/sci/ - Science & Math

No.
"..." and infinity are not real numbers and shouldn't be treated as such.

http://en.wikipedia.org/wiki/0.999......

>>2613744

wiki says its real but im really confused.

>>2613743

Why this isnt correct?

it is correct as a statement, but not as a proof as you haven't proved the hard part

which is things like 10 x 0.999... = 9.999...

and 9.999.... = 9 + 0.999...

to prove this hard part you have to learn a branch of maths called real analysis

>>2613766

so in real world u cant take like 10 cm of material and cut it into ten 0.(9) cm pieces. coz there will be some material left?

>>2613766
in fact, you haven't even defined yet what 0.999... actually means

>>2613778

it means 0.(9) -> 0.999999999999999999999999999999999999999999999999999999999999999999etcetcetc

0.99999 = x
10x = 9.99990
9x = 8.99991
x = 0.99999
reoccurring numbers have to end somewhere since infinity is just a concept. and when it does...

>>2613777
nothing to do with real world. forget the real world.

real world you can't cut 1 cm, nothing is 100% accurate.

point is you need to define what 0.999... actually means, (an infinite sum, so what does infinite mean, etc), then show you can do arithmetic with infinite sums and it works as you expect.

all this is possible for decimal numbers. but it takes a bit of work.

think about it
what's the number that needs to be added to 0.999.. to reach 1
logically it's <span class="math">\frac{1}{\infty}[/spoiler]
if you believe that <span class="math">\frac{1}{\infty}=0[/spoiler]
then there is no problem
if not.. i'd say it's close enough for scientists :)

>>2613785
>>2613782
define "etc". in maths you can't justr say "oh, you know"

the "concept" of infinite decimals can be handled precisely by using limits and something called epsilon-N notation

>>2613792
this is imprecise nonsense talk. ignore it. there is a way to do this properly

>>2613785
i guess you think natural numbers 1, 2, 3, 4, 5, ... have to end somewhere, herp derp

>>2613799
>doesn't know shit about nonstandard analysis
>i don't know it, therefore it's bullshit
>typical /sci/tard

Look at it this way:
1/3=.333333333...
(1/3)*2=2/3=.6666666666.....
(1/3)*3=3/3=.9999999999.....
3/3=1

Fuck.
Why do people keep making these threads? They always start huge endless arguments.

Hell, at least come up with a new question. Like, is infinity/infinity = 1?

>>2613799
it's logically correct
what is the thing you find imprecise?

>>2613806
of course i know about nsa, but what you posted is non-rigourous shittalk.

nsa is also pretty shit tier pure maths, though loved by people that don't want to do the work of real analysis and just read pop science books

>>2613805
if nobody's counted to infinity then I don't see how that's such a ridiculous claim. You could say that they go on forever but in this instance, it explains the proof.

>>2613815
you haven't defined what 0.999.. means, or what infinity means, or how division works on infinity, you've just said shit basically

got 10 pencils.
cut each the 0.(0)1 part
i got 10 = 0.(9) pencils + the 0.(0)1 part

>>2613820
you are confusing a set of infinite size and a number called infinity.

easy to prove they go on forever. if they didn't there would be a highest number n. but n+1 is higher, therefore our assumption they stop is wrong.

you can make a map between 1, 2, 3, 4, ... and decimal places, this proves the decimals go on forever

i'd get out this thread though dude, it's way over your head

>>2613808
this is a problem caused by the counting system we use
base-10
if you divide 1/3 in base-3 there will be no such problem
but then dividing other numbers cause similar problems

>>2613785
>reoccurring numbers have to end somewhere
it's been proved they don't. if they did all numbers would be rational numbers
>since infinity is just a concept.
all of maths is concepts

>>2613822
the same definition that you'll find in every math book

1/9= 0.111.....
1/9*9=1
0.111... * 9 = 0.999....
so 1=0.999

>reoccurring numbers have to end somewhere since infinity is just a concept. and when it does...
>0.(0)1
>MFW people are retarded near me

>>2613834
I understand the concepts of infinity and continuous sets but if you multiply something by 10 it will be 10 times larger, hence for this proof, the number shouldn't be continuous.

Even if it went to infinite places, multiplying by 10 still moves everything up 1 decimal place and adds a 0 at the end. You can argue that it goes on forever and the final digit (n) will be 9 but n+1 will then have to be a 0 etc. It ends somewhere and when it does...

0.99999999999999999........... Is just another way of writing 1.

>>2613848
there are a few definitions of infinite decimals
two main ones, sum of an infinite series
partition of the real line

or do you mean high school books herp derp

there are many precise definitions of "infinity", all different. in the 0.999... case it is a map between the natural numbers and the terms of a series, and a limit of that series to "infinity".

in your case of 1/infinity, it's not at all clear. any maths book with that in would be retarded.

>>2613861
>i understand

you've just proved you don't with the rest of your post

the x is now diamonds

>>2613861
lol, retard math

lrn2analysis

>>2613869
suck my penis. you know it's true

>>2613861
>0 at the end

only if there is a last decimal place. there isn't.

you don't really understand infinite sets.

>>2613869
i like how he introduced continuity, without really knowing wtf he was talking about. probably thought it meant infinite or something

>>2613889
infinite isn't even a word. dumbass

>>2613865
with sum you may represent an irrational number
or how fast a series grows but not infinity itself
to have this representation means that you have a concept of infinity
but use this:
http://en.wikipedia.org/wiki/Infinity

0.999.. is just a number

an analogy would be 0.9 how much does it lack to reach 1
it's 0.1
we use this analogy with 0.999... (which has an infinite number of nines behind the decimal point)
to represent 0.1 in the case of 0.999... we need to use infinity
so 1/infinity = 0.000...01 logicaly infinite number of zeros before 1
but we say that 1/infinity = 0 because of the nature of infinity itself we can never reach 1 the last number
there is no last number with infinity

logical right ;)

bout all theories lemmi tell a joke.

you+dog=3 legs each by the power of statistics!

>>2613912
no one claimed we were representing infinity as a number except the retard who put 1/infinity

you keep using the word "concept" as if it meant anything. everything is maths is concepts.

>quotes wikipedia on infinity when i have a B.Sc. and M.Sc. in maths

lol

>uses analogy in maths

herpity derp

>>2613908
word isn't even a word, underagedbanned

>>2613925
>i have a B.Sc. and M.Sc. in maths
find it hard to believe

>uses analogy in maths
this is what applying axioms and theorems means

please go back to >>>/b/

>>2613908

¹Infinite - adj \ˈin-fə-nət\

1: extending indefinitely : endless <infinite space>

2: immeasurably or inconceivably great or extensive : inexhaustible <infinite patience>

3: subject to no limitation or external determination

²Infinite noun

something that is infinite (as in extent, duration, or number)

>>2613955
>analogy = axioms and theorems

i find it hard to believe you are over 14

i've given the two 1st year uni standard accounts of infinite decimals. i don't see anyone else here except retards

>>2613739
Correct, especially if you use "intuition", but there are a few problems with it. First, you need to define in what system you're operating, that is, is it real numbers or something else. When you define real numbers (and limits), you'll need to use limits to prove that 0.(9)=1 within that axiom system. Without defining the axioms (and the system in which you're operating), the proof is somewhat meaningless, and within some axiomatic systems, the proof is even wrong. However, for layman's understanding of math (which is usually implies real numbers), the statement 0.(9)=1 is true.

>>2613912
>logical

not really. you've introduced infinity as a number without any justification. you said analogy, but analogy falls down a lot in maths.

to do this successfully you need rigor.

Numbers cannot into infinity.

>>2613964
Haven't read anything else in this thread, but this guy seems on the level.
Modern maths works like this: we define inference rules and axioms.

We infer.

That's all there is to it, just some first order predicate logic wankers, the whole bunch of them.


Btw I wouldn't trust someone who claims to have a masters in maths and still fails to capitalize consistently, not entirely sure, though.

mfw people seriously talk about numbers "reaching" other numbers, and the end of infinite sequences.

giuse, prove that <span class="math">1.\bar{0} = 1[/spoiler]. They're not written the same way so they are obviously different numbers.

>>2613978
well, that's me.

here i go not capitalising with my M.Sc.

do you think there's some magical time when we move from being someone who knows how to use upper case letters, but chooses not to on message boards, to someone that decides he really ought to, given that he has letters after his name?

i don't think so.

>>2613963
i'm 25 not that it matters but i have my masters degree in CS

>>2613967
is it that hard to see the idea
yeah i've written 1/infinity instead of limit_(x->infinity)(1/x) cause it's easier to read

and the analogy here is quite obvious
just like any other number you have to add something to reach the other number

i would like you to explain to me why this fails in this case?

the point that i make is that you add nothing to reach 1

maybe i'm no good at explaining things..

One note on this thread and to the OP:
the "proof" may not be mathematically rigorous.
But it works in the following sense:

if you want your "numbers" to obey the "laws" used in the proof, i.e.
10 * 0.9999... = 9.9999..
and
9.9999.. = 9+0.9999..
and the distributive law, then it follows that 0.9999... must be equal to 1.

That has been proven.

Now, only a complete fucktard would argue against the rules used above, I'm sure.

>>2613978
I'd go on the content of their posts.

If we judged mathematicians by such superficial things we would have ignored so many mathematicians.

grigori perelman looks like a tramp

pic related

>>2613978
he may have the degree you don't need to be genius to have a degree
and lot's of people don't capitalize first letters of sentences(like me)
considering the people from my group graduated you don't even need to be smart to have a masters degree

>>2613997
>CS

explains a lot

basically i will criticise a proof whether what it is trying to prove is true of false if the proof lacks rigor.

else you could say "1+1 = 2 because potatoes", and to anyone who criticised this proof you could ask if they were denying 1+1 = 2.

maths would be a horrible mess if you kind of nonsense was allowed

>>2614001
yes, I've looked over the posts of the guy who claims he has a masters in maths, and I haven't found anything that leads me to believe he isn't full of shit, basically.

Add to that the fact that he needs to make his posts more important by noting that he has a masters (which in no way can ever be proven) and his inconsistent capitalization.

So I agree with you completely. People on 4chan should be judged by the content they're posting. And judging from that I'd say he's full of shit.

>>2614007
I hope you misquoted, in case you didn't...

I just stated that the "proof" rigorously works if we take for granted the rules it uses to infer the result. This is how rigorous proofs work, deal with it.

Now most highschool idiots won't be exposed to any rigorous notion of even a real number, so trying to prove the trivial result 0.9999... = 1 in true rigour will be doomed to failure, while using the OP's proof will work, unless they want to deny that any of the rules used don't work for their intuitive notion of numbers as decimal representations of reals.

>>2614008
>content of they're posting
oh my fucking god, I spent too much time on 4chan, that's for sure..
Or perhaps I meant to write
>the content that they're posting

I always believed that 0.9999 =

<div class="math">\lim_{n\rightarrow \infty} ( 1 - \frac{1}{10^n} )</div>

<div class="math">\sum_{n=1}^{\infty} (9 \cdot (\frac{1}{10})^n)</div>

and i'm only a first year math student, but I was told during analysis that infinity is the number that is greater than all others, and is the surpremum of a diverging series

>>2614011
I don't know, if a highschooler's intuition on dealing with infinite decimal representations of reals was sufficient, this thread wouldn't be so popular.

>>2614017
It's not usually considered to be a number, but that's pretty close to the mark.

>>2614017
That's the "infinity" that's viewed as "the end of the number line".

I think a more rewarding global definition for "infinite" is "unbounded".
The natural numbers are "unbounded", thus they're infinite.

However, the interval [0,1] is most certainly bounded, but the *amount* of real numbers in that interval is not, so again, infinity comes into play.

Also note that there are cardinalities and you can define an arithmetic on them.
Not sure, but I think in that context the "end of the real line" infinity would probably be aleph_0, or perhaps there's no connection between these 2 notions of infinity at all, I don't know.

>>2613739 here. Here's my proof:

0.999... = 0.9 + 0.09 + 0.009 + ...
= 9/10 + 9/100 + 9/1000
= 9 * ( 1/10 + 1/100 + 1/1000 + ... )
= 9 * (sum(from 1 to infinity) 1/*(10^n))
= 9 * (lim n-> infty (sum(from 1 to n) 1/*(10^n)))
= ... ( read the proof after this to see how to make the transition past this point)
= 9 * (1/9)
= 1
Proof for the 0.(1) = 1/9:

Now to calculate that sum of a geometric progression (power series), you need to show that 1+a+..+a^(n-1) = 1-a^n/1-a, for a!=1. We do that by proving the base case, then assume n and prove n+1 is true:
To prove P(n): 1+a+a^2+a^3+...a^(n-1)=(1-a^n)/1-a, for a!=1
P(1): 1 = 1-a/1-a = 1 (true)
P(2): 1+a = (1-a*a)/1-a = 1+a (true)
Assume P(n), and prove with this P(n+1):
S(n) = 1+a+a^2+a^3+...a^(n-1) = (1-a^n)/1-a (assumed)
P(n+1): 1+a+...+a^(n-1)+a^n = S(n) + a^n = (1-a^n)/1-a + a^n = (1-a^n)/1-a + (a^(n)-a^(n+1))/1-a = (1-a^n+a^n-a^(n+1)/1-a = 1-a^(n+1)/1-a = S(n+1), thus P(n+1) is also true, and by induction P(n) is proven.

>>2614023
> continued
Now when 0<a<1 (a fixed), let's see what the value of is:
lim n-> infty (sum(from 1 to n) 1/a^n)) = lim n->infty (1-a^(n+1)/1-a) = 1/1-a - 1/(1-a) lim n->infty a^(n+1) = 1/1-a
( lim n->inf a^n = 0 for 0<a<1 (can be proven using the definition of limit)).

Anyways, applying that to our current value:
lim n-> infty (sum(from 1 to n) 1/*(10^n)) = 1 + (1/(1-(1/10))) - 1 = 9 * 10 / 9 - 1 = 10/9 - 1 = 1/9.

PS: I'm sorry for sucking at TeX, so I hope it's readable without it.

Erm, >>2614023 here, and I meant I'm >>2613964, not >>2613739.

>>2614023
nobody's gonna read that shit. Everyone who already "believes" the result probably knows several ways to prove it himself and everybody who doesn't know will have a lot of points of attack to your "proof".

Protip: you can't start proving anything about real numbers without even defining what a real number is.

>>2614028
I could do that, but it's going to take me a lot more posts than that to define real numbers and limits. Why not just read a mathematical analysis book instead of having me post all that?

>>2614032
well you could take the high-route and use an axiomatic approach.

R is a Dedekind-complete ordered field, there you go. In fact, all you really need is that the reals are an ordered field, because from this it follows at once that there is a real number between every to non-equal real numbers.

And easy reasoning will soon yield that if 0.999.. really is a real number, then there can't be any real between 0.999.. and 1.

the formal proof, and if you don't understand it, then GTFO

>>2614028
Also, it's not a matter of "belief", math is either true or false (or unprovable), given some system of axioms. 0.(9)=1 is true as far as real numbers in base 10 are concerned.

>>2614040
every *two non-equal real numbers

Why do we need to have this thread so often?

>>2614049
Quite obviously, which is why I wrote "believe".

But I mean it's all the same really, you *can* express disbelief in a rigorous proof, even though you understand every single step, simply because you don't agree with the overall result intuitively.

Which would mean there's either a fundamental problem in the axiom system you're using or the proof is in fact botched.

The thing about proofs is that they can get complicated. Most proofs nowadays are reviewed using a heuristic approach. There are proofs that some parts of the mathematical community doesn't "believe" (I think the classification of finite simple groups or something? It was some proof that heavily relied on computers doing special cases.)

>>2614057
0.(9)=1 makes sense intuitively for me as intuitively, I imagine 1-0.(9) = 0.(0)1 = 0. In the sense that there's no infinitely small number different than 0, however the problem here is that given an entirely different axiomatic system, such infinitely small numbers could exist, they just don't exist in R.

>>2614045
wow, I'm amazed time and again by the retardation displayed in these so called "explanation" pics.

The "explanation" why OP's proof fails is WRONG. Utterly and completely wrong.

>on the right side the assumption to be proven ("c = 0.999..") is inserted immediately.

But that's WRONG you motherfucking retard. The assumption is "c = 0.9999...", the thing to be proven is "c = 1", yielding "0.999... = 1".

Fucking idiots, copy and pasting other idiots.

>>2614065
>infinitely small number
I'm fairly certain that contradicts order-completeness.

>>2613739
Where did I put that image....ah! Here it is!

>>2614087
How is that a proof by induction?

OP IS WRONG

1=1
0.9999... = 0.9999...

simple put 2x0.9999... =1.8888...

>>2614087
this is fugly and obviously written by a /b/-tard.

I would actually bother to make a pic, just so that these retard-explanation pics don't circulate as much, IF THERE WEREN'T A COMPLETE FUCKING WIKI PAGE ABOUT THE TOPIC. Seriously..

http://en.wikipedia.org/wiki/0.999......

Although I don't like that page too much either. Basically everyone who needs "proof" for 0.999... = 1 is not really ready to understand said proof, because he probably doesn't know what a real number is (rigorously).

>>2614093
it's obviously not. These pics are all written by attention-seeking children.

>>2614095
nope, learn2longdivision.
You forgot to "carry the 1s", idiot.

You get 2*0.9999.. = 1.9999999...
for example 2*0.999 = 1.998 (to illustrate your failure in an example which doesn't involve "infinity" (hurr))

>>2614104
obviously I didn't mean "longdivision". Not sure what the actual word I wanted to use is, though. (Long multiplication?)