/sci/ - Science & Math

>>7475207
The missing part that stops it from being one is infinitely small.

Because the difference between the two isn't any number bigger than 0. If you accept this, you'll find you can't use 0.9999... in any way that leads to an inconsistency.

>>7475214
But there IS a missing part. So CONCEPTUALLY, it is 1.
Similiarily, how can 3/3 be 1? isn't it 0.99.........???????

>>7475216

the difference between the two is an infinite gap.

since 0.999... never finishes, thus it never actually reaches or becomes or is equivalent to 1.

>>7475218
>conceptually it is not 1*

>>7475218
The infinitely small missing part cannot be computed. Rounding down produces a less accurate number than rounding up to 1.

>>7475220
>>7475225
So it never ACTUALLY becomes 1, we only assume so?

>>7475227
It does become 1. If the missing part is infinitely small, that means it's 0.

>>7475207
Keep this for the next 0.999... thread

It is a rational number, meaning that you can describe it either with a fraction or by a number itself.
Square root of zero is irrational, because the operation never ends.

It's a quirk of the decimal number system. Numbers do not always have a unique decimal representation.

See: https://en.wikipedia.org/wiki/Decimal_representation#Non-uniqueness_of_decimal_representation

>>7475230
but
>since 0.999... never finishes, thus it never actually reaches or becomes or is equivalent to 1.
So are we either,
assuming that it is the same as 1
or we simply neglect the infinitely small part,
or 1 is both the same as 0.99..... and 1?

>>7475242
See >>7475235
Infinitely small numbers don't exist.

>>7475239
I meant sqaure root of two sorry.

>>7475242
We are saying that for the layman who doesn't understand maths, he feels 0.999... being 1 is "wrong" and his feelings get in the way of accepting the truth.

Similar to people disbelieving in almost any scientific theory (again, not a layman or colloquial theory)

>>7475247
I get it now.
Sorry, some incompetent faggot was unable to explain it to me.

>>7475207
It's not necessary equal.
What you should first think about is what number set are you using, because 0.9999... and 1 are just strings of characters, not actual mathematical numbers in abstract sense.
In real numbers these numbers are equal, because they have to be represented in the same way = equal.
However in Surreal numbers they are different numbers.
Read more:
https://en.wikipedia.org/wiki/Real_number?oldformat=true
https://en.wikipedia.org/wiki/Surreal_number?oldformat=true
You might also be interested in set theory and ways how you can define various numbers and compare them using logic symbols and sets.

/thread

>>7475255
Exactly, this is the problem, it's not evwn shitposting, "feelings" get hurt.
Some people need to learn about what they are talking about, if he disagrees in the end, close the darn thread.

Just to make something sure, because somebody in this room is confusing me here.
We are not actually neglecting the difference between 0.99....... period and 1, because there exists no difference, right?

The real answer is that there's a lot of natural, logical cases where 0.9999... = 1 is helpful but few good reasons for 0.9999... not to equal 1.

Generally, things in mathematics are defined in ways which seem natural and useful.

If there was a situation where 0.999... not equalling 1 was useful then you could you define a number system to do it and solve problems with it. Turns out it's not remotely useful, so nobody bothers.

Consider a sequence such that, for every positive integer n, x_n = 1-10^-n. This is the same, as 0.9, 0.99, 0.999, ...0.9999...9 (the last one has n 9's), a strictly increasing sequence such that, for any positive integer n, x_n<x_(n+1)<1.

It is easy to prove there are no upper bounds less than 1, so 1 is the least upper bound for this sequence. By the monotone convergence theorem, this sequence has a limit, and this limit is 1, when n goes to infinity. But, as n goes to infinity, the number of 9's also goes to infinity. We have our sequence tending to 0.999999... too. But, for any sequence having a limit, the limit is unique, so it must be that 0.9999...=1.

>>7475218
contrary to what you think, 3/3 is definitely 1 in the common algebra. 1/3*3 might be 0,99 on your calculator, but that is a rounding error. 3/3 is definitely equal to 1.

>>7475207

It's BY DEFINITION. There is no proof and it's done so decimals satisfy the real axioms

you can prove it using globes in real analysis, right?

>>7475207
I dunno anon, seems pretty simple

>1/3 = 0.333...
>2/3 = 0.666...
>3/3 = ?

>>7475227
You're talking about the number like it's a function.

>>7475235
That's not a proof by induction...

>>7476050

Actually in base 12, 1/3 is .4, 2/3 is .8, and 3/3 is 1.

>>7475235
OMG THANK YOU I LOVED THE INTUITIVE EXPLANATION

<span class="math">\displaystyle
1=\frac{3}{3}=3\cdot \frac{1}{3}=3\cdot0.\bar{3}=0.\bar{9}
[/spoiler]

>>7475207
x = 0.999999....
10x = 9.99999...
10x-x = 9.99... - 0.999...
9x = 9
x = 1

filename related

>>7476335
but what happens to the 0.000...1 ?

>>7476345
This is not how limits works.
<span class="math">\lim_{n\to\infty}f(x)[/spoiler] is not a value of f when <span class="math">x=\infty[/spoiler] but a value this function approaches as n approaches infinity.
And this value is 0.

>>7476330
>10x-x = 9.99... - 0.999...
>9x = 9

sorry if this is a retarded question, but why is the decimal part of 9.999..... necessarily = x, after you've multiplied it by 10?

wouldn't all of those 9's have moved up 1 place, making it a different number?

wouldn't it be more like 9.999...990?

>>7476491
see
>>7475235
pic related(number 3)
it doesn't terminate
there are infinitely many 9 after the decimal point. thus when you take it times 10 you still have infinite many trailing 9.

Heres a great proof you can work out on your own.

We have the standard euclidean metric in 1-space, and we are trying to determine if 0.99999.....=1. Let d(x, y) be our distance function. d(x, y)=0 iff x=y, so if 0.9999... doesnt equal 1 then there should definitly be a real number between 0.9999.... and 1 that is not equal to either of the numbers. Good luck finding it retard

>>7475207
Who is this lickidy splickidy

>>7475207
>ohlookitsTHISthreadagain.gif

>>7475258
you're the only incompetent faggot here if you can't understand something as simple as this

>>7475207
The decimal system is flawed.

Look at it this way, the ... after .999... mean that there's infinite repeating 9s after it.
Using basic logic, the closer you are to something, the closer you are to touching it.
So basically .999... is infinitely close to 1 and infinitely close is also referred to as touching it, which means it's 1.

Every 9 after .9 represents how close you are to to 1. You can easily calculate how close to 1 you are with a finite amount of 9s, but if it's with infinite then you're infinitely close to 1, therefore 1.

There are many ways to prove it too, logic, math, etc.

1/3=.333..
2/3=.666...
3/3=.9999... =1

>>7476784
3/3 != .999 though.

>>7475235
>1/3=.333..

All of these are circular logic, and thus invalid proofs. The .333... one is particularly insulting because anyone questioning the identity of .999... is really asking what does it mean for a decimal to repeat infinitely and thus wouldn't accept 1/3=.333...

The truth is that we simply choose to make .999... = 1 by how we define real numbers. The reasons are practical, and do not arise from logic the way other mathematical principals do.

A good use for .999... ≠ 1 is in defining exclusive ranges:

0 - .999... = red
1 - 1.999.. = green
2 - 2.999... = blue

Another related use is that it gives an answer to this equation:

x < 1
x+n ≤ 1
Solve for n.

>>7475207

For two numbers to be different, there has to be a difference.

There is a difference between 5 and 3 and the difference is (5-3) = 2, so 5 and 3 are not the same number.

So: If you can find a difference between 1 and 0.9..., then they are different numbers.

But there is no difference between the two numbers, so they are the same.

>>7476784
1 + 2 + 3 + ... = -1/12

1 = -12 - 24 - 36 - ...

0.(9) = 9/10 + 9/100 + 9/1000 + ... =
9 x ( 1/10 + 1/100 + 1/1000 + ... ) =
9 x ( -12/10 -24/10 - 36/10 ... -12/100 -24/100 - 36/100 - ... ) =
9 x ( -11x12/100 - 11x24/100 - 11x36/100 - ... - 101x12/100 - ... ) =
99 x ( -12/100 - 24/100 - 36/100 - ... ) = 99 x ( -12/11 ) =
-108

>>7476858
No, this is not a choice. It has been mathematically proven that 0.9999.... = 1. Wikipedia has a dearth of proofs of this.

>>7476858

>The truth is that we simply choose to make .999... = 1 by how we define real numbers. The reasons are practical, and do not arise from logic the way other mathematical principals do.

Twice wrong.

>The truth is that we simply choose to make .999... = 1 by how we define real numbers.

No, it's =1 by how we define "..."

>The reasons are practical, and do not arise from logic the way other mathematical principals do.

It arises from the logic of the definition of "..."

Additionally:

>0 - .999... = red
>1 - 1.999.. = green

This implies red = 1= green. If you intended to say "all values from zero and up to but not 1 are counted as red," you should have written it

0 ≤ x < 1 => red
1 ≤ x < 2 => green

Additionally:

>Another related use is that it gives an answer to this equation:

>x < 1
>x+n ≤ 1
>Solve for n.

n = 1-x, no reason to include the ... operator anywhere in this.

>>7476819
but 3/3 = 0.999...

>>7475207
0.999... just, by definition, a way of writing down the infinite sum:
<div class="math">\sum_{n=1}^{\inf} 9(\frac{1}{10})^n </div>

Which makes sense, right? After all, 0.999... is obviously 0.9 + 0.09 + 0.009 + 0.0009 + .... and so on and so on, forever.

This is a geometric series of the form

<div class="math"> ar + ar^2 + ar^3 + ar^4 ... </div>

and since | r | < 1 (r = 1/10), we can apply the convergence theorem for geometric series:

If <span class="math"> | r | < 1 [/spoiler], then the sum of the geometric series is equal to <span class="math"> \frac{ar}{1-r} [/spoiler]

If we plug this in, we find that

<div class="math"> 9(\frac{1}{10}) + 9(\frac{1}{100}) + 9(\frac{1}{1000}) ... = \frac{9(\frac{1}{10})}{1 - \frac{1}{10}} = \frac{\frac{9}{10}}{\frac{9}{10}} = 1 </div>

And therefore

<div class="math"> 0.999... = 1 </div>

If you don't believe this, think about it this way - if you keep getting closer and closer to 1 for literally forever, then you will eventually reach 1. In the real numbers, at least, "0.000...1" is simply nonsense; such a number does not exist; a number divided by infinity is zero.

>>7476873
>Wikipedia has a dearth of proofs of this.

That wikipedia article is notorious for it's endlessly huge talk pages. I've participated in them. Also look up "Dearth" in the dictionary. Also, see pic.


>>7476883

>No, it's =1 by how we define "..."

But you haven't defined "...".

>This implies red = 1= green.

Circular logic again. Even given my stated premise that .999... isn't 1, you keep assuming it is in order to "disprove" it.

>0 ≤ x < 1 => red
>1 ≤ x < 2 => green

Clear as mud. You knew what I was saying with the first equation - it succeeded in communicating the idea. Drop this on some random person and see how long it takes for them to figure it out without your help.

>n = 1-x, no reason to include the ... operator anywhere in this.

No. X is a singular fixed value.

>>7476873
>Wikipedia has a dearth of proofs of this.

That wikipedia article is notorious for it's endlessly huge talk pages. I've participated in them. Also look up "Dearth" in the dictionary. Also, see pic.


>>7476883

>No, it's =1 by how we define "..."

But you haven't defined "...".

>This implies red = 1= green.

Circular logic again. Even given my stated premise that .999... isn't 1, you keep assuming it is in order to "disprove" it.

>0 ≤ x < 1 => red
>1 ≤ x < 2 => green

Clear as mud. You knew what I was saying with the first equation - it succeeded in communicating the idea. Drop this on some random person and see how long it takes for them to figure it out without your help.

>n = 1-x, no reason to include the ... operator anywhere in this.

No. n is a singular fixed value for any possible x.

is there one word or phrase that basically describes that theres an infinite amount of numbers between 0 and 1????

>>7475207
because 0.9999999... <span class="math"> = \sum_{n=1}^{\infty }\frac{9}{10^n} = 1 [/spoiler] .

>>7477415
The real numbers don't contain infinitesimals as actual numbers. You have to go to bigger and stranger number models for that.

There actually are mathematical systems where "0.000...1" actually means something and is a completely valid number- the surreals, for instance. But under the rules of the reals, 0.999... is just a different way of writing 1, and all infinitesimals are zero.

>>7477430/.
"Fractions exist"

>>7477437
good one

>>7477434
>But under the rules of the reals, 0.999... is just a different way of writing 1, and all infinitesimals are zero.

So then why did you assume that OP was talking about the reals? That is an obnoxious assumption. Imagine someone discovering the idea of fractions, only to have the experts insist that all discussions happen in the natural numbers.

You can "prove" that 1/2 does not exist if you refuse to think outside the natural set.

>>7475207
Let us redefine .999.... as limit as x approaches 1 from below on the graph x=y. Since there are no discontinuous moments on the graph x=y, the limit of x as it approaches any number will be equal to value of x at that point on the graph.

>>7477469
Same reason that if somebody asks whether 10 or 100 is larger, I don't first ask "well, are we talking about p-adic numbers here? What base?"

Real numbers are the "standard" numbers. If you want to know whether 0.999... = 1 in the hyperreals, sure, ask away, but for God's sake specify.

(And there's real reasons not to use the surreals, hyperreals, etc - expanding much further than the real numbers makes them far less well-behaved and in turn less useful)

>>7477471
>Let us redefine .999.... as limit as x approaches 1

We can also redefine .999... as a soft drink, but it isn't one. It's a single number. If x<1, what is the limit of a process rising within x? The limit might be 1, but x itself is defined as never being one.

>>7477493
>but it isn't one

Well, that's where you and the definition of recurring decimal notation will just have to agree to disagree. A repeating decimal is by definition a convergent infinite sum.

>>7477484
>Real numbers are the "standard" numbers. If you want to know whether 0.999... = 1

Obnoxious.

Anyone who is questioning the meaning if an infinitely repeating decimal is by definition thinking outside the Real set. What you are doing is exactly the same as using the Natural set to deny the existence of fractions. If they knew the existence of infinitesimals and hyperreals, they wouldn't need to ask the question.

>>7477509
>is by definition thinking outside the Real set.

Yes, but usually that is because they are wrong. It is very easy to think outside of conventional theories by asking questions like "If evolution is real, why are there still monkeys?"

Within the particular model of mathematics that has proved most useful and become a priceless tool in analysis, 0.999... = 1 by definition. Of course there exist exceptions in other models. *everything* is true in at least one consistent model, except for contradictions. Numbers are games and puzzles we have built for ourselves and it only makes sense to talk about whether something is true within a set of rules for that game.

Like, well, this discussion. 0.999... could be considered a separate number, but it also could be not. In the kind of mathematics taught to undergrads, it is absolutely wrong to say it is a separate number. In many less-useful but still logically consistent and valid systems, it is.

>>7477550
>Yes, but usually that is because they are wrong.

It is wrong to think outside of the real set? Are infinitesimals immoral?

>"If evolution is real, why are there still monkeys?"

This is a perfectly reasonable question. It has a good answer. Give it with some patience and you might build a better world. If instead you start using appeals to authority and insults, you've just created a creationist, and you have no one but yourself to blame.

>Within the particular model of mathematics that has proved most useful and become a priceless tool in analysis, 0.999... = 1 by definition.

Useful does not mean right. When the apollo missions went to the moon, their computers used newtonian rather than einsteinian equations. This was a good practical choice, but it doesn't "prove" anything.

In real life, no one uses real numbers for anything but mathematical proofs. We use a subset of the rationals, called floating point or fixed point. For an engineer building a wheel, Pi has maybe 5-10 digits at most.

>Like, well, this discussion. 0.999... could be considered a separate number, but it also could be not.

Then you have just admitted that it is wrong to give a blanket 0.999... = 1 without asking the questioner what his/her "a set of rules" are. why impose your own set of rules without even telling OP that it is indeed a game?

>>7477711
There *is no* One True Mathematics. Numbers are a thing we made up! To claim there's an objectively correct mathematical structure, and that reals are incomplete for not incorporating infinitesimals, is nonsense.

The only way in which a mathematical system can be "correct" is by what people are agreeing to use to talk about math in a particular context. In the vast majority of these contexts, the reals are the default assumption. Claiming something conventionally considered true is false and then going "Ha-ha! I meant the *surreal* numbers, of course!" when somebody calls you on it isn't clever, it's deliberately poor communication.

Also, "0.999..." isn't even proper notation for any infinitesimal-containing system, anyway. So you're not only poorly communicating and then acting smug about it, you can't even be bothered to learn how to write it correctly to avoid being misunderstood. 0.999... is conventionally understood to mean "the infinite sum of 0.9 + 0.09 + 0.009 + ... and so on to 9 * 10^-infinity" The number you're trying to refer to is written <span class="math"> 1 - \varepsilon [/spoiler]. Same way that if I wanted to note I was talking about 3-adic numbers instead of regular rationals, I'd have to say <span class="math"> |5|_3[/spoiler]

In base 2,
0.111... = 1

>>7477792
Proof.

>>7477743
>To claim there's an objectively correct mathematical structure, and that reals are incomplete for not incorporating infinitesimals, is nonsense.

I never said there was anything wrong with the Reals or the Naturals. I said YOU were wrong for denying the existence of anything outside of the Reals when you knew it wasn't true.

>>7475214
It still can't be 1 if there's something missing.

>>7476345
There's no such thing as 0.000...1. What do these ... mean? Infinite number of zeros? Then you cannot have 1 after infinite sequence, you can't have anything at the end of something that has no end. Big but finite number of zeros? Then it has nothing to do with .999...

>>7475247
yes they do. They're called infinitesimals.

0.999... = 1-infsml

>>7476330
nice

>>7476902
even nicer

>>7478247
They don't exist in the deals, faggot.

If you want to do math with hyperreals, go ahead, no one here will stop you. Good luck!

>>7478255
>deals
Reals

because it works better with the rest of math that way

>>7475207
We have to accept that 0.99999... = 1 (that the idea of convergence is valid) because otherwise we can't resolve Zeno's paradox, OP.

His paradox states that a man walking towards a wall will never reach that wall, because first he has to walk half the distance, then half that distance, then half again, for all eternity, so he'll never get to the wall. However, we know by common sense that he will.

His movement can be conceptualized in the following series:

1/2 + 1/4 + 1/8 + 1/16 + 1/32...

If we accept the idea of limits and convergence, then we can say that the above series is equal to 1. Otherwise, the man never gets to the wall. He just keeps moving closer and closer for all eternity. In the same way, we can say that 0.9999999999... = 1. 0.99999... converges on 1.

>>7476330
winner...

>>7478361
What if the man is shrinking as he is walking? He would never reach the wall if he just kept getting smaller.

>>7478388
>>7478361
SPIRAL STAIRCASE, RHINOCEROS BEETLE, A GHOST TOWN, A PEAR TART, RHINOCEROS BEETLE, VIA DOLOROSA, RHINOCEROS BEETLE, HYDRANGEA, RHINOCEROS BEETLE, SINGULARITY POINT, GIOTTO, ANGELS, and THE SECRET EMPEROR.

By definition, 0.999... = lim x->infinity 9/10 + 9/100 + .. + 9/10^x
By the definition of limit, this equals 1.

>>7478137
>infinitely small

There is nothing missing. Use your brain.

>>7475235
>x = 0.999...
>thinking you can multiply this by 10 and simply move the decimal place

Casuistry.

>>7478467
There's always going to be a little bit missing.

>>7478475
doesn't that extrapolate to 999...

>>7477550
if evolution is real, how come an ape like you is here?

can't find the comic i'm looking for so have this one instead

>>7478486
Wait, so you find more comfort in thinking that some all powerful being created some insufferable prick that you disagree with?
As opposoed to be able to assign his emergence on the scene of life to some qvasi-random natural process gone awry?

Shouldn't you curse the tripple/singular sky-father for his careless handiwork everytime you happen upon a creature flawed by it's design?

>>7475207
it isnt.

>>7477805
<span class="math"> \displaystyle
1(\frac{1}{2}) + 1(\frac{1}{4}) + 1(\frac{1}{8}) ... = \frac{1(\frac{1}{2})}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1
[/spoiler]

>>7476858
The second one is circular, but the other two are perfectly valid. When dealing with infinite quantities you are allowed to shift them, and still have infinity. That is an axiom based on the idea that infinity minus 1 is still infinity. If you don't agree with that, THEN you can argue.

>>7477413
>But you haven't defined "...".

No, I'm on /sci/. I also haven't defined "+", and instead just assumed I don't need to.

>>7475207
This might help explain it a bit.
www.youtube.com/watch?v=s86-Z-CbaHA

>>7478854
That was the most amazing thing I've seen in a year or so. Thank you so much...

>>7478247
>0.999... = 1-infsml
Wrong. 0.999...+e>1 for all positive infinitesimals e.

Which number is between 0,9... and 1? Every pair of real numbers has a number between it.

>>7479612
Every pair of real numbers that aren't equal to one another*

>>7475207
Congrats, OP, you have restarted one of the many most brutal fights between idiots and geniuses.

These threads are simply testing our ability to create proofs

>>7475235
/thread

>>7475207
There is no number system which includes real numbers in which every real number has a unique representation.

The 0.999... = 1 is not saying that they're equal, like some equation. These are literally the same number, written two different ways.

Lets do this in Value 6 by 2

6/6=1

Divide the "glass" by 3 -> 2/6, 2/6, 2/6

Combine the pieces, 2+2+2=6

Wooaah, decimal system is flawed, there you go.

what is a synonym for uncountable infinity

>>7475207
if we cannot represent enough decimals that are nine, then we may assume that the one after the last one we represented is higher than 9, then its 1.