/sci/ - Science & Math

1/3=0.333...

3*(1/3)=3*(0.333...)

1=0.999...

Well, since your claim runs on nothing concrete but your own personal belief, I raise you my personal belief with no concrete evidence that .999999....=1.

1 > 0.999...

Real numbers are in fact not defined by decimals, but rather by Dedekind cuts. Each real number is essentially equated with the set of all rationals less than that real number. It is clear that if, given x and y, the set of all rationals less than x is the same as the set of all rationals less than y, then x is the same real number as y by definition. This clearly applies to the case of 1 and .999...

There is no difference between the two, thus they are the same... bitch.

1 - 0.999... = 0.000...1
So how can 1 equals 0.999 now?

>>4405616

The right-hand side of your equation is an infinitely small number.

>>4405616
Please see #2 in the yellow panel.

This pic also answers any other questions and confusion.

/thread

>>4405623
>>4405628
So is 0.999...

>>4405583
This is a number theory question and it should be answered with number theory, so let's take a look at this in the sense of Peano arithmetic. The real numbers are constructed from the rationals (which're constructed from the integers, which're constructed from the natural numbers, which're constructed straight from the Peano axioms, but I'm not doing all of that).

First a note on notation here, I'm not texing this up, so or = union, and = intersection, !0 = the empty set, Q = the rationals, R = the reals

Let's start by defining a Dedekind section then and the real numbers then, so we all know what we're talking about.

A Dedekind section of the rationals is an ordered partition (L, U) of Q such that L or U = Q and L and U = !0 such that if x is in L and y is in U then x < y and L has no greatest element. Since L uniquely determines U and vice versa let's just refer to this to this as L in place of (L, U)


Now we define the set of real numbers as the set of all Dedekind sections of Q. We can show (I'm not going to) that this is indeed what we mean when we speak naively of the set of real numbers.

Now all that remains to be seen for the proof that .999... = 1 is that the Dedekind section defined by one is exactly that section defined by the other, then they are the same number by definition.

Consider the Dedekind section defining 0.999... this is precisely the set of rational numbers p such that p<0.9^n for any natural number n (so those less than 0 or less than 0.9 or less than 0.99 etc) that is p is less than 1-(1/10)^n for some n. Every element of 0.999... is also an element of 1 then (since 1 > 1-(1/10)^n for any n)

....

>>4405648
Now for the converse

Consider the Dedekind section defining 1. This is precicely the set of rational numbers a/b such that a/b < 1. But then a/b < 1-(1/10)^n (I will show why this is so in a moment, see ***) so then every element of 1 is also an element of 0.999... But then 1 and .999... contain the same elements, and thus are equal by definition.

***
Since a/b < 1, we consider the minimum a/b (so for example 2d/2c = d/c, its the d/c we're interested in) and a is strictly less than b, clearly since otherwise a/b would not be less than 1. Now consider the LARGEST of these for each b, that is a = b-1, so we have (b-1)/b < 1, but (b-1)/b + (1/b) = 1, and its certainly true that there is some number of the form 10^n which is larger than b, and thus (1/10)^n is less than (1/b), so (b-1)/b + (1/10)^n < 1, but this was for the largest a for a given b, and is true for any b, so then its true for all a, b.

Please take the time to read this whether you believe 0.999... = 1 or not, this is the proof from the definitions, this is why it is true.

>256 posts and 10 image replies omitted

let x=0.9999.....
10x=9.99999.....
10x-x= 9.99999.... - 0.9999999...
9x=9
x=1

0.999999999 = 9*0.111111111111

= 9 * [ (1/10) + (1/100) + (1/1000) + ... ]
This is an infinite sum of

9 * (1/10)^(n+1) since we have to start from n=0
= 9/10 * (1/10)^n
This is a geometric series, with
k = 9/10 and
r = 1/10
since r < 1, this converges to
k/(1-r)
(9/10) / (1-1/10) = (9/10) / (9/10) = 1

0.99999... = 1

>>4405669
>>4405656
Looking at this from an analysis perspective is like tying one's shoe laces with a crane - its not the tool meant for the job. This is a number theory question and it should be answered with number theory. See
>>4405648

It's all made up shit anyway so who cares.

Taylor series, dude.
0.999...= 9 * 0.111...
= 9 * sum(i=1, infinity, (1/10)^i)
= 9 * ( sum(i=0, infinity, (1/10)^i) - (1/10)^0 )
= 9 * ( 1 / ( 1 - 1/10) - 1 ) // Taylor series for x^i
= 9 * ( 1 / (9/10) - 1 )
= 9 * ( 10/9 - 1 )
= 9 * ( 1/9 )
= 1

>>4405656
internal consistency is pointless if it lacks applicability.

I have a number of atoms which ultimately equate to the existence of an apple (a rather tasty one at that).

1

I remove a singular atom from this delicious apple.

1 - 0.00000000000...1 (or whatever the proper ratio of atoms in an apple equates this subtraction to actually be).

= 0.999999..9

Do I still have an apple? Yup.

>>4405813
Pragmatism has no place in mathematics.
Neither do strawmen or poor understanding of infinity.
Please GTFO.

I'm the samef­ag from a couple days ago who proved it was 1.

x = 0.999..
10x = 9.999..
9x = 9
9x divided by itself = x
9 divided by 9 = 1
x = 1

You take the effort to write a proof from the definitions which provides an actual insight into the nature of numbers and no one bothers to read it... What a shame.

>>4405833
Sorry. I was going to read it, but then realized I already studied Peano systems, Dedekind cuts, and Cauchy sequences.
So I was like meh. Too tired to read what I already know.

Can we just agree that any proof would only get within an epsilon?

the limit of -1/x + 1 is 0 as it goes to infinity, but it never actually equals 1. The best most proofs would be able to do is say "how close do you want to get"

Hey...hey mods...yeah, I actually need something from you....
PUT FUCKING SHIT LIKE THIS BULLSHIT AND THAT A=B BUT OMG DO ALL THIS SHIT AND A=2B EQUATION WHERE THE FAGGOTS DIVIDE BY ZERO INTO A STICKY THAT EVERYBODY SHOULD READY; ANY MORE FAGGOT BULLSHIT LIKE THIS COMES UP, YOU PERMABAN THE FUCK OUT OF THEM, EH?
I KNEW THAT 0.999...=1 WHEN I WAS FUCKING 14, WHAT THE FUCK

>>4405822

9x divided by itself is 1 you dumbass.

>>4405813

Shit analogy. 0.999... implies that there isn't a lower limit in terms of what you can add/subtract. The 9's 0.999... extend to infinity. You can't just write 0.999...9 and say it's the same as 0.999...

>>4405849
It bothers me that you're technically right.

>>4405854
y u so mad

.999..........9

doesn't exist prove me wrong faggots

>>4405846
Fair enough, but most of the high school students here haven't studied even the beginnings of number theory. If they actually took a cursory glance at some nice Peano simplifications they might start to see why arguments like :"1/3 = .333... ..." are retarded, and might learn an actual appreciation for mathematics.

>>4405849
>>4405857

How is he technically right? 0.999... is an irrational number. -1/x+1 is a rational number. You wouldn't expect the rational "version" to ever equal 1. But, you're right (I think you made a typo) that as it approaches infinity it approaches 1. 0.999... is also a number that that function will always approach, but will never be!

>>4405858
SHOULDN'T I BE? YOU AREN'T MAD BECAUSE YOU ACTUALLY LEARNED SOMETHING YOU SUBHUMAN NEANDERTHAL

>>4405849
Fucking no we can't you ingrate. The argument doesn't even require the existence of limits, limits are fucking a mile away from relevant to this question. Why are you all so obsessed with analysis? Broaden your mathematical horizons.

PS - I know why you're all obsessed with analysis (you've all only studied calculus).

>>4405870

I don't see how those arguments are incorrect. They don't take any understand of the underlying mathematics into account, but they do resolve the question itself.

>>4405877
PROTIP:
Even equivalence classes of cauchy sequences are defined in terms of limits.
The number theory argument stems from that too.

>>4405878
They're not incorrect, but I its not whether an argument is correct or incorrect that makes it good or bad, don't you agree? I'd rather see a beautiful argument which happens to be wrong than one that shows no understanding of the mathematics.

>>4405887
PROTIP:
You're painfully stupid. Please fuck off. The number theory argument doesn't require the existence of sequences.

>>4405772

Your proof is horribly flawed in your third line. You cannot separate an exponential expression into a subtraction (i.e. 0.1 ^ i = 0.1 ^ 1 - 0.1 ^ 0). You could do a subtraction on the exponent itself, then separate the expression into a division (i.e. 0.1 ^ i = 0.1 ^ (i - 0) = 0.1 ^ i / 0.1 ^ 0). This particular example, however, will do nothing because, as everyone knows, anything to the zeroth gives one; and anything over one returns the numerator.

>>4405871
Why is 0.999... irrational?

>>4405905
It isn't, ignore that bloke.

>>4405892

Absolutely, I do agree. The simple proof ought to be enough to put this thread to rest, though. But, I do like your approach better. The elegant proof would be much more valuable over the simple one in any serious discussion.

>>4405905

Go ahead and make it a fraction of the form (-1/x+1) and you'll see why immediately. Hell, the definition of irrational is it cannot be made into a fraction. You obviously cannot make it into a fraction!

>>4405915
Please quit while you're ahead. I can think of a million and one fractional forms for it, let me list some for you...

1, 2/2, 3/3, 4/4, 5/5, ...

>>4405921

Right, and that would validate the point that 0.999...=1.

But okay, I'll admit that that's a mistake. I'll change my argument to: You cannot possibly put 0.999... into the form (-1/x+1) with some number x, which was the argument I was alluding to.

Well I'm going to stress once more before I leave that people actually look into the number theory. See my posts below for a dip into it, but if it sounds interesting just google Peano Arithmetic, its a good starting point for getting real mathematical insight as far as number theory is concerned.

>>4405648
>>4405652

Sometimes its nice to think about why things are true, rather than just using argument based in mathemagic.

>>4405871
0.9999.... is rational since it is repeating, but for argument's sake,
e is irrational, and is expressed as a limit of a sum of rational numbers

>>4405877
integrals are limits

>>4405930
I only used that function because it approaches 1 from below as x goes to infinity, and starts to look like 0.999...

>>4406092
>integrals are limits
FUCK REALLY!?! Good observation! Luckily there were no integrals in the argument! Phew.

I blame #3

Where is your god now?

>>4406103
>Fucking no we can't you ingrate

Forgive me, you were difficult to understand. Thought you were distinguishing integrals and limits or something.

>>4406127
Your forgiven. Ingrate, not Integrate.

>>4406135
Oh wow, I don't know why my mind just slipped a t in there. Too much advanced calc homework tonight.

>>4406135
>assburgers math faggot detected

>>4406158
Because going around on the internet calling people 'assburgers' sure isn't a sign of social retardation.

ahem.... let's just say 0.9999999999999...=1. As the 0.000......1 is so infinitely small and immeasurable that we don't really give a shit. Let's say that that 0.000....1 is the total length of your dick. Might as well say it doesn't exist. 'your dick'=0

This is my dad's argument. Take it down.

Lim (n-1/n)=1
n->∞

for what real value of n is (n-1/n)=1?
GO!

>>4407530
It's a sound and valid argument. Accept that you are wrong.

bump

>>4407530
(n-1)/n is never equal to 1, no matter what real number n is. But he's right that the limit is 1.

Behoald!

0.9999>1

>>4405588
shockingly convincing it is.

>>4407544
So... you're saying that my dad is right?

>>4407558
Of course he is. Now educate yourself.

>>4407562
Oh great one. I bow down to your awesomeness, and I would like you to educate me.

>>4407544
infinity is not a real. n-1/n is never equal to one for a *finite* number n.

>>4407568
He's wrong, as is your father. 0.9 recurring equals one, as a proper mathematician would be able to confirm for you.

The sum to infinity of any geometric progression is a/(1-r). The geometric series with a=0.9 and r=(0.1) therefore sums to 1. As you can see summing the series to a finite number of places (let's say 5) the GP gives 0.99999. where n= infinity, you get an infinite number of 9s, otherwise known as 0.9 recurring.

>>4407576
Any geometric progression with |r|<1*

>>4407576
>as a proper mathematician would be able to confirm for you

So we need arguments by authority now?

10x - 9 = x

What is x?

>>4407582
Not at all. In fact, I gave you the argument right there (I see you ignored it).

All I'm saying is that all of you could clear this up by asking someone better qualified, such as a career mathematician.

>>4407586
1.

>>4407586

x = 9/10

>>4407586
What does this prove?

>>4405648
>>4405652

Thanks for posting that; I'll be looking more into Dedekind cuts and related topics due to it.

It only works when you accept the conceit of infinite number of parts (which we do in math -- it's just convenient).

It's a little moot because if you have have integers that divide to make 1, such as 4/4, you won't produce any nines by the standard methods of decimalization of fractions. That is, you have to fuck up to produce the nines (uhhh let's see, how many 4s in 40, I guess its nine, subtract 36, get a 4, repeat the fuckup).

Because 1 > 0.999..., whereas 1=1

>>4407642
Except the math authorities rigged the game by saying that the value is the limit. (Unless you play with the infinitesimals concept alternatively.)

>>4407657
1 > 0.999...

>>4407657
The number can be as close to one as it wants, but it will never be one. Therefore 1 > 0.999

I will prove your wrong.

Pick any
<span class="math"> \epsilon > 0 [/spoiler]

Notice that
<span class="math"> |1 - 0.9\Sigma_{i=0}^n 0.1^i| < \epsilon [/spoiler]

after at most <span class="math"> ceiling\left{\frac{ln|\epsilon|}{ln|0.1|}\right} [/spoiler] steps.

Since the step size is infinite, we can take any number of steps we want and no matter what number not equal to one you claim it is equal to, I can tell you when it will have become closer to one than that.

>>4407663
Please quantify the difference between 1 and 0.999...

Then please quantify the difference between that number and 0.000...001

>>4407663
I pretty much agree, because I don't have the conceit that I understand having a whole partitioned into infinitely many parts, but the math authorities accept and embrace the idea, and to participate in their work, we accept the conceit, which is useful after all.

>>4407679
>mfw he's never looked for the area under a graph

>>4407683
Right, another place where it's useful to imagine we partition a whole into infinitely many parts and have the limit of the increasingly tiny parts' sum give us the value of the whole.

http://qntm.org/pointnine

A few proofs..

>>4407712
What do you think of >>4407576 ? I didn't get any proper responses, and you look entertaining.

>>4405600
>>It is clear that if, given x and y, the set of all rationals less than x is the same as the set of all rationals less than y

What the fuck am I reading.

x = 2
y = 100

>>MFW people cannot into basic English

>>4407678
Consider the line y = (-1/x^2) + 1

Y will never be one. As x approaches infinity, y approaches one; but y will never be one. There's something called an Asymptote at one. One will always be greater than the value of Y.

Therefore 1 > Y and 1 > 0.999...

>>4407733
>trolls being defeated by a 10th grader
Mark today, for it will forever be known as the day /sci/ beat the trolls.

>>4407729
>It is clear that if, given x and y, the set of all rationals less than x is the same as the set of all rationals less than y, then x is the same real number as y by definition

>>4407733
The operative being Approaches.

For any finite recursion, there is a finitely small difference between the number and 1. For any infinite recursion (not a finite recursion that approaches an infinite one, rather one that is already infinite) there is an infinitely small difference, IE 0.

In your specific example, y is not 1 for any finite value of x. However, for x = infinity, -1/x is zero, and therefore f(x)=1.

Thanks for having a decent conversation.

>>4407744
If x = infinity then it would be 1/infinity, not zero. It would be an infinitely small number, sure, but it wouldn't be zero.

>>4407722
Sure, that's where students usually first encounter the idea of limits, and they may be giving exercises where they take decimals such as 0.35353535..., re-express as 0.35 + 0.0035 + ..., apply the sum formula and thereby derive the ratio of integers. They may be shown that applying the process to 0.9999... gives 1/1 also. Now, because we learned to derive the decimals from long division of the numerator and denominator of fractions, and the process is progressive, one digit generated at a time, that is how students' minds are trained to understand what is meant by decimal notation, and if done correctly, will never generate 0.9999... by such a process, so 0.9999.. itself is novel, and the idea of infinitely many parts is novel, and so that where it gets weird for people.

>>4407751
I think your confusion comes from not understanding how infinity works.

If there is a finite difference between two numbers, then they're not the same. If there is an in-finite distance between two numbers, then there's no distance between them. Please tell me what the difference between 1/x and 1/2x is where x=infinity.

Also, do you understand geometric progressions? If so, check >>4407576

>>4407765
Oh, I see your point.

In the case that x = a finite number, 1>0.999...
In the case that x = an infinite number, 1=0.999...

Is this correct?

>>4407772
Absolutely.

>>4407606
No worries, I'm glad someone actually took the time to read it instead of just talking about limits, lol.

>>4407777
Then the debate of whether or not 1 = 0.999... is meaningless because the OP didn't clarify whether or not we 0.999... is a finite or infinite number.

Thanks for your explanation, I truly appreciate you taking aside time to explain it to me.

>>4407785
He's wrong though, please see.
>>4405648

>>4407657
As has been said many times above, you don't need the concept of limits to define infinite decimals. But you do need a definition. It doesn't do to pretend that an infinite decimal is the same kind of beast as a finite decimal. You cannot define something as the final result of an infinite sequence of arithmetic operations. Such a sequence of operations never has a final result because it is infinite. You can talk about the supremum or the limit of the intermediate results, but not of the final result itself, which doesn't exist.

>>4407803
Students typically first encounter 0.999... becoming evaluated as 1 through the concept of limits in their sequences and series units in high school. If they never move on to calculus, that may be all they hear of it. That's why it's worth bringing up.

>>4405628

That isn't a proof by induction....

Though the picture is correct otherwise.

>>4407785
0.999... is used to represent <span class="math">0. \dot{9}[/spoiler], in lieu of a better form of notation. OP could have used LaTeX and written it as \dot{x} but that's pretty advanced knowledge for someone who doesn't use tex.

>>4407842
Oh, so he did define it as a never-ending number?

Alright then. Thanks /sci/.

>>4405639
i think you need to check your math.