/sci/ - Science & Math

0.999...
vs
1.000...

First digit is off by +1
All the rest of the digits are off by-1

>But infinite times makes them all off by 0

Proof it

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...

>>14778075
>First digit is off by +1
But if I add 1 I get 1.99999...

>>14778085
Thats not what offset means

That would just make every other digit -9 off then

>>14778102
Sry +9

>>14778084
>1/inf = 0
nice bait

[math]0.999\ldots \cdot \pi = 3.1415926535\ldots[/math]

[math]0.9999999999\ldots \times 3.1415926535\ldots = \pi[/math]

>>14778057
Cry harder, midwit crank.

>>14778057
>pi
>0.999...
not numbers

>>14778084
>= 0.9... + 1/inf
wrong
it should be 0.9...+ 1/10^inf

e^iπ=-1

>>14778144
>implying it's not true

Claim: [math] 0.999_{\dots} \neq 1[/math]

Proof: We use induction. The base case is trivial: [math] 0.9 \neq 1[/math]. Next we introduce the notation that [math]0.9_n = \underbrace{0.9999999}_{n-\text{many nines}}[/math] is the decimal with n-many 9s.

Now the inductive step: we assume [math]0.9_n \neq 1[/math]. Then trivially [math]0.9_{n+1} \neq 1 [/math]. It might help to notice that [math] 1 - 0.9_{n+1} \neq 0[/math].

This implies that [math]0.9_n \neq 1 \qquad \forall n\in \mathbb{N}[/math]

Finally, we define [math] 0.999_{\dots} := \lim_{n\to\infty} 0.9_n[/math].

[math]\therefore 0.999_{\dots} \neq 1 \qquad \square [/math]

>>14778075
Give me an example of a number smaller than 1 but larger than 0.999...

>>14778581
Lmao I've seen gas stoves better at induction than you

>>14778610
What's wrong with my induction?

>>14778581
>not 1 for finite n
>thus also not 1 at the limit as n -> inf
Anon, do you have brain damage? Do you know what a limit is? You defined 0.999... as the limit and then concluded that it's still not equal to 1?

>>14778631
https://en.wikipedia.org/wiki/Transfinite_induction

>>14778625
[math] 0.999... > 0.9_n \forall n \in \mathbb{N} [/math] which is easily proven by [math] \epsilon [/math]-proof.

Your induction only works for real numbers. You don't have [math] n \in \mathbb{N} [/math] for [math] n \rightarrow \infty [/math]

https://en.wikipedia.org/wiki/Transfinite_induction

>>14778057
>YOU NEED TO WRITE OUT LITERALLY EVERY SINGLE DIGIT OF PI IT'S ABSTRACT IT'S NOT REAL IT'S NOT REAL NOOO
wtf?
>i dont fucking get it
yeah, we neither. wtf are you on about?

>>14778705
Dude we are literally calling you out on this exact type of proof. Instead of repeatedly posting the link, how about you try to find flaws in our counterarguments? Ya'know.. like a mathematician?

>>14778639
>>14778705
https://en.wikipedia.org/wiki/Red_herring

>>14778729
https://en.wikipedia.org/wiki/Transfinite_induction

>>14778437
10^inf = inf
duh

>>14778625
Induction can only prove stuff about "finite things", and doesn't (immediately) generalize to infinite stuff, such as this infinite string of nines.

Otherwise, let me prove the set of natural numbers is finite by induction. Base case: empty set, clearly finite. Now suppose {1..n} is finite. Adding (n+1) to it clearly keeps it finite, so {1..n+1} is also finite. Then the set of all integers is finite (wrong).

Look up transfinite induction for a generalization.

As for the question in the OP: someone please tell what is the distance between 1 and 0.9 periodical? It cannot be anything other than zero, hence they are the same. If you don't accept that, then you're not talking about the real numbers and no one gives a shit.

>>14778852
Then how do you prove infinite things?

>>14778864
By literally any other method? For instance I gave a proof about the question in the OP. Or look up the proof on the infinitude of primes. Or how about the fact there are arbitrarily large gaps in the primes (for every natural number N, there's a sequence of N consecutive numbers that are *not* prime, easy by considering N! and its successors). Those are all proofs about infinite stuff. But if you want to be brainfucked look up the Löwenheim–Skolem theorem.

>>14778639
What the fuck is this reply implying? You did not consider the limit case which is precisely why your proof fails.

>>14778873
>>14778864
>>14778852
>>14778631
>>14778581

Idiots, just use the epsilon lambda definition for the limit to see that it's dependent on other terms such as epsilon and delta, in itself it's not representable as 9 repeated infinite times, it's just the definition of the limit.

And infinity has nothing to conflict here with induction.

>>14778911
>Idiots, just use the epsilon lambda definition for the limit to see that it's dependent on other terms such as epsilon and delta, in itself it's not representable as 9 repeated infinite times, it's just the definition of the limit.
what? You mean the limit of the sequence 0.9, 0.99, 0.999, etc? That's clearly 1, what's the issue?

>>14778581
Lel, I got a better bait for you :
We know that :
[math]0,111111111...=\frac{1}{10}+\frac{1}{100}+\frac{1}{100}+...=\sum_{n=1}^{\infty }\frac{1}{10^{n}}[/math] = (A)

Also :
[math]\frac{1}{9}=0,111111111...[/math] = (A)

And :
[math]\frac{8}{9}=0,888888888...[/math]

Obviously :
[math]1=\frac{9}{9}=\frac{1}{9}+\frac{8}{9}[/math]


Then with (A) :
[math]1=\frac{1}{9}+\frac{8}{9}=1\sum_{n=1}^{\infty }\frac{1}{10^{n}} +8 \sum_{n=1}^{\infty }\frac{1}{10^{n}}[/math]

[math]1=\frac{1}{9}+\frac{8}{9}=(1+8)\sum_{n=1}^{\infty }\frac{1}{10^{n}}[/math]

[math]1=9\sum_{n=1}^{\infty }\frac{1}{10^{n}}=9(0,111111111...)=0,999999999...[/math]

Checkmate >>14778057

[math]\begin{array}[c]\\
\dfrac{1}{9} = 0.111\ldots\\\\
\dfrac{2}{9} = 0.222\ldots\\\\
\dfrac{3}{9} = 0.333\ldots\\\\
\dfrac{4}{9} = 0.444\ldots\\\\
\dfrac{5}{9} = 0.555\ldots\\\\
\dfrac{6}{9} = 0.666\ldots\\\\
\dfrac{7}{9} = 0.777\ldots\\\\
\dfrac{8}{9} = 0.888\ldots\\\\
\end{array}[/math]

Then just extrapolate one step further, and what do we get? You guessed it.

[math]\dfrac{9}{9} = 1 = 0.999\ldots[/math]

What about surreals

>>14778057
Because .999... is clearly rational, while pi clearly is not.
[math]x = 0.999...\\
10x = 9.999...\\
10x - x = 9.999... - 0.999...\\
9x = 9\\
9(x) = 9(1)\\
x = 1[/math]
Since pi is not a repeating decimal, the above method will not work.

>>14779271
The three dots can be used to indicate non-repeating irrtional decimals

>>14779294
They are not being used that way here.
A less ambiguous notation is
[math]x = 0.\overline 9\\
y = 1.2\overline {345}[/math]
which shows exactly which digit(s) repeat. But this is not a universal standard afaik (but it should be.)

>>14779169
>nine lines repeating the same tautology
what exactly do you think you are extrapolating?

>>14779302
I think [math]0.\bar{9}[/math] or [math]0.(9)[/math] is more of an universal standard than [math]0.999...[/math]
The triple dot notation is a mental shortcut at most.

>>14779271
the leap from
>x = 0.999...
to
>10x = 9.999...
is a disgusting piece of abuse, which demonstrates that only God can be trusted with infinity

>>14779402
Not him but it really isn't. The number of 9s is infinite so there's no problem moving the decimal point by an arbitrary number of places either way. It's not like you will run out of infinite 9s.

pi >~ 3.14159...
1.0 >~ 0.9999...

faithlessly assume the "equals" ( = ) button on a calculator is actually true rather than meaning "attempt to solve"
1/3 = 0.333...
0.333... x 3 = 0.999...
something was lost

correctly assume ( = ) means attempting to solve, combined with logical inferences
1/3 > 0.333...
0.333... < 1/3
3(0.333...) < 3(1/3)
0.999... < 1

>>14779422
>attempting to encapsulate infinity with your feeble human mind

everyone in this thread is a retard except me

>>14779501
That's exactly what a retard would say

it actually follows from the MVT that its =/= 1...

>>14779434
What part of [math]0.\overline 9 = 0.9\overline 9 = 0.99\overline 9[/math], etc. is so hard for you to wrap your feeble American brain around?

>>14779619
I'll bite, explain how

>>14778144
he's right tho

>>14779169
1/9 = 0.111...
+
8/9 = 0.888...
=
9/9 = 0.999...

>>14778547
>>14779668
Infinitesimal is not zero, which is proven by the fact that infinity times zero is zero.

>>14779737
>which is proven by the fact that infinity times zero is zero.
jesus fucking christ the stuff I have to read from you niggers

>>14779737
>infinity times zero is zero
lolno

>>14779837
but yuo see, 17.999... is ambiguous. She could have been [math]17.\overline 9[/math], or [math]18 - \frac 1 \infty[/math].
the former is legal (since [math]17.\overline 9 = 18[/math]), on the other hand, the latter is objectively immoral (since [math]18 - \frac 1 \infty < 18[/math])

>>14779621
what part about "thou shalt not bear false witness" do you not understand?

stop implying beyond your means
how many 9's can you imagine?

>>14779895
>>imagine
>t. 12yo fairy tale fan

>>14779884
>18−1/∞<18)
wrong
1/inf=0

>>14779900
Wrong. Infinitesimal is not zero.

>>14778590
0.999...1

>>14779899
>not knowing what imagination is
you are fucking sub-animal legitimately

>>14779900
if
[math]\frac{1}{\infty}=0[/math]
then
[math]1= 0 * \infty[/math]

does it?

>>14779923
yeah, s'pose so

>>14779923
0 times anything is 0

>>14779884
>1/inf
1/inf is undefined. You mean ε.
>>14779923
Infinity is not a number, so both multiplication and division by infinity are undefined.
>>14779928
0 times infinity is undefined.

>>14779899
>>imagine
Your right.
Some self taught filthy swines would tell you about "imaginary numbers", [math]\sqrt{-1}=i[/math] or [math]i^{2}=j^{2}=k^{2}=ijk=-1[/math]

They have obviously a number of unreal complexes where they imagine to fuck their num.

>>14779923
if 22/∞=0
then
22=0∗∞

this is how stupid you sound like

>>14779903
under R it is

>>14779932
>so both multiplication and division by infinity are undefined.
wrong
inf-inf and inf/inf and 0×inf are undefined but 1/inf and 2×inf are 0 and inf

>>14780012
>1/inf and 2×inf are 0 and inf
Wrong. Both are undefined.

>>14780020
>source: my ass

>>14779302
>But this is not a universal standard afaik (but it should be.)
its not really feasible to do with plain text, not that many fonts render number+overline(U+0305), example 9(don't know if 4chan will filter the overline), that's why it isn't more common+its its a chore to type out

>>14779402
>>14779434
>>14779895
i bet that you read the scofield bible

>>14778057
ching chon bin bong, herro preez, no calcium, ty

>>14779910
Imagining it as an array of decimal digits, what's the index of that 1 at the end?

>>14780074
infinity

>>14780074
>>14780075
-1

>>14780075
If both of them have infinity digits then how can you compare digits after infinity?
This is so dumb and inconsistent. On one hand you people are "um actually you can never reach 1 because it takes infinite steps/time/computation" but then turn around and say even dumber shit like "actually there's [some number] after infinity (like 0.000...01)". Do you even hear yourselves? Do you have dementia that you can't even remember your own thoughts and opinions?

>>14780109
[math] \displaystyle
0. \bar{0}1
= \lim_{n \to \infty} 0. \underbrace{0 \dots 0}_{n ~ \text{times}}1
= \lim_{n \to \infty}
\left [
\left (
\sum_{k=1}^n \dfrac{0}{10^k}
\right )
+ \dfrac{1}{10^{n+1}}
\right ]
=0
[/math]

>>14779910
You can't append a digit to an infinite sequence, fa.m

>>14780142
Correct

>>14778728
If 0.99999... is 1 then pi is 3.14 and that's it.

>>14780159
nice proof, bro

>>14778057
>well you see, 0.999..... is just another way of writing 1! It's just notation! Our notation isn't unique!
I disagree. They are equal, but 0.999... is notation for a certain limit, which in most number systems is indeed equal to 1.

>>14780162
If 0.999999999... is equal to 1
then why isn't 3.141592653... equal to 3.14

>>14780183
because 3.14 < pi
duh

>>14780185
and 0.999... < 1
so what

>>14780186
what's the difference.
3.14 is about 0.0016 less than pi

>>14780183
3.141592653... is clearly larger than 3.1415, right? Then how can it be 3.14 if 3.14 is smaller than 3.1415?
The analogy doesn't work because for 0.999... you can show that for any number smaller than 1 there exists such n of 9s in 9.999... n times that is larger than this value.
In your example there is such an approximation of Pi that is larger than 3.14 so 3.14 is not the value of 3.14159...

>>14780109
>If both of them have infinity digits then how can you compare digits after infinity?
you're gonna flip when you hear about infinite ordinals, but you're right anyway in this case

>>14780109
hopefully i understood this right in high school:
to put it simply anon,
[math]a = \infty + 1 = \infty\\ b = \infty\\ a > b[/math]
how can [math]\infty[/math] be different from [math]\infty[/math] right?
well because it's not a variable with a finite number, it's a quantity...
this is why (i believe) indeterminate forms like [math]\frac{\infty}{\infty}[/math] or [math]\infty - \infty[/math] are indeterminate.
because [math]\frac{\infty}{\infty}[/math] might be more or less than one and [math]\infty - \infty[/math] might be more or less than zero.
any math guy here please correct me if im wrong

>>14780243
https://youtu.be/SrU9YDoXE88?t=23m5s

>>14780329
omega and infinity are not the same thing tho

>>14779670
...but 9/9 = 1
So 9/9=1=0.99999...

>>14780410

>>14779323
It's no more or less of a tautology than anything in math.

>>14778590
±½N

.999… isn’t even a real thing. There’s no physical way to represent this in the universe. It’s a logic-only concept that can never be manifested. So who gives a shit about this argument at all? You’re arguing about something that has no practical applications in anything.

>>14780756
>physical
the fuck does that have to do with math

>>14778590
they're adjacent

>>14778581
*clap* you used the big word "limit" like the big boys (and girls ;-) ).
Now you just need to learn what it means! We are already halfway at the finish line! Then you get a cookie and can stay up an extra hour this evening! :festive_emoji:
>>14780243
inf - inf is not indeterminate, it's infinity. (inf/inf is honestly a retarded concept/notation.)
(a > b) <=> false
It's truly not easier to dumb down why uncountable infinities are bigger than countable ones than talking about bijections. If you can pair any natural number (apple numbers) to something (e.g. a digit in a notation), it's countable.
But bijection breaks down when the formula kinda incestuously, kinda like an Ouroboros, loops around and applies to itself. Because then, you can't non-arbitrarily pair an element of the domain to (exactly one/any) in the codomain, because you already need another infinite bijection for that.
Think about it like this: a string is recursively augmentable with arbitrary symbols. Countable infinities only use strings of length 1. Uncountable infinities can use ANY string length, including infinite.

>>14780858
https://www.wolframalpha.com/input?i=inf-inf

>>14780896
Frankly, I knew I was giving more my opinion than some written in stone slab rule of math. Arithmetic operators like we used in these posts make zero sense, but people are slobs when it comes to implicitly including some | | around them.

But undefined is very much not the same as indeterminate.

Why ∞-∞ is not 0 ? but undefined..
Is it because it sounds blashpemous?

>>14780954
After all
If it were so that 0.999... = 1
And 1-1=0
Therefore 0.999... - 0.999... = 0
Right?

>>14780954
it's because it sounds cursed

>>14780954
For any given infinity, there is an even greater infinity, so it makes no sense to treat infinity as a number.

>>14780986
Yea i mean i get that, But then somehow 1/inf is cohesive
Weird thats all

>>14780935
>undefined is very much not the same as indeterminate
>Result: (undefined)
>Other indeterminate forms
>Other
>Limits with [math]\infty - \infty[/math] indeterminate forms
we're not saying the result is indeterminate, the form is. the result is undefined because it doesn't have one.

>>14780834
Then (1 + 0.999...) / 2 is larger meaning the 0.999... doesn't have infinite number of digits because it would be larger if you added a single 9 to the end.

>>14780834
No such thing in the reals.

>>14778610