/sci/ - Science & Math

Oh god not again.

>1 is the smallest real number greater or equal to every member of the sequence 0.9, 0.99, 0.999...

>>8568371
>10 is the smallest real number greater or equal to every member of the sequence 9, 9.9, 9.99...
REALLY MAKES YOU THINK

IF YOU COMMENT AFTER THIS POST YOUR MOTHER WILL DIE IN SLEEP TONIGHT

>>8568373
no, that would be 9

This thing has become such a meme that as someone who doesn't know math I don't know if it's equal to one or not anymore.

>>8568360
no, everyone knows that .999=-1/12

>>8568390
DELET

>>8568390

no

it's 0.1 + 0.2 +0.3 .... = -0.1/0.12

>>8568456
or something like that

>>8568360
That guy just oozes autism.

/His/torian here, for you guys is watching Numberphile the equivilant of Lindybiege for us?

>>8568504
I don't even know I'm OP and from /g/

>>8568504

What's the overall consensus in /his/ on Lindybeige?

>>8568360
sage

>>8568390
Kek

>>8568504
the guys at numberpedophile actually know math. isn't Lindy speaking out of his ass half the time?

>>8568533
lindy is obviously just a token female

>>8568383
if you are interested...
https://www.youtube.com/results?search_query=0.9999+equals+1

>>8568546
>lindy
>female

>>8568650

>>8568376
..

>>8568504
Numberphile is a good channel. The only reason /sci/ hates it is they like feeling superior to them. It's pretty basic math (for obvious reasons sci cant seem to grasp) so they feel smart for knowing it.

They make bulshit claims about the rigor, without taking even a second to consider that you can't really bring math to the people if you spend the first 10 minutes of a video explaining why Z/5Z is "technically different from the integers 0 through 4."

It's a good channel.

>t. BS in math

is Hannah Fry a tranny

>>8568766
she is legit gorgeous, though.

>>8568360
When will you idiots learn that .99999 is not equal to 1?

>>8568390
You might be on to something, anon, have some dots.

>>8570436
Yeah, you are right:

.99999 is not equal to 1, but 0.999...=1.

Here's the deal.

Part of the reasoning for .999... = 1 is because you can't have a number like .000...01.

That itself doesn't make sense to me. How can you have an infinitely precise number, but not have an infinitely small number?

[math]x = .999...[/math]
[math]10x = 9.999...[/math]
[math]9x = 9[/math]
[math]x = 1[/math]

The problem with this reasoning is they assume 10x = 9 + .999..., when it actually equals 9 + .999...90, which apparently can't exist for some reason, so they say fuck it.

I'm fine with saying .000...01 is not a real number, but then neither should .999... be a real number either, in which case the entire thing is meaningless.

Heres the deal
1/3=0.3333...
2/3=0.6666...

Therefore it follows that:
3/3=0.9999...

But we know that 3/3=1

X=.9999...
10X=9.9999...
10X=9.9999... -0.9999...
9X=9
X=1

Proof, unless you have some proof to claim otherwise

>>8570546
** 10X-X=9.9999... -0.9999... **

>>8570545
>>8570546
It only works because mathematicians made up a rule that this works.

They treat .333... and .999... as real numbers because it's convenient, not necessarily because they are.

We can have irrational numbers. I don't understand why we can't have ir-digital numbers.

>>8570547
Yes.
9.99999...
+
0.99999...
=
>** 9 **

Another problem. When you guys say .999..., you're treating that like a number, and not a limit.

It's the limit of 9/(10^x) as x approaches infinity. You tell that to anyone and they'll immediately agree with you. Instead you leave that part out and then are frustrated why you have trouble convincing people.

>>8570550
You dont have ir-digital numbers because of mathematical rules either. They represent the absolute value of an integer. An integer does not include fractional components by default.

>>8570550
.333... isnt a real number, regardless of what position you hold on the subject. Go back to grade school before you start inventing more conspiracy theories and start offering some proof to contradict 1=0.999...

>>8570536
3*.333... = .999... = 1, do you really need to go further?

>>8570568
Do you even know what you're saying? If .333... isn't a real number, how can .999... be a real number? Or are you going to start saying 1 is also not a real number?

>>8570567
First off, .333... isn't an integer. Second off, some numbers cannot be expressed exactly without deviating from simple decimal form. (Irrational numbers like squareroot of 2). So I'm not sure what your point is.

>>8570569
I feel like you didn't read my post. I don't think .333... should be considered a real number. Further thought, I'll accept .333... as a real number only if you consider .333... to be a limit and not the actual root, final form of a number, like 2.

>>8570571
That actually equals .99999999996..., your calculator just rounds up.

>>8570536

This is the whole reason I think it's just math mumbo jumbo. Like sure, let's assume that it's true, because it pragmatically useful and I'm a big fan of the pragmatic theory of truth, but if we take it too seriously, then all of math breaks down.

Because if 0.99... = 1, then this implies that 0.00...1 = 0. Consequently 0 = 0.00...1 = 0.00..2 = 0.00...3, repeated over an infinite number of times until you reach such absurd conclusions as 0=1, and eventually 0=2, and so on. The end conclusion of this assumption would seem to be that all numbers equal one another.

>>8570580
Wait a minute.


Does 0.00...1 + 0.00...2 + 0.00...3 + ... = -0.00...1/12

>>8570580
I'm not seeing how that implies that.

>>8570582
You might be on to something here...

>>8570580
>0 = 0.00...1 = 0.00..2 = 0.00...3
This is true, insofar as 0.00...# means anything.

1 divided by (basically) infinity equals 0.
2 divided by basically infinity equals 0.

>The end conclusion of this assumption would seem to be that all numbers equal one another
All numbers do equal one another, when they're divided by infinity. They all equal 0.

if 0.999...≠ 1 then 1 – 0.999... > 0
so call it ε = 1 – 0.999...
and divide 1 by ε via long-division;
you get a series that diverges,
therefore ε = 0 , QED

>>8570606
>divided by infinity
division is a binary operation on numbers
infinity is not a number
therefore no division by infinity

>>8570606
I feel like you can stretch this because of math rules.

If you take a truly random, real number, the probability of you selecting any specific number is 1/∞. You can consider the odds of picking any real number to be zero because you're dividing a finite number by an infinite number, however you do pick a number. Why? Because 1/∞ is only zero because math doesn't allow ifinitesimals. It's a rule that just works because math says so.

>this thread again

[eqn]
0.9999... = 0.9 + 0.09 +0.009 + ~... \\
0.9999... = \sum_{n=0}^\infty \frac{9}{10}(0.1)^n = \frac{9}{10}\sum_{n=0}^\infty (0.1)^n \\ \\
\frac{9}{10}\sum_{n=1}^\infty (0.1)^n = \frac{9}{10}\left(\frac{1}{1-(1/10)}\right) = \frac{9}{10} \cdot \frac{10}{9}=1
[/eqn]

>>8570633
oops, made a mistake there, that last sum should be

[eqn]
\frac{9}{10}\sum_{n=0}^\infty (0.1)^n
[/eqn]

>>8570621
>math doesn't allow infinitesimals

https://en.wikipedia.org/wiki/Infinitesimal#Number_systems_that_include_infinitesimals

>>8570619
I said basically infinity.

0.000...1 = 1 * 10^(negative infinity).
It's 1 divided by a power of 10 with infinitely many zeros.

And yeah, 10 with infinitely many zeros is not a number and therefore no division by it. In other words 0.000...1, the quotient of that division, isn't a number. I'd say that's more a matter of what you want to call things than anything else.

>>8570621
Math does allow infinitesimals, they're a vital part of things like differentiation. But infinitesimals are infinitely small and functionally equal to zero. If you take a paper twice as long as it is wide, and shrink it to a point, it's now an infinitesimal of its former size. And it still has its own shape, the long edge is still twice as long as the short edge, but they're both functionally zero. (Zero equals zero times two.)

>>8570637
If you're using hyperreals or something similar, then .000...1 does not equal zero. What's your point!

>>8570643
>I said basically infinity
therefore basically retarded
gtfo fgt pls

>i can prove you that an orange is actually not a full orange

Infinity is just a stupid concept in general.

The odds of me flipping a coin and it ending up as heads is 50%. If I flip a coin 10 times, the odds of me not getting a single heads is 1/1024. If I flip a coin an infinite number of times, the probability that I do not get a single tail is 0%, meaning at some point a 1 turns into a 0 as you go to infinity. At some point, the odds of flipping a coin and landing on tails is 100%, even though the odds are always going to be 50%.

There's no real world applications here, it's shit that only happens in math, so they need to make up a bunch of rules about it.

>>8570644
It was a side note. He said math doesn't allow infinitesimals, I provided evidence to the contrary.

But to actually address the assertion in >>8570580:
>Because if 0.99... = 1, then this implies that 0.00...1 = 0. Consequently 0 = 0.00...1 = 0.00..2 = 0.00...3, repeated over an infinite number of times until you reach such absurd conclusions as 0=1, and eventually 0=2, and so on.

Your problem is you're assuming you can reach numbers like 1.00 and 2.00 by following the sequence of """numbers""" 0.00...1, 0.00...2, 0.00...3. You can't.

Essentially what you're doing in this sequence is taking a number and progressively dividing it by 10. Consider the following sequence of sequences:

1, 0.1, 0.01, ... , 0.00...1
2, 0.2, 0.02, ... , 0.00...2

...

47, 4.7, 0.47, ... , 0.00...47

We can immediately see that this is following the process you described. Generalizing this sequence, we obtain the following:

[eqn]
a, ~\frac{a}{10},~\frac{a}{10^2},~\frac{a}{10^3},~\frac{a}{10^4},~...
[/eqn]

...where one element of the sequence is [math]\frac{a}{10^n}[/math].

We want to examine what happens as this sequence goes to infinity, so we take the limit as n approaches infinity:

[eqn]
\lim_{n\rightarrow\infty}\left(\frac{a}{10^n}\right) = a\cdot\lim_{n\rightarrow\infty}\left(\frac{1}{10^n}\right)= a\cdot 0 = 0~ \rm for~all ~\it a
[/eqn]

We can very clearly see that no matter what [math]a[/math] is, this sequence will always approach zero. Thus, following this process 0 = 0.00...1 = 0.00..2 = 0.00...3 will not lead to any absurd conclusions at all, since nonzero numbers can simply not be written in the form [math]\lim_{n\rightarrow\infty}\left(\frac{a}{10^n}\right)[/math].

>>8570670
Why does the limit as x approaches infinity for 1/10^x equal 0? If we take infinitesimals off the table, that doesn't make sense to me. On one hand you're saying you .000...1 does not equal 0, but on the other hand, you're saying it does.

>>8570670
Obviously the limit will approach 0 but that doesn't mean 1 / inf = 0.

Remember when taking limits that you aren't calculating the value at a point but rather the value you are approaching while calculating values close to that point with an approximation of arbitrarily large precision.

Generally limits are introduced as a way of calculating instantaneous rate of change of something (usually distance) and that such calculations are undefined because "instantaneous" velocity implies delta y units of distance divided by delta x = 0 units of time (undefined), and only by using arbitrarily small (infinitesimal) widths of delta x can you approximate velocity at an instant in time to an arbitrarily large precision and you need to be extraordinarily careful about what you mean by instant rate of change. I'll simply end by saying that this sort of subtly of x -> c, x =/= c when taking limits and derivatives is more thoroughly explained in most rigorous introductory courses of calculus and a more detailed explanation here is beyond the scope of my post.

>>8570682
>Why does the limit as x approaches infinity for 1/10^x equal 0?
I'm not going to explain it fully, since the formal definition of a limit is not something you learn in an hour on /sci/, but here's a place to get started on why this is true:

https://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit#Precise_statement_for_limits_at_infinity

I hope you like real analysis.

>On one hand you're saying you .000...1 does not equal 0
Where am I saying this? Please point it out.

What I'm saying is that 0.000...1, 0.000...2, 0.000...3 and so on are all zero. You can't write whole numbers like 1, 2, and 3 in that form, so your logic as to why they are zero too (>>8570580) is false.

>>8570700
What I get out of this is normal limits don't apply to systems where infinitesimals are allowed. In that article there's even a section on calculating limits in the hyperreals, which tells me it's a plot hole that math had to fill by making up rules

>>8570696
>1 / inf

infinity is not a value in the real numbers, ffs stop treating it like one

>>8570702

The general concept of my post is that you can't treat it like a real. I apologize if my presentation was unclear.

>>8570701
>What I get out of this is normal limits don't apply to systems where infinitesimals are allowed.

Of course. Adding infinitesimals to the mix throws a bunch of stuff relating to infinity out the window. Assuming that we're only talking about real numbers and using the traditional interpretation of an infinite decimal expansion (for the reals) then 0.999... = 1 and 0.000...1 = 0. However, adding infinitesimals would most assuredly change that. How, I'm not sure; I don't have any experience to draw on for hyperreals.

>>8570708

Infinitesimals are used frequently in calculus and can be manipulated algebraically (among other things).

I would say delta x or other forms of notations expressing ideas similar to the epsilon-delta definition simply don't have decimal representations (in general) because such notation would be pointless arithmetically. IE: Okay I'll just divide 4 by 0.000...1 but I don't know how to complete this operation, let me just divide by 0.00001 as an approximation, then 0.0000001 to see what value I am approaching, etc. Actually let me just define limits with epsilon-delta notation and use calculus theorems and scrap my decimal representation.

>>8570720
>I would say delta x or other forms of notations expressing ideas similar to the epsilon-delta definition simply don't have decimal representations (in general) because such notation would be pointless arithmetically.

Pretty much this.

IIRC infinitesimals were deemed "not rigorous enough" and for a long time and calculus was actually formulated in such a way that it avoided the use of infinitesimals, and they were re-added later when some smart guys came along and formalized the concept.

When you say...

> Infinitesimals are used frequently in calculus and can be manipulated algebraically (among other things).

...do you mean things like this: [math]\frac{\rm d \it f}{\rm d \it g}\frac{\rm d \it g}{\rm d \it h} = \frac{\rm d \it f}{\rm d \it h}[/math]

Because that's just a notational trick that happens to work. I believe there is a formalization of the ideas at play when you do stuff like that that involves infinitesimals, but it's possible to do completely avoiding infinitesimals altogether.

>>8570580
>an endless row of zeros with a one at the end
yeah makes total sense

>>8568360
x=0.99999
10x=9.999999
10x-x=9x=9.9999999-0.9999999
9x=
x=1
=-1/12 or something

>>8570727
Correctly typesetting differentials. Instant boner.
Thanks.

>>8568360
There's a fundamental assumption that no one ever talks about when they're trying to explain this, and it's that a number that's infinitely small is equal to zero. Once that's established, all you have to do is this:

1 - .999... = infinitely small = 0 so .999... = 1

>>8568390
That can easily be shown with some algebra.

>>8568360
0.99999... = 0.9999... + 0..09999 + 0.009999 + ...

Geometric series with r = 1/10. Just sum it up.

>>8572099
no.....
0.999...=9+0.9+0.09+0.009+...

>>8572217
anon... no.
0.999... = 9+10+11+12+...

>>8572219
im sorry mate but no.
0.999...=2+3+5+7+11+13+17+19+23+29+...

>>8568510
That he's a passionate amateur historian, who often gets things wrong and make speculations in areas he doesn't know about. Also a huge anglophile.

I still watch his videos now and then for entertainment.

>>8570667
>There's no real world applications here, it's shit that only happens in math, so they need to make up a bunch of rules about it

Now you know what pure mathematicians do. Literally glorified fingerpainting.

>>8572223
just... no.
0.999...=2+3+5+7+13+17+19+29+31+41+43+59+61...

>>8571260
.999999.... =/= .999999

not technically at least.

>>8570518
M

E

T

A

.9999... = 1-0.0...001

/thread

if you all used the superior non-standard analysis instead of the elitist ε-δ approach, you'd realize that by 0.999... one quite clearly means 1 - ε which quite clearly does not equal 1

The real question is why is this true:
1/1 + 1/2 + 1/4 + 1/8 +1/16 +1/32 +1/64 + 1/128... =/= 2

>>8573780
but it equals two

>>8570568
" [math] 0.1_3 [/math] isn't a real number "

hurr durr

ur all retarded. you can easily prove this with series.

>>8570727
>Because that's just a notational trick that happens to work.
What.

>>8570536
try this instead anon:
>Two numbers x,y are equal if there exists no number z such that x < z < y.
there is no number between 1 and 0.999..., hence they are equal
we don't even have to deal with 0.0...01 here

>>8574902

Rigorous treatment of infinitesimals wasn't done until after WWII, before the only rigorous treatment of limits was the still used (ε,δ)-definition of limit. Check out this historylets

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

>>8568360
1 != 0.99999...

Proof: http://aynrandlexicon.com/lexicon/infinity.html

>>8568360
I can hear his voice already. Omfg.