Let [math]a = 0.999999...[/math]
[eqn]
\begin{aligned}
10a &= 9.999999... \\
10a - a &= 9.999999... - 0.999999... \\
9a &= 9 \\
a &= 1
\end{aligned}
[/eqn]
[eqn]
\begin{aligned}
\frac{1}{3} &= 0.333... \\
\frac{2}{3} &= 0.666... \\
\frac{3}{3} &= 1 \\
\frac{1}{3} \cdot 3 &= 1 \; \therefore \; 0.333...(3) = 0.999... = 1
\end{aligned}
[/eqn]
[eqn]
\begin{aligned}
0.999... &= \left( \frac{9}{10} \right) \left( \frac{1}{1 - \frac{1}{10} } \right) \\
&= \left( \frac{9}{10} \right) \left( \frac{1}{ \frac{9}{10} } \right) \\
&= \left( \frac{9}{10} \right) \left( \frac{10}{9} \right) \\
&= 1
\end{aligned}
[/eqn]
>>7948603
This is all true, but also unlikely to convince anyone who doesn't already understand the reasons of why 0.999... = 1. For the bottom one in particular, I'd like to see an intermediate step through the series interpretation.
Quit posting this slut
>>7948629
Shit taste goes to >>>/lgbt/, Anon-kun~!
>>7948603
>10a=9.999999...
Ironically enough that was the first step that confused me. I forgot that multiplication can move the decimal place like that.
>>7948620
Think about this... [eqn]
\begin{aligned}
0.999... = \sum_{n=1}^{\infty} \frac{9}{10^n} &= \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \; ... \\
\sum_{n = 1}^{\infty} a_n r^{n-1} &= \frac{a}{1 - r} \quad \left( \because -1 < |r| < 1 \right)
\\
&= \frac{ \frac{9}{10} }{1 - \frac{1}{10}} \\
&= \left( \frac{9}{10} \right) \left( \frac{1}{1 - \frac{1}{10} } \right) \\
&= \left( \frac{9}{10} \right) \left( \frac{1}{ \frac{9}{10} } \right) \\
&= \left( \frac{9}{10} \right) \left( \frac{10}{9} \right) \\
&= 1
\end{aligned}
[/eqn]
>>7948666
Our notation wouldn't even get that far if we'd just standardized continued fractions already.
>>7948666
trips checked
>>7948620
Top one seems pretty idiot-proof though. If they understand the concept of repeating decimals they'll understand that one.
just use your sig figs guys!
.9999... = 1
but not 1.0
>>7948603
>All such interpretations of "0.999…" are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999….
>>7949355
This is really only true if you allow infinitesimals
>>7949355
This is true for limits only 2bh. As you get infinitely closer to [math]c[/math] whilst not actually approaching to it fully.
>>7948631
kurisu was always a big walking SHIT
>>7950165
(You)
>>7948603
>9.999999... - 0.999999...
is actually
9.999999...90 - 0.999999...99
or
8.999999...91
>1/3 = 0.333...
This simply shows the decimal representation of 1/3 isn't perfectly accurate.
>[math]\left( \frac{9}{10} \right) \left( \frac{1}{1 - \frac{1}{10} } \right)[/math]
Here you are just parroting formulas without proof like a good authoritarian.
>>7950203
>what is infinite geometric series
underage b&, do your precalc homework
>>7950235
>what is infinite geometric series
It's the thing you're supposed to be proving but instead regurgitated from your textbook.
>>7950203
>9.999999... - 0.999999...
>is actually
>9.999999...90 - 0.999999...99
Yeah, no. Those are infinite series of nines. They don't have an end, the moment you try to show they have an end (appending a zero) they're finite, and you're changing the value.
Go back to your blog where you belong:
https://yngthlet.wordpress.com/2016/03/19/4chan-and-me-a-science-war-of-posts/
>>7948603
>[eqn]
Stopped reading there.
>>7950278
lulz
>>7950272
Again that post simply cited the formula without proving it. You can do better than this, anon.
>>7950276
The problem is that you are trying to prove things about infinite decimals based on pencil-and-paper calculation algorithms that are used with finite decimals. Obviously you cannot do pencil-and-paper calculations with infinite decimals. If you want to argue by analogy to pen-and-paper calculations, the burden is on you to prove that the result of your pseudo-algorithm is correct.
>>7950355
You're insufferable.
The formula for the sum of a geometric series is a well known fact. It really doesn't need to be proven every single time it's used, but since you're being a little bitch about it, read these:
https://en.wikipedia.org/wiki/Geometric_series#Formula
https://en.wikipedia.org/wiki/Geometric_series#Proof_of_convergence
That should give you all the proof you need that this formula is valid.
All of this is on the /sci/ guide, you insufferable bastards.
>>7950355
you're a moron
>>7950355
Provided that [math]|r| < 1[/math]
[math]S_n = a + ar + ar^2 + \; ... \; = ar^{n-1}[/math]
Multiply [math]r[/math] onto [math]S_n[/math]...
[math](r)S_n = a + ar + ar^2 + \; ... \; = ar^{n-1} + ar^n[/math]
Subtraction of the second equation from the first yields...
[math](1 - r)S_n = a(1 - r^n)[/math]
Rewrite...[eqn]S_n = \frac{a(1 - r^n)}{1 - r} \\
\lim_{n \rightarrow \infty} S_n = \lim_{n \rightarrow \infty} \frac{a(1 - r^n)}{1-r} = \frac{a}{1 - r}[/eqn].
>>7950178
nice image of A SHIT you posted there
>>7950469
>>7950533
lmeo you're a L I T E R A L shitposter
>>7950541
(You)
>>7950533
gas all weeaboos
>>7950557
Gas is my favourite state of matter because muh vapours and fractionating columns!!
>>7950561
Why are anime girls so cuddly
>>7950584
Because [math]0.999... = 1[/math] !!
just say that there is no 1 at the end as its infinitely repeating
>>7950594
But does this tell us why kids love cinnamon toast crunch?
>>7950597
Show us all some arithmetic on that, anon-kun.
I'll assume for now that you're implying [math]0. \bar{9} = 1[/math].
>>7950609
Why do kids love cinnamon toast crunch? Surely there must be some mathematical proof for this.
>>7951829
what does pointing out shitposting have to do with legbutt
>>7952096
You mean BLT. :^)
>>7948603
>return to /sci/ after nearly 2 years
>this shit is still around
>leave forever
>>7952750
At least it isn't [math]\zeta (-1)[/math]. :^)
>>7948629
>>7950165
(You)
>>7952804
Fug, who is that cosplayer.
I want to know b-because...R&D.
bumpu-desu~!!
>this meme again
