1 = 0.99999~

Discuss.

oh no not this again :(

teh answer is: yes

END OF THREAD!!!

NO IT IS NOT. IT IS A JEWISH CONSPIRACY.

>>2
LOL MATHS FAILS!

limit            N
n -> infinity    ---
                 \
                  \       (9/(10^N)
                  /
                 /
                 ---    
                 1


= 1

also = .9999999~

QED

ick that sigma looks hella bad... should look like this:

-------
\
 \
 /
/
-------

0.000~1

God damn it.  I said that if I saw this question one more time I'd burn down the internet.  You brought this all upon yourselves.

PS: 0.999~ = 1.

Thread is over.

a real number is actually an equivalence class of the limit of sequence of rational numbers

Take the sequence .9, .99, .999, .9999, .99999, ...
Obviously it converges to .9 repeating.
However, it also converges to 1.
Thus the limit of the sequence converges to both .9 repeating and 1, so by definition the two equivalence classes (and thus the real numbers) are equal.

now gb2/food you fail at maths

>>8
>>9
0.9999~ < 1 formally.
Yet 1 - 0.999999~ = 0.00000~ = 0 thus 0.9999~ = 1 empirically.
Thus, maths fails as a sound model, but succeeds as a pragmatic one.
That's why so many maths geeks hate philosophy at first, until they study it and realize so much of it defines the strength of cases like this.

there is no 'empirically' in math, shithead.

I'm a Japanese student.
I'm from 2ch in Japan.
Nice to meet you.

And sorry for my poor English ability.

I think 1=0.999999....

>>11
Ah the good old naive "maths is an apriori thing" crowd.
Pray tell, then, why we prove so many theorems by showing that there is no counter example, and disprove standing ones by trying to produce counter-examples?

I hate shitheads like you who are randomly dismissive. You guys like to feel comfortable with your epistemic model, even when it leads you to be blatantly wrong. That's why you guys never make it anywhere in academia and have to get dead-end jobs as software developers.

日本人は無視ですか？

>>14
nonononononononononononono
SHARAMANDOUMAL to you

>>14
give me a counter example for this:
42 = 42 -> Math pwns Humanities or go back to your humanities!

a = a is always apriori motherfucker.
42 = 39 + 2 isn't.

>>17
Oh, and maths doesn't really mean anything without philosophy.
More to the point, I stand by the fact that you fail academia. Probably got a second rate maths degree, or are working on failing one.

>>18
1/0 is

let (math ∈ philosophy) = true
then (math ∩ philosophy) = math

>>14
http://en.wikipedia.org/wiki/Minimal_counterexample
this doesn't seem empirical

>>21
Yup.

>>14
>>17
lol? mathematics is a formal system, and everything in it is derived from the ten ZFC axioms. there is absolutely no empiricism or philosophy involved.

Go back to masturbating to Spinoza and stop thinking you know anything about mathematics, shithead.

>>24
>>> Whine whine whine whine whine
I reckon there's just one maths emo kid who's randomly pissed off at philosophy because his girlfriend dumped him for some humanities person, and keeps on flaming here. The rest of us, thankfully, know better.

Spinoza does suck though.

whatever makes you happy, shithead.

king氏ね

kill the king

"king" is one of the most fucking HN in math category of Japanese 2ch.
He(/she?) is a serious pedophilia.(however,in Japan, I think
most mathematicians are pedophilia.... )

All this bullshit about philosophy and empiricism doesn't matter, since the guy was originalyl wrong anyway.  Here is a formal proof that 0.999~ = 1.

x = 0.999~
multiply both sides by 10
10x = 9.999~
subtract x from both sides (remember x = 0.999~)
9x = 9
divde both sides by 9
x = 1

ZOMG 0.999~ = 1

Now shut the fuck up.

oh lol, I did do the orginal one wrong.....

the thing in the sum should be:

(9/10^1)+ (9/10^2) + (9/10^3) .... (9/10^N)



Haet forgetting the easy ways just because I learn some fancypants limits.

>>31

Well, I was referring to >>10, but whatever.  Really, in good sigma notation, it would look like:

         N
lim   -------
N->inf\
       \
       /      (9/10^k)
      /
      -------
       k = 1


I hope that displays right.

Fuck, it messed it up.

FuckFuckFuckFuckFuckFuckFuckFuckFuckFuckFuckFuckFuckFuckFuck

>>32
>>6
you can always use unicode 8721dec
         N 
lim      ∑  (9/10^k)
x→∞     k=1

Be cool.All right?

>>10
.99~ is not less than one, dipshit, they are literally the same number! That's like saying 1 < 1. You need to stop thinking of real numbers as "numbers"

>>37
I know, when will people learn, right? But this is fourchan, and includes a lot of the "fine" american values in its userbase (thank you american education ministry for limiting people's selves for them) so you have to come to expect these things, maybe just ignore the stupid comments and work on your self so that one day maybe you won't have to read world4ch to relieve boredom. Of course, it makes for less gifted minds, and more
SSSSCCCIIIIEEENNCEEEEEEEEEEEEEEEEEEEEEE
but again it's 4chan, so who cares if it goes to shit!

>>37
>>38
LOL @ arrogant dipshits losers.

lol >>10
1/9 = .1~, ok
1/3 = .3~, ok
1/1 = .9~, OMGNO IMPOSSIBLE

>>38
gb2yurop education ministry

42GET! OHR

Is 42.99999~GET = 43GET?

1= 0.8

It's called a limit
GTFO

o.999~ < 1

>>30
is a lying troll who doesn't know shit. your calculation is wrong and you know it, if you don't, gb2college math 101

>>47
QFT

>>47

Then prove it wrong, fuckhead.

>>49
He doesn't really need to, kid.

You can just see it by contradiction. Suppose 0.999~ is not 1, then you can find a real number strictly greater than 0.999~ and strictly less than 1. But you can't.

It just means that the irrational number k = 0.9999~ is infinitely close to 1. By definition, this means nothing is closer to 1 than k. That doesn't mean k = 1.

You can prove the contrary through induction:
k = lim(n->+∞) ∑(n=1) 9*10^n
Take nmax = 1: k1 = 0.9
Take nmax = 2: k2 = 0.9+x = 0.99 where x=0.09 and x < 1-k1
Take nmax = 3: k3 = 0.99+x = 0.999 where x=0.009 and x < 1-k2.

If you see k as an arithmetic series, for every rank, the number added is always inferior to 1-k(n-1) where k(n-1) is the previous rank. Therefore it will grow infinitely close to 1, without ever reaching it, ever.

>>30 was therefore wrong. He was just playing with the misrepresentation that (0.99999~ * 10) - 9 is still 0.9999~, which is a circular proof.

>>47 is therefore correct, but should have provided us with the above truth instead of saging.

Good, you all have it now, you can all let the thread die.

>>52
haha oh wow, stop pretending you know shit all about maths.
First of all 0.999~ IS a rational and not an irrational like you claim. If you know anything about decimal expansions then you know that a number is rational iff it has a decimal expansion which is periodic after a certain point.

I am not even going to bother with the rest of the stuff you said as you seem to be a total fuckhead.

SO DON'T LISTEN TO THIS GUY. HE DOESN'T KNOW WHAT THE FUCK HE IS TALKING ABOUT.

>>53 has attacked one terminological point in the post, without attacking the rest. He has not even tried to refute the proof. In fact, the point he attacked has nothing to do with the proof as >>52 could well have just posted:

It just means that the irrational number k = 0.9999~ is infinitely close to 1. By definition, this means nothing is closer to 1 than k. That doesn't mean k = 1.
You can prove the contrary through induction:
k = lim(n->+∞) ∑(n=1) 9*10^n
Take nmax = 1: k1 = 0.9
Take nmax = 2: k2 = 0.9+x = 0.99 where x=0.09 and x < 1-k1
Take nmax = 3: k3 = 0.99+x = 0.999 where x=0.009 and x < 1-k2.
If you see k as an arithmetic series, for every rank, the number added is always inferior to 1-k(n-1) where k(n-1) is the previous rank. Therefore it will grow infinitely close to 1, without ever reaching it, ever.

QFT. >>53 doesn't know shit about what he's talking about, and is probably the same person as >>30.

My God, you people are assholes. I know you're just trolling, but some poor kid might read this and believe you. So let's get something more authoritative:
http://mathworld.wolfram.com/RepeatingDecimal.html
"Numbers such as 0.5 are sometimes regarded as repeating decimals since 0.5=0.5000~=0.4999~."

"are sometimes regarded" that sounds very authorative/final.

>>55
How the fuck are they trolling? >>30 is just being a bit of an asshole, and being wrong.
>>52 is making a good point. If you guys aren't going to bother coming up to an argument to a common proof by recursion, then you don't really have any say in how unauthorative this thread is.

>>56
Welcome to English. The claim after 'since' is completely independent of that.

>>57
>>52 is too retarded to understand limits apparently.

Seriously, I'm not going to waste my time on a bunch of trolls. Here's a couple of people with actual qualifications if you'd care: http://www.newton.dep.anl.gov/askasci/math99/math99167.htm
And of course there's a Wikipedia page you can go and edit if you're so convinced of your correctness: http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1

>>54
I am not going to take someone who claims 0.999~ is irrational very seriously. It's clear that the guy knows nothing about maths by the usage of his terms. "infinitely close"?? If you can't find a real number between two real numbers then they must be the same. His so called proof relies on "it's true for n=1, it's true for n=2, it's true for n=3, so it must be true for countably infinite summation". I am sorry, but that simply doesn't work that way. By that arguement you would be able to prove countable intersections of open sets is open, but there are plenty of counterexamples to that.

LOL @ Mathematicians thinking maths is real.
This is all theoretical shit with no grounding in reality.
It's just like philosophy: intellectual masturbation.

Theorem: 1=0.999~ in R

Proof: we recall that R defined set of equivalence classes of converging rational sequences in the topological competion of Q with regard to the relation a ~~ b iif lim(n->inf, a) = lim(n->inf, b).

we note that:

0.999~ ~~ lim(n->inf, 1 - (1/10^n)) = lim(n->inf, 1) ~~ 1

therefore 0.999~ = 1 as required.

qed.

>>60

Oh right. Ever heard of number theory? The stuff that makes all the transactions on the internet secure? Get your ass out of your head thanks.

>>62
I wonder why it's called number THEORY then.
KTHXBYE

I guess you dismiss the "THEORY of relativity" or other physical and chemical theories. I guess Gravity doesn't exist for you.

>>64
Just because you have reasonable doubt about the reality of theories doesn't mean they don't exist, or that we can't take a pragmatic stance on them.
Go back to epistemology 101 motherfucker.

>>65
oh ok, so you doubt gravity. fine.

Nerd fight, nerd fight!

1 = 9/9 = 9*(1/9) = 9*(0.11111~) = 0.99999~

QED

>>68
something is subtly wrong with this proof (albeit the result 1=.999~ is right) but I just can't figure out what

kaka poopoo peepee butt ass lesbian

>>69

Perhaps this part: 9/9 = 9*(1/9)

or:
9*(0.11111~) = 0.99999~

where one assumes you can work with digits separately even if there are infinitely many of them

>>71
What's wrong with that part?  Nine ninths makes a whole, last I checked.

>>73

The proof is circular.

0.999~ = 9(1/9)=9(0.111~)=0.999~

>>74
lol

i'm a mathamagician

>>74
QFT.

>>74

It's not circular, it just relies on the fact that 0.111~ = 1/9.

Which can be shown in the exact same way as I proved 1 = 0.999~ in >>30.

It's a false question because 0.9999... never occurs from division of any two integers, and therefore is a decimal representation of a fraction that does not exist.  (You could get 0.9999.... from 9/9 if you fuck up the first step of long division, and pretend that there are zero nines in 9.)  You have to bullshit about rational numbers having to equal a limit, or pretend that you start with 1 and infinitely partition it in ever smaller parts in order to compensate for the fuckup.

>>79
But 0.3333333~ does happen and it equal 0.4

>>80
no. .399999999999~ = 0.4

>>81
That only applies to 0.9999999999999~ = 0.1 , am I rite?

heh, you lost 8 tenths and a lot of 9's after that, what is your equal sign doing?

0.9999... never occurs from division of any two integers

Actually, it occurs every time you divide an integer by itself. You know, what with it being the same thing as 1 and all.

>>83
He probably redefined the = operator

0.999~ is an element of the open set (0,1)
1 is an element of the closed set [0,1]
The intersection of these two sets does not in include 1
<=> 0.999~  =/= 1

0.999~ is an element of the open set (0,1)
Assumption here being that 0.999~ < 1.
Your result is simply weaker form of that assumption.
Congratulations on a shitty circular proof.

hey that anonymous guy posts ALOT. and he seems to contradict himself over and over.

Disregard that, I suckle the cocks of anonymi.

>>86
Fail for what the fuck, that was the faggest fuck up ever.

>>90
Just because you don't understand it doesn't mean you should dislike it...

>>91
0.999~ is an element of the open set (0,1)
Yeah, right, that was like a gay pride claim you know

.999~ is represented by the limit as n approaches infinity of the sum from i=1 to n of (9/10^i) = L
this is equivalent to the limit of this series
a(1) = .9
a(n+1) = a(n)/10 + .9

or maybe
1/9 = .111~
.111~ * 9 = .999~
1/9 * 9 = 9/9 = 1
i dont know if youd accept that though!

this is >>93 claiming an identity. <3

im pretty awesome at reposting what other people said btw.

>>95
Hey, i own an note pad too

This is some deep shit.

I own a cock

>>98
excellent!

I regret writing that script now.

not so tough now eh?

DICKS = COCKS

>>64

The Theory of Gravity has been dismissed since GR, there is no force, gravity does not exist. You would know this if you were not retarded.

Now, before the keys start flying off of your keyboard, know that this does not mean that Newton's equations do not offer a pretty close approximation to true gravitational acceleration, or that they are not useful, just that nature does not work that way, and gravity is a fictitious force (hence, Theory of Gravity is wrong).

1/3 = 0.3333~

1/3 + 1/3 + 1/3 = 3/3 = 1 OHSHI-

>>103
The Theory of Gravity has been dismissed since GR, there is no force, gravity does not exist
Dear god you're a moron.

The fact that you don't know the difference between the theories of relativity superceding Newton's theory and said theories meaning ``gravity doesn't exist'' suggests that you get your facts from skimming half a Wikipedia article while also being extremely bad at reading comprehension.
Please stay out of these discussions until you make it to high school.

>>105


Anonymous...I wish you would sometimes think before you post. Gravity simple does not exist, its clear. In the Theory of Gravity, gravity is caused by a force, since Einstein we know that there is no force and our old perception of gravity was flawed because we take a noninertial frame (accelerating) as inertial. Really, gravitation is not a force but acceleration (due to space time geometrics, yes). I suggest you read up on the equivalence principle, if you can understand physics that far.

>there is no force

>>106

"equivalence principle" if they're equivalent, surely you're just arguing semantics

Gravity *simply* does not exist

Came back to clear up this mistake in my posts, I know you 4chan kids are not the brightest so I didn't want this confusing you. English is not my first language, appologies.

P.S. I would like to know what exactly is this special knowledge about GR that 105 apparently knows. Either there is a force exerted by all things with mass (gravity), or there isn't. Lol.

>>108

Well, its more than semantics. GR and gravity are two fundamentally different theories for explaining the same phenomenon, acceleration toward objects of mass. Just because in an accelerating frame, people can percieve gratitational acceleration as a force does not mean it really exists.

>>106
In the Theory of Gravity, gravity is caused by a force
Bullshit.

since Einstein we know that there is no force
Also bullshit.

Until you make an attempt to demonstrate either of those things, you are a worthless douchebag troll not worth more than those replies.

Principle of least action.
/forces
/thread

>>111

Well, I would expect you to be the one giving me explainations since you are going against current scientific theory. But, alas, I will feed you once at least.
H
=On question 1:

The theory of Gravity was produced by Newton, the mechanics of which were discribed by Newtons law of universal gravitation. Here is a translation of Newton's own words: "Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses"

There you go, gravity is a field force, according to the Theory of Gravity. This is the equation/relationship used in most introductory phy textbooks, and the definition of gravity. If you were not a 4channel I would not have even bothered with that, but some of you guys are uh, search engine challenged.

=On point two, why gravity does not exist:

Since you seem to be very handicapped in this area, let me define for you a term called the Inertial frame of reference:

 "If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K."

In the physical law we will consider is Newtons Laws, specifically F=ma. This law says that any force acting on an object will produce an acceleration dependent not only on the force, but on the mass of the object that is being accelerated. Now, lets see if we say an object falling on Earth follows this law. We will see that acceleration from "gravity" is constant, and does not change with the mass of the object.

This is not a property of true forces, as I have shown you, but it IS a property of what are known as fictitious forces.
As Einstein astutly noted, "gravitation force" is actually the product of a noninertial frame (accelerating frame) being taken as an inertial frame. This produces an apparent "force" in which all objects are accelerated by this force at the same rate, what is known as "gravity".

Einstein then noted:

"we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system." (Einstein 1907)

Saying that any reference system experiencing acceleration is physically equiv (indestinguishable) to a "gravitational field" within that frame.

"But 112! What causes this apparent acceleration that causes us to invent forces like gravity?", you ask?

Well, this is where GR comes in, in it Einstein showed that mass inertacts with space-time (discribed by different tensors) and changes it, space time in turn has an effect on mass and causes objects within it to accelerate toward the mass. See, no forces involved! (This is a kindergarden explaination, and I really am not so educated in the mechanism of GR, but I thought you might benifit from it)

Now, I hope I have showed you finally how you are wrong, how the Theory of Gravity is wrong, how gravity is not a force, and how gravitation is what we percieve as "gravity (doesnt exist)".

use the force

0.999... is not a well-defined number, you fucking nerds!