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/sci/ - Science & Math

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8982267 No.8982267[DELETED]  [Reply] [Original]

0.999... DOES NOT EQUAL 1

>> No.8982272
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8982272

Yes it does.

>> No.8982276

It does not, stop being so base 10 you fucking queer

>> No.8982279
File: 8 KB, 308x336, 1323000a7690456ccea9.jpg [View same] [iqdb] [saucenao] [google]
8982279

Except 0.999... does infact equate to 1

>> No.8982284

>>8982279
You cannot add infinitely many things, and if you add finitely many things, there will always be a tiny cube or rectangle missing in the top right corner.

>> No.8982294

>>8982284
>calling a square a cube
>not understanding infinite series

>> No.8982295

>>8982267
by its very definition 0.999... is a limit of a certain sequence. this limit equals 1.

>> No.8982312

you retards are basically saying you know how the universe works. only god knows dofus

>> No.8982316

>>8982276
In binary, [math]0.\bar{1}=1[/math]
In decimal, [math]0.\bar{9}=1[/math]
in base b, [math]0.\overline{{b-1}}=1[/math]

Stop being a brainlet

>> No.8982320

>>8982316
0.999... = 0.999...
0.999... is the highest number before 1, but isn't 1
If you want to say 1, just say 1 instead you fucktarded pile of bitch

>> No.8982325

>>8982272
I dont get #2 in the yellow part

>> No.8982326

>>8982312
if you even came close to comprehending it your head would explode

>> No.8982330

>>8982320
There is no "highest number before 1 that isn't 1" for the same reason that there's no "smallest positive number greater than zero"
If your shitty attempts to convince people that [math]0.\bar{9}\neq1[/math] is your best attempt to contribute something to mathematics then you should probably pick a different line of work.

>> No.8982332

>>8982330
Yes there is, just not in base 10
This is like trying to prove something is true in the world based off of the English definitions of words instead of forming a coherent argument

>> No.8982333

>>8982320
>0.999... is the highest number before 1, but isn't 1

>successor function
>densely ordered set
pick one

>> No.8982345

>>8982332
[math]0.\overline{b-1}=1[/math] is true in any base b, not just base 10. If you're talking about numbers in any base this is always true.

If you consider the set of rational numbers, there is no smallest rational number. You can always find a smaller one. This argument does not involve bases.

Go learn some algebra instead of being a retard.

>> No.8982381

>>8982332
"hurr I can just assign properties to objects and the object always exists just because I assigned a property to it"

Alright faggot I'll play your game and see where it leads.

(P1) Let [math]\omega[/math] be the largest positive finite number.
(P1) [math]\implies \frac{1}{\omega}[/math] is the smallest finite number.

Now lets see if [math]\omega[/math] is a rational number. That is, it's a member of [math]\mathbb{Q}[/math].
We know that [math]\mathbb{Q}[/math] is an extension of [math]\mathbb{Z}[/math], the integers. We also know that for any integer z [math]\in \mathbb{Z}[/math], z+1 [math]\in \mathbb{Z}[/math]. This is the definition of a ring.
Therefore, [math]\mathbb{Z}[/math] doesn't have a largest element. You can always find a bigger one. So [math]\omega \not\in \mathbb{Z}[/math]. We'll call this (P2).

(P2) [math]\implies \omega \not\in \mathbb{Z}[/math]. Since [math]\mathbb{Q}[/math] is the extension of [math]\mathbb{Z}[/math] which includes all the multiplicative inverses of [math]z \in \mathbb{Z}[/math], this implies that [math]\frac{1}{\omega}[/math] is not a member of [math]\mathbb{Q}[/math] either.

A similar argument can be used to prove that [math]\omega \not\in \mathbb{R}[/math] as well, but that would involve constructing [math]\mathbb{R}[/math] which I do not have the patience to do. Just know that it hinges on the fact that [math]\omega \not\in \mathbb{R}[/math], which implies that its inverse isn't there either.

We only care about [math]\omega[/math] because [math]1-\omega = 0.\overline{9} \neq 1[/math] But it's not real.

>> No.8983002

>>8982267
Here is a proof simple enough for any brainlet to understand:

http://journalofsci.weebly.com/home/simple-elegant-proof-of-0999-1


>>8982272
Too complicated for brainlets.

>>8982295
Too complicated for brainlets.

>> No.8983010

Approaching something means you're that thing. So if I approach a bear somehow I magically turn into a bear.

>> No.8983026

If they're not equal you should be able to find a number between then

>> No.8983027

>>8983010
>So if I approach a bear somehow I magically turn into a bear.

It's not "magic", the bear would eat you and digest you and you would, in effect, become the bear.

>> No.8983029

>>8982284
>You cannot add infinitely many things

>> No.8983030

>>8983026
That's easy. This number is exactly half way:

[eqn]0.99999... + \frac{\epsilon}{2} [/eqn]

>> No.8983051

>>8982272
How is green a proof by induction? I mean it's a totally reasonable proof, but I can't see the induction-part.

>> No.8983056

>>8983030
ε is not a number

>> No.8983082

>>8982284
Wild Burger pls go

>> No.8983250

>>8983030
Ok now add

[eqn]0.11111...[/eqn]

To that number and then subtract

[eqn]0.11111...[/eqn]

Watch what happens. Go here to see it worked out: >>8983002

>> No.8983527

>>8983002
"elegant"? that proof is trash and makes use of all sorts of hand-waving. try this instead:
>x,y are equal iff there exists no z such that x < z < y
>consider x= 0.999... and y=1. if they are not equal, then there is a z such that 0.999... < z < 1.
>let z = z0.z1z2z3... and note z0 must be 0. note also that there exists n such that xn != zn. consider the smallest such n. >suppose zn != 9. then z < x, a contradiction. hence no such n exists, and consequently no such z exists. qed

>> No.8983559

>>8982272
If x = 0.999,
then 10x = 9.990 (or 9.99), not 9.999
That image is a good troll image. The green box is bullshit, didn't read the rest, why bother.

>> No.8983616

>>8983027
But if you are what you eat the bear would be a human.

>> No.8983624

>>8982325
It is saying that there is no number between 0.999... and 1. Therefore, 0.999... = 1

>> No.8983625

>>8983527
The only assumption that proof makes is that those decimals converge, other than that it is completely rigorous.

I've used this proof to convince complete normies that 0.999.. = 1, and they believe and understand it. Your proof is nice but it won't convince the brainlets that argue otherwise.

>> No.8983634

>>8983002
>>8983625
it's shit. the whole point of proving 0.999... = 1 is proving that decimals converge, that you can't be "infinitely close" and not be the same

>> No.8983648

>>8983634
Wrong.

People accept that the number represents a finite value, but they don't accept that that value equals 1.

One doesn't imply the other, just because something is finite doesn't mean it has to be 1.

>> No.8983653

>>8983026
[eqn]0.9999... 5[/eqn]

>> No.8983659

>>8982284
>You cannot add infinitely many things
Why's that Sherlock?

>> No.8983706

>>8982294
>>8983029
>>8983082
>>8983659
He is actually right. "Infinite sums" are not really sums, they're the limits of sequences of sums.

>> No.8983857

>>8983706
>I literally don't understand limits

Infinite sums are sum. And you find that SUM by taking the limit of a sequence.

>> No.8983907

>>8983625
What do you mean by "assuming the decimals converge"? There is no convergence there.

>> No.8983953

>>8983907
>What do you mean
Assuming that 0.9999... equals a finite number.

>> No.8984596

>>8983857
No, they aren't. The limit is just the value that the sum gets arbitrarily close to if you keep adding terms, it's not actually a sum of infinitely many terms because there would be no way to calculate that.

>> No.8984604

>>8982284
>You cannot add infinitely many things
Yes in a sense you can't do that, but that's literally what 0.999,,, means. You add 0.9+0.09+0.009+... .
It is the limit of it.

>> No.8984767

>>8984596
>u get close 2 the value but u nevar reach it xD xD
>I literally don't understand limits

>> No.8984778

>>8984767
Sounds like you're the one who doesn't understand limits, kiddo.

>> No.8984831

>>8982284
>You cannot add infinitely many things

Correct. That's one of the big reasons why the limit was invented -- to assign a well-defined result to an infinite series.

Example:

Adding finitely many things together:
9/10 + 9/100 + 9/1000 = 0.999

This process ends, so we can get a final result with no problem.

However, adding infinitely many things together:
9/10 + 9/100 + 9/1000 + ... (endlessly) = ?

This describes an endless process, therefore there there can be no end result. To fix this, we apply the limit:

lim(n=1..infinity) Σ(k=1..n) 9/10^k

Roughly speaking, the limit is the answer to this question: "What is the smallest number that's greater than all the partial sums?" (That's an oversimplification of what the "limit" really is, but for newbies, it's okay for introducing the concept.) We use that question to provide a well-defined "result" for the infinite series.

This is *really* *really* tricky for students, because of the fact that the limit notation doesn't appear explicitly when you write down "0.999...". For "0.999...", the need to apply the limit is assumed *implicitly*, because there is no way to get an answer otherwise. This is because you're describing an endless process for which there can be (obviously) no end result -- so you *must* then apply the limit to get a well-defined result.

>> No.8984852

>>8983559
but x doesn't equal 0.999 you dumb fucking retard

>> No.8984871

>>8983953
Are you trolling?

>> No.8984921

>>8984831
>This describes an endless process, therefore there there can be no end result.

Kek. You are working in an abstract world, to think that you are physically confined is laughable. Read a book about Emergent Abstractionism if you can't wrap your tiny brainlet brain around these concepts.

According to your logic:

>You cannot add 100,000,000,000,000 things together because this would take longer than the life of the universe.

>this process takes too long, that's why there can't be a result

>this is why multiplication was invented. To assign a value, but remember that value isn't the real answer!

>You can calculate the answer to adding "x" to itself 100,000,000,000,000 times, the answer is x*100,000,000,000,000, but this is only the answer it WOULD be if you COULD add that many "x"'s together. It's not the real answer guys, trust me!!

>> No.8984946

>>8982333
>0.999
see that 0?
It's not 1.
Have a nice day.

>> No.8984957
File: 40 KB, 853x519, Capture.png [View same] [iqdb] [saucenao] [google]
8984957

>>8984921
this is bait

>> No.8984960

>>8982284
>>8982279

the proof that .99999999999999... equals 1 comes directly from the construction of the real numbers

You can definitely play Wildberger and say the real numbers are fucked but don't try to do a geometric proof or make some kind of intuitive argument you dummy..

>> No.8985065

>>8983653
0.9999...5 < 0,9999... = 1

>> No.8985070

>>8982284
>you cannot add infinitely many things
Maybe you can't, brainlet
hahahaha

>> No.8985183

More than [math]1\neq 0.9...[/math] faggots I hate human vermin that claims [math]0.9...=1[/math], but the equality doesn't hold in some spooky field of hyperreals or other meme field

>> No.8985209

>>8983559
The "..." after the 0.999 means the nines keep on going.

>> No.8985239

>>8982284
>You cannot add infinitely many things
Please take calculus.

>> No.8985310

>>8985239
Please take analysis.
Infinity is a very finicky thing, and at the end of the day we DO NOT allow the addition of infinitely many things. (also, "infinitely many" or "infinite amount" are oxymorons when considered as two separate terms (though they've adopted their own meaning as single terms)).
We define constructs referencing finicky infinity rigorously in terms of limits. That is, we take a point and a ball around that point, and see which FINITE sums of terms of a sequence can be brought within that ball. We are not actually "adding infinitely many things" but talking about what the sum "converges" to using keywords like ALL and ALWAYS.
"Infinite" should honestly only be used in casual conversation to mince words and removed from legitimate mathematical discussion.
>tl;dr brainlets need to stop using "infinity" carelessly

>> No.8985330

mathematicians should've just agreed that beyond a reasonable limit a number cannot get any smaller meaningfully and numbers are indistinguishable beyond that degree

all this infinitely large infinitely small shit is nonsensical farce