016
>0.999...
there's no such number.
QUAD EPSON DEMONSTRATUM.

>>12355020
>0.999... = limit of the infinite series
that's how 0.999.. is defined
>If some anon interprets 0.999... to mean the infinite series itself
then they interpret it wrongly, because it's not what the symbol means

>>12355038
Others can have other interpretations. You have no power to condemn. For those that in their lack of conformity in their naivety interpreted differently; they can rest assured that the infinite series itself does not equal one.
Only its limit equals one.

>>12355037
there is such number as 0.(9)
there is no such number as 0.(0)1
you're either very stupid or a troll

>>12355064
>Others can have other interpretations
>I'm gonna interpret 2 as 1+1+1+1 and go argue that 2 > 3
>Others can have other interpretations

Redone without the "by iteration" step.

>>12355088
I am talking about operational definiton of things, versus what you suppositorize as being a part of a coherent system. One can have different operational definitons, and still have the same underlying structure (think category theory).
You are so zealous that you don't realize the level at which this conversation is taking place.
It is about the underlying notions, not you zealous forcing of certain symbolism.

>>12355108
Kek will choose the winner here.

A little better here.

>>12349408
How can 0.999... exist in that case, when there is no end?
Infinite series cannot be multiplied and divided like integers... which is what infinite decimals are shorthand for...not the LIMIT of the infinite series.
lim 0.999... = 1.
0.999... != 1.

>>12355108
0.999... has an unambiguous universally accepted meaning and it coincides with the meaning of 1. there's no place for personal interpretations.

>>12355153
what is better? It is still wrong.

>>12355165
What error do you see?

>>12355161
Is it the limit of the infinite series, or the series itself?
Proper english grammar and word formation would indicate it is the infinite series itself, and not the limit of the series.
It is like saying a pianist, is a piano player with institution power. You can deconstruct proper word formation but the truth will come out of your chaos, tranny.

>>12355154
>How can 0.999... exist in that case, when there is no end?
The same way e constant has decimal notation of infinite "length", but is a single value.

>>12355161
I interpret it to mean how many times I will pay you mother, never quite once. Just shy of being a hooker.

>>12355185
So are you some idiot trying to prove euler's constant = 1?
0.999... != 1.
e != 1.

>>12355180
[math]0.999\dots = \lim_{n\to\infty}\sum_{k=1}^n \frac{9}{10^k}[/math]
I dare you to find one source which says otherwise

>>12355185

1 - 0.999... = 0.000...1.
0.999... + 0.000...1 = 1.
0.999... != 1.
1 - 0.000...1 != 1.
1- 0.999... != 0.

>>12355203
no, i proved a value can have decimal notation with no end
0.(9) exists and is equal to 1
0.(0)1 does not exist

>>12355216
>0.000...1
there is no such number
you failed at the first step

how old are you?

>>12355082
0.999... + 0.999... = 1.999...8

>>12355212
\displaystyle \lim_{x -> \infty} \displaystyle\sum_{k=1}^x \frac{9}{10^k} \ != \displaystyle\sum_{k=1}^\infty \frac{9}{10^k}

>>12355253
>1.999...8
there is no such number

>>12355253
This is correct. It either stays or goes with the whole system.

>>12355225
It is operationally defined.

>>12355269
Wrong.

https://en.wikipedia.org/wiki/Series_(mathematics)

>>12355225
>(1+ 0.999...) / 2
There is no such number.

>>12349372
so some fractions are well-defined. others are just symbolic representations of indefinite repeating decimals? why do we not differentiate between the two types of fractions?

>>12354032
based
>>12349562
sure you can think of it that way. Try doing the same operation in a difference base, instead of base 10 use base 3 for example, and it will become even more clear.

>>12355291
Exactly. We should differentiate.
Completeness of the reals and limits of infinite series uses circular reasoning to create a bastard number system.
Proof lies in the order of operations regarding partial sums, and the treatment of the series at infinity.

>12355286
>12355284
no more (you)s for the troll

>>12355304
>Completeness of the reals and limits of infinite series uses circular reasoning to create a bastard number system.
>circular reasoning
lol brainlet

>>12355270
For practical purposes pretending it doesn't exist makes sense. For accuracy it exists.
0.9 + 0.9 = 1.8
0.99 + 0.99 = 1.98
0.999 + 0.999 = 1.998
and so on
0.999... + 0.999... = 1.999...8

>>12355291
One can say 2 * 0.111... is defined,
but not 3 * 0.777....

>>12355313
what reasoning did you use to arrive at the last line?

>>12355328
THE SAME INFINITE BULLSHIT SUCKAS USE TO SAY 0.999... = 1.

>>12355305
REAL-NUMBER FAGS WILL REALIZE THEY ARE WORSHIPPING A FALSE GOD!
THERE CAN ONLY BE ONE TRUE NUMBER SYSTEM!

>>12355313
2(0.999...) + 2(0.000...1) = 2.
Makes perfect sense.

>>12355345

>>12355325
>but not 3 * 0.777....
[eqn]3\cdot 0.777\dots = 3\cdot \sum_{n=1}^{\infty}\frac{7}{10^n} = \sum_{n=1}^{\infty}\frac{21}{10^n} =\sum_{n=1}^{\infty}\frac{2\cdot 10 + 1}{10^n} = \sum_{n=1}^{\infty}\left( \frac{2}{10^{n-1}}+\frac{1}{10^n}\right)=\\= \sum_{n=1}^{\infty} \frac{2}{10^{n-1}} + \sum_{n=1}^{\infty}\frac{1}{10^n} = 2 + \sum_{n=2}^{\infty}\frac{2}{10^{n-1}} + \sum_{n=1}^{\infty}\frac{1}{10^n}\\
= 2 + \sum_{n=1}^{\infty}\frac{2}{10^{n}} + \sum_{n=1}^{\infty}\frac{1}{10^n} = 2 + 0.222\dots + 0.111\dots = 2.333\dots[/eqn]

OP, trolling on /sci/ won't make your father come back, stupid son of a bitch

>>12355336
limits, you say? limit of (1.8,1.98,1.998) is 2 though...

>>12355328
Simple base 10 Math. What made me think of it was infinite expansion. How can it expand if there is nothing after infinity.

>>12355387
Wrong.
The correct answer is 2.333... - 0.000...2 = 2.333...1.

>>12355401
You forgot the 0.000...2 idiot.
Otherwise 1 = 2.

>>12355424
you should be able to point out which step is false

>>12355428
>Otherwise 1 = 2.
prove it

>>12355386
The Jews worshipped Mammon the devil while they were on this earth.
Continued fractions and infitesimals are the one true number system.

>>12355435
You cannot multiply and divide infinite series, just like division by zero is undefined.
You system would require a syllogism chain of infinite length, which you conveniently dispense of.

>>12355458
>You cannot multiply and divide infinite series
source: my ass

Proposition: there are no nonzero infinitesimals in the real numbers (as defined by Cauchy sequences of rational numbers x=(x_i) ).
Proof:
Assume x=(x_i) is a Cauchy sequence such that for all natural numbers N, |x|<1/N. [Remember that the absolute value of a real number is just the sequence of absolute values, i.e. |x|=(|x_i|), and that for real numbers a,b, a<b means that there exists a rational number q>0 such that eventually a_i + q < b_i].
We prove that such an x must be 0, which means that for all rational e>0, eventually |x_i|<e.
If x is not 0, then there must exist a rational e>0 such that there are infinitely many |x_i|>=e.
But then take some natural number N such that 1/N < e.
Then |x|<1/N means eventually |x_i| are below 1/N. But there are infinitely many i such that |x_i|>=e>1/N. Contradiction.
So x=0.
QED.
Corollary: 0.999...=1.
Proof: 1-0.999... is an infinitesimally small real number, so by the proposition it must be zero. Thus 1-0.9999.... = 0. Adding 0.999... to both sides, get 1=0.999.... + 0 = 0.999....
QED

>>12355387
Let me guess... 1+2+3+... = -1/12?
Totally correct. No problem with appyling arithmetic operations to infinite series at all...

>>12355458
Multiplication is just recursive addition, division is just recursive subtraction with a counter. Division by zero equals infinity.

>>12355489
This is flawed. Cauchy is flawed.

>>12355507
no problem with applying arithmetic operations to infinite series which are absolutely convergent
1 + 2 + 3 + ... is not absolutely convergent
9/10 + 9/100 + 9/1000 + ... is absolutely convergent

>>12355489
You are flawed and cannot extend those arguments for infinite sequences.

>>12355526
Convergence itself is a problem.

>>12355534
what?

>>12355547
he doesn't understand it, so it's a problem

>>12355534
Cauchy sequence is circular reasoning.

>>12355566
The real number system is not real, the ultimate pilpul embedded in the name itself.

Do we really need the same bait post every. single. day?

>>12349297


I guess .99999=1. But really .9999999999 is a made up number.

There's no way to divide numbers and come up with .99999999.., like you can divide 8 by 9 and come out with .88888888...

>>12354343
>>12354403
the infinite series and its limit are isomorphic, it's entirely reasonable to state that they are equal

>>12355629
>entirely reasonable
no proof.

>>12355628
1/9 = 0.111...
+
8/9 = 0.888...
=
9/9 = 0.999...

>>12355797
1/9 != 0.111...
8/9 != 0.888...
9/9 != 0.999...

>>12349297
I know you are ayyy lmao. This is a test. I win.
0.999... < 1.

>>12349297
It's close enough, faggot, just like 0.4999=1

>>12355902
lol

If cauchy sequences are flawed, then they must create some inconsistency. Show me an inconsistency of the real numbers derived by cauchy sequences

>>12354328
>>12354328
>whats 0.99999... times 2?

You tell me what it equals.

>>12357007
1/[18,18,18,........]

>>12354409
Wrong.

You can make operations on them provided covergence is proven at some point. Which totally makes sense. A convergent infinite series is equivalent to a number. So operations on a number can apply to a convergent series.

>>12349297
Its not. Its a fantasy concocted by retards to deal with the concept of infinity. Numbers are tangible reality, infinity is not. The two dont mix, but retards tried to force them to mix and so you get these sorts of absurdities, together with equally forced "proofs" which are no more than a prior assumptions. Essentially "Let assume it works so here is the proof derived from that assumption that it works, oh! Oh! Look see, it works so its proof! "

Fucking retards, man, too fucking conceited and stupid to even realize just fucking stupid they are.

This bullshit about .999.. being equally to 1 is very much on the same level as someone saying petrol combustion engines work because there is a Pink Flying Elephant in the combustion chamber doing all the work, only it can only be seen when you are NOT looking at it. Can you see the Pink Flying Elephant when the engine is running? No? WELL THEN! THERE YA GO! PROOF THAT THE PINK FLYING ELEPHANT MAKES THE ENGINE WORK! FUCK I AM SO CLEVER! SUCK ME! SUCK MY TINY INFINITE MATHS DICK! YEHAAAA!

>>12349297
>but I'm just not convinced.
read chapter 1 of baby Rudin. A lot of the misconceptions about 0.9... = 1 come down to the fact that people don't understand how the real numbers are constructed

>>12349429
based

>>12349297
space is not infinitely divisible; at some point 0.999=1, empirically.

>>12357497
>space is not infinitely divisible
source: my ass

>>12357416
>read chapter 1 of baby Rudin
This is the low IQ post.

>>12349297
Mathfags are contradictory retards. Sometimes they accept infinitism, sometimes finitism. Unironically they're just mainstream low iq retards who dogmatically accept previous teachings .

>>12357773
>Sometimes they accept infinitism, sometimes finitism
give example

If i was a mod I would permanently ban .9999... = 1 denialists from this board.

1 - 0.999... = 0.000...
It's literally impossible for 0.999... to be anything other than 1.

>>12357778
Here's a philosophy graduate illustrating how retarded mathematicians often are when it comes to infinity:
http://milesmathis.com/square.html

>>12359146
>strawman arguments
he's completely missed the point of squaring the circle. The fact that transcendental numbers are excluded from the other numbers under certain conditions, demonstrating the concept of algebraic closure, is meaningful and well demonstrated by the "squaring the circle" exercise. He is essentially implying that mathematicians don't believe in circles

>>12354017
>= 0.3... + 1/inf
There's no justification for removing the " + 1/inf ".

>>12349297
Here's a clear way thinking of it as a geometric series with a sum to infinity.

>>12361465
>There's no justification for removing the " + 1/inf ".
lol of course there is. it's just shorthand for [math]\lim_{n\to\infty}\frac{3}{10^n} = 0[/math]

>>12361465
>>12361664
[math]\lim_{n \to \infty}\frac{1}{3\cdot10^n} = 0[/math], my bad

>>12361667
[math] \displaystyle
\lim_{n \to \infty} \dfrac{1}{3 \cdot10^n} = 0
[/math]
Optimized.

>>12361638
nice

>>12361726
Thanks. We don't have to overcomplicate things. Next rebuttal is probably going to be: "There's no empirical evidence infinity actually exists REEEE!"

*sighs*

We could have colonised Mars and the Moon by now.

Why should 0.999... have a value at all? And how can we tell what value it should have? Most people who think it should not equal 1 haven't really thought these questions through.

>>12357413
What number below 1 is greater than 0.999...?

>>12361638
>>12349297

BASED.

OP is a denialist faggot.

>>12354014
dum

>>12361839
fight it out with WA

>>12361839
nice argument

>>12349297
0.999... != 1
raise both sides to the power of infinity
0.999...^inf != 1^inf
0 != 1

x=0.99999....
10x = 9.9999999....
10x-x = 9
9x = 9
x = 1

>>12361875
BASED!

>>12361861
>0! = 1
only correct line

>>12349297

x= 0.9999999...
10x = 9+x
x=1

Think about it like this.

1/10 = 0.1
1/100 = 0.001
1/1000 = 0.0001
1/infinity = 0.000...1

0.999... + 0.000...1 = 1


understand now?

>>12363163
infinity = infinity
infinity = infinity + infinity
infninity = 2*infinity
1/infinity = 2/infinity
1/infinity = 2*1/infinity
0.000...1 = 2*0.000...1
0 = 2*000...1 - 0.000...1
0 = 0.000...1

0.999... + 0.000...1 = 1
0.999... + 0 = 1
0.999... = 1

>>12354343
No you idiot. 0.9 + 0.09 + 0.009 ... would be a series. 0.99... is the limit of that series. It isn't in itself a series. That doesn't even make sense. A single number isn't a series.

>>12363239
Checked. You can express it as an infinite geometric series with a sum to infinity like anon (>>12361638) did.

>>12363393
0.999... = 1 is the SUM of that series. It isn't the series itself. (Though defined as a real number, you could say 1 is really an equivalence class of Cauchy sequences of rational numbers.)

let thy assume that 0.999... equals to 1
then
floor(0.999...) = floor(1)
0 = 1
contradition
q.e.d

>>12363621
Your proof requires floor(0.999...) < floor(1), and therefore requires 0.999... < 1. It begs the question.

>>12363621
>Assume 0.999... < 1
>Therefore floor(0.999...) < floor(1)
>Therefore 0.999... < 1
Circular reasoning ftw.

floor of 0.999... is 1 not 0

>>12349297
Can you add any more to it, without it being greater than one? If so, show me.

>inb4 0.999...+0.000...1

In hexadecimal, it is certainly not the case that 0.999... = 1.

>Replies: 278

>>12364539
in hexademical, it is certainly not the case that 9+1=10

>>12365632
[math] \displaystyle
\begin{align*}
1 = \left (\frac{15}{16} + \frac{1}{16} \right )
&= \text{0x0.F} + \frac{1}{16} \\
= \text{0x0.F} + \left ( \frac{15}{256} + \frac{1}{256} \right )
&= \text{0x0.FF} + \frac{1}{256}\\
= \text{0x0.FF} + \left ( \frac{15}{4096} + \frac{1}{4096} \right )
&= \text{0x0.FFF} + \frac{1}{4096} \\
= \text{0x0.FFF} +\left ( \frac{15}{65536} + \frac{1}{65536} \right )
&= \text{0x0.FFFF} + \frac{1}{65536} \\
&\vdots
\end{align*}
\\ \displaystyle
\Rightarrow \text{0x}0.\overline{\text{F}} = 1
[/math]

>>12365641
your point?

all the math seems like bullshit.
0.999...~ doesn't = 1 because they are two different numbers.
It doesn't matter that there is a ascii character for infinity. 0.999 repeating will never equal 1. that's what makes it what it is.

>>12365650
>0.999...~ doesn't = 1 because they are two different numbers.
>0.999... doesn't equal 1 because 0.999... doesn't equal 1

>>12365643
just before the F

>>12365650
>0.999...~ doesn't = 1 because they are two different numbers.
>1/2 doesn't = 2/4 because they are two different numbers.

>>12365650
>will never
0.999... is static, the length is aleph_0 from the get go.
Your naive cartoon vision of a diesel engine chugging along is ridiculous. Embarrassing even.

>>12365641
Why are your numerators and denominators written in decimal?

>>12365650
But they aren't different numbers. They are the same number. What leads you to believe that they are different numbers other than the fact that they are written differently? Do you even know what a repeating decimal means?

>>12365662
why not?

>>12365651
>>12365655
>>12365657

the number 0.999 repeating is a number that logarithmically increments. It cannot equal 1 because it is 0.999 repeating.

>>12365667
>increments
>>12365657

>>12365670

Like I said before, it doesn't matter if you can wrap up a particular infinity into a single character. That number isn't 1. If saying any number is not equal 1 except for 1 is against an axiom, let me know. :)

>>12365686
Again, how do you *define* a repeating decimal? Obviously any terminating decimal 0.999...9 < 1. We can define such a terminating decimal by 9*(1/10) + 9*(1/10)^2 + ... + 9*(1/10)^n, where n is the number of nines. But what if the decimal repeats? Then there are "infinitely many" nines. How do you define it then?

I'm being dead serious. How do you propose we define this infinite sum? I have literally never seen a consistent proposal for which it is not exactly 1.

For that matter, how do you define the sum of *any* Cauchy series that isn't eventually constantly 0?

>>12365667
>the number 0.999 repeating is a number that logarithmically increments.
lol no. a number doesn't "increment". it's a constant value.

>>12365700

it is infinite. You answered your own question. You can represent the concept any way you want. But the truth is the number will never reach 1. To deny that is silly.

>>12365711

It increments if your lifespan was infinite and you decided to count it. Or plot it. Either way, in your eternity of writing nines, you will never write a 1.

>>12365712
you're right, it doesn't "reach" 1. it already is 1.

>>12365719
>It increments if your lifespan was infinite and you decided to count it.
but my lifespan is not infinite and I don't decide to count it. therefore it doesn't increment.
nice logic kid

>>12365720


you're right, a number that is not 1 is 1.

I agree completely and this is not sarcasm at all :)

>>12365727
>a number that is not 1 is 1.
agreed.
0.999... is 1 though, so I don't know how is that relevant.

>>12365731
You're arguing infinity is finite. When does infinity stop being infinite?

>>12365731

This is a misunderstanding of the concept of infinite and by extension the concept of numbers. It is an Achilles race. Spoiler alert, Achilles cannot win.

0.999... is a finite number

>>12365742
0.999... is an infinite representation of a finite object.

>>12365745
Spoiler alert, Achilles wins every time.

>>12365686
>let's pretend infinite is finite
if it's infinite, it is 1
if it's finite, it isn't 1

>>12365686
1 = 9/10 + 1/10
= 0.9 + 1/10
= 0.99 + 1/100
= 0.999 + 1/1000
:
= 0.9... + 1/inf
= 0.9... + 0
= 0.9...

>>12365667
>the number 0.999 repeating is a number that logarithmically increments. It cannot equal 1 because it is 0.999 repeating.
oh no, you misunderstood what logarithmically incrementing number means. it increments but it does so like this:

it takes 1 second to generate 0.9
it takes another 1/2 second to generate 0.99
it takes another 1/4 second to generate 0.999
it takes another 1/8 second to generate 0.9999
it takes another 1/16 second to generate 0.99999
it takes another 1/32 second to generate 0.999999
etc.

so it's indeed equal to 1, but for comparision you have to wait at least 2 seconds until it calibrates. you might run into problems otherwise.

>>12365782


Oh no, the computation will never reach 2 seconds because it is always dividing the the time of the computation.

>>12365803
>>12365755

>>12365803
>time will never reach 2 seconds

>>12365712
>it is infinite.
0.999... is infinite? How can infinity be less than 1?

I think what you mean is that it is irrational. But that doesn't answer the question. How do we define it?

Maybe you haven't taken enough math to come across this idea, but all numbers have definitions. 3 isn't just some fuzzy concept floating out there.3 is the successor of 2. 2 is the successor of 1. 1 is the successor of 0. And 0 is the empty set. The successor function S(n) can be any bijection from O∪N to N, where N is any infinite set (whose existence is guaranteed by the axiom of infinity). But conventional choices are often made that, for instance, would define 0={}, 1={0}, 2={0,1}, 3={0,1,2}, etc. From there we can define addition recursively. Once we have addition, we can define a sort of subtraction where (a,b) (read a minus b) is equivalent to (c,d) if there exists a natural number n such that a+n=b and c+n=d or a natural number m such that a=b+m and c=d+m, and also all pairs (x,x) are equivalent to each other. Every equivalence class is an integer. In the first case, we call the integer positive, in the second case, negative, and in the third case, zero, and we say a - b = (a,b) if a and b are natural numbers. Then we can extend addition, subtraction, and multiplication to the integers. Next we can do something similar with division and define the rational numbers.

So it isn't some law from God that a/b + c/d = (ad+bc)/(bd). That is literally how we define addition of rational numbers. All of this is to say that we need a RIGOROUS foundation for math. There is no other way to prove theorems. So when you have these vague ideas like "it is infinite" (when in fact it is so finite that it is not even greater than 1), you need to make those ideas rigorous.

And it turns out that the way we MAKE real numbers rigorous (indeed the whole point of having real numbers) is to define them as values to which sequences converge.

>>12365848
>0.999...
>irrational
literally any integer divided by the same integer is 0.999...

this guy >>12365848 is either first time on /sci/ or he's taking trolling to the next level

>>12365875
What he means is that it is irrational. (I think? Not really sure. He keeps calling it "infinite.") Obviously it is just 1.

>>12365987
That's what I gathered. I was just pointing out that 0.999... is as rational as it gets.