/sci/ - Science & Math

[Anonymous No.15288263]

0.999... is not equal to 1 because there is no digit '9' in 1.000...

[Anonymous No.15288270]

Surely if they're not equal then you should be able to name a number between them?

[Anonymous No.15288273]

>>15288263
Yeah but it’s like really really close man. I mean SUUUUPER close. Can’t we just call it 1 and let that 0.00000…000…0001 be our little secret?

[Anonymous No.15288278]

1 / 3 = 0.333....

0.333... * 3 = 0.999...

0.999... = 1

[Anonymous No.15288323]

>>15288278
>1 / 3 = 0.333....
high school moment

[Anonymous No.15288326]

Anddd once again /sci/'s intelligence has fallen drastically

[Anonymous No.15288353]

>>15288326
it hasn't since the only reason .999... can be equal to 1 is because they defined that notation to use limits which is misleading notation, and if you have a sum [math]\sum_{i=1}^{n}\frac{9}{10^i}[/math] it can never equal 1 for any natural number n. Normally when people think of decimals, sums is how they think of that. that's why limits are misleading. because limits of sums might not equal the true value of the sum, as in this case. If you want to prove me wrong, let 1 = that sum I gave, and see how that turns out. It leads to a contradiction.

[Anonymous No.15288356]

>>15288353
I meant [math]\sum_{i=1}^{n}\frac{9}{10^i}[/math].
it didn't show up right.

[Anonymous No.15288358]

>>15288356
it's bloody over.

[Anonymous No.15288394]

>>15288263
Here is the reasoning of 0.999... = 1

If you look at two parallel railway lines going off into the distance you will see they get closer and closer to each other. At an infinite distance from you they do in fact meet!

Mathematics will indeed be a smarter place once these drooling morons are neutered and shipped off to permanent isolation on some deserted island.

[Anonymous No.15288401]

>>15288394
your eyesight isn't continuous, it's discrete, so it wouldn't be able to tell.

[Anonymous No.15288407]

x = 0.999...

10x = 9.999...

10x - x = 9.999... - 0.999... = 9

[Anonymous No.15288414]

>>15288323
>noooo how can it be this simple we need a 430 pages long PDF proving it

[Anonymous No.15288431]

>>15288263
What do you mean by 0.999...?
"..." can be ambiguous.
If "..." means "arbitrarily many 9s", then it's not a fixed number, so of course it's not equal to 1.
If "..." means "infinitely many 9s", then you're a retard.
If "..." means "the limit as the number of 9s increases", then it's unambiguously equal to 1.

[Anonymous No.15288434]

>>15288431
When you see a number written out as A.BCD… that means by definition that the sequence goes on forever.

[Anonymous No.15288438]

>>15288278
>0.999... = 1
No. Just typing (...) as an appeal to some actually infinite process is not the same as an infinite process actually having happened. An actually infinite process CAN'T ever happen. And so you can only ever get closer to one, but getting 'closer' to one in a continuous sense is meaningless, since no matter how close you get, you will always still be infinitely far away from 1. This (...) is un-specific. This is related to the concept stated in pic.

[Anonymous No.15288445]

>>15288434
If you have a finite sequence of digits, then the number it represents is unambiguous. You just multiply each digit by the appriopriate power of 10 and take the sum.

What number does an infinite sequence of digits represent?
Note: I can't perform an infinite amount of work.
So I can't do infinitely many additions.
But I can take limits since proving that a number is the limit os a sequence only requires finitely many steps.

[Anonymous No.15288459]

>>15288263
its a geometric series anon that converges to 1

[Anonymous No.15288471]

>>15288353
I prefer defining infinite decimals in terms of nested intervals. Seventh grade concepts should not be defined in terms of calculus concepts that high school students struggle to grasp. An easy way to explain it is to write (part of) and infinite decimal, cover up all but the first n digits, and ask what's the largest and smallest the number could be. (This requires accepting that 0.999... = 1 of course; the explanation for that is that the real numbers are designed to ignore infinitesimal differences.)

[Anonymous No.15288474]

>>15288431
0 followed by an infinite sequence of 9's after the decimal point.

[Anonymous No.15288476]

>>15288471
Also it's good to explain how infinite decimals locate a point on the real number line by a process of repeated subdivision.

[Anonymous No.15288482]

>>15288471
>> Seventh grade concepts should not be defined in terms of calculus concepts that high school students struggle to grasp.

I agree, but I'd argue that anything involving infinity, including infinite decimal representation, is not a high school concept.
They still ask for answers rounded to the nearest hundreth in high school.
They still round numbers like pi and square roots.
They can stick to approximations and finite decimal representation.
There's absolutely no need for infinite decimal representation.

[Anonymous No.15288489]

>>15288270
0.99999.... with 1 less nine

[Anonymous No.15288533]

Taking "0.999..." to mean infinitely many 9's after the decimal - which is reasonable - we have the following. d(x,y) = 0 iff x=y and d: XxY -> R both by definition of metric. The "standard" metric on R is d(x,y) = |x-y|. |1-0.999...| < epsilon for all epsilon > 0. Hence |1-0.999...| = 0 iff 1 = 0.999... . So yes 1 = 0.999... .

[Anonymous No.15288609]

>>15288263
I got 0.999... problems but 0.999... ain't 1.

[Anonymous No.15288679]

>>15288278
claiming that the ellipsis after 0.333 means the same thing as the one after 0.999 is intellectually dishonest.

[Anonymous No.15288687]

>>15288679
It means exactly the same thing though

0.333... <- 3 repeats forever
0.999... <- 9 repeats forever

The fact that 0.999... is equivalent to 1 is simply a logical deduction. You're thinking about it in terms of symbols rather than quantity, and that's why you don't get it. Once you do get it, it's extremely intuitive

[Anonymous No.15288700]

10 > 9
100 > 99
1000 > 999
10000 > 9999
.
.
.
No matter how many times you keep doing this, the numbers will never become equal.

[Anonymous No.15288703]

>>15288700
>what is convergence to limit

[Anonymous No.15288704]

>>15288703
A fairy tale.

[Anonymous No.15288712]

>>15288704
Yeah it's not like calculus works or anything

[Anonymous No.15288723]

Copes ITT:
>The "..." in 0.3... is NOT the same as the "..." in 0.9...! Don't ask why 3 times 0.3... changes it!
>An infinite sequence of 9s TERMINATES!
>There IS a number between 1 and 0.9..., I define it as the number between 1 and 0.9... and it is an infinite string of 9s but distinct from 0.9... because 0.999 contains one less 9 in its infinity!
>If you subtract 1 from an infinity, it is now smaller than an infinity of the same magnitude!
And funniest of all
>1/3 does NOT equal 0.3...!

[Anonymous No.15288849]

>>15288263
It's just by convention, relax.

[Anonymous No.15288858]

0.999... = 1 - ε,
where ε > 0.

If you're allowed to invent things like infinity, then I'm allowed to invent ε.

[Anonymous No.15288907]

>>15288858
>0.999... = 1 - ε,
>where ε > 0.
Just because you can state something doesn't mean it's true. There is no ε > 0 where this is true. If you think there is tell me. ε = 0.01? No bc then 1-ε = 0.99 < 0.999...

[Anonymous No.15288924]

kek
1/3 = 0.1/3
2/3 = 0.4/6
3/3 = 0.9/9

[Anonymous No.15288950]

>>15288907
Just because you can parrot deprecated proofs which didn't account for infinity doesn't mean it's true.

[Anonymous No.15288984]

>>15288907
No, ε is smaller than every positive real number but greater than 0.

[Anonymous No.15289302]

>>15288687
> extremely intuitive
if it repeats forever, what doesn't it repeat to not be 1 but just more 9s?

[Anonymous No.15289303]

>>15288687
> extremely intuitive
if it repeats forever, why doesn't it repeat to not be 1 but just more 9s?

[Anonymous No.15289308]

>>15288263
it's not a simple arithmetic thing. it's a theoretical concept that is accepted to mean "if it coverged to that than it's that".

it's similar to the infinity theory and limits and the same thing.

[Anonymous No.15289409]

>>15289308
>"if it coverged to that than it's that"
precision at iteration will never be equivalent to [math]\textbf{exactness}[/math]

[Anonymous No.15289431]

Who gives a shit, there's more 9s than atoms in the universe

[Anonymous No.15289461]

ITT: 1st year math students and cranks getting filtered by literal introduction to logic material

>>15288263
In the standard reals, 0.999... is equal to 1.0 because there is no real number between 0.999... and 1.0, so they cannot be defined as different numbers. Also, decimals are imprecise representations of transcendentals and certain rationals. Also also, the number 1.000.... is equal to the number 1.0.
In the hyper-reals, where there are infinite numbers and infinitesimals, 0.999... isn't equal to 1.0, and 1.0 is not equal to 1.000.... You need the construction of the hyper-reals to make this true, because the axioms of the reals do not allow for infinite numbers or infinitesimals.

>>15288273
There is no such thing as a number 0.000...001, where there are an infinite number of zeros between the decimal and the 1. That the number terminates with a 1 contradicts the idea of infinity itself. There must be a finite number of zeros in such a number.

>>15288438
>for no finite number
That's true. Sure is a good thing finite numbers aren't the only numbers that exist. The concepts of limits, continuity, derivation, and integration you learn about in the contemporary style of calculus rely on approximating the concept of an infinitesimal using epsilon-delta analysis. These things were initially defined using infinitesimals, because it is perfectly logical to imagine that, physically speaking, there must be an absolutely smallest, non-zero change from one state to another. However, the axioms of the reals do not allow for such a thing. We have since proven that infinitesimals exist in extensions of the reals, e.g. the hyper-reals.

[Anonymous No.15289462]

>>15289431
imagine how much money has been misallocated since unix time started due to programmers not appreciating the difference?

[Anonymous No.15289472]

>>15288431
You're the retard. "the limit as the number 9s increases" would only equal one when the number of 9's is represented by n and we let n tend to infinity. In other words, the number of 9's becomes infinite. In other words, infinitely many 9's.

[Anonymous No.15289480]

>>15289461
>0.999... is equal to 1.0 because there is no real number between 0.999... and 1.0, so they cannot be defined as different numbers
>r 0.000...001, where there are an infinite number of zeros between the decimal and the 1. That the number terminates with a 1 contradicts the idea of infinity itself.

This is counter logical, if statement 2 is true, than statement 1 hangs indefinitely trying to resolve 0.999... . 0.999.... is always between 0.999... and 1 If it doesn't terminate it is never defined to perform an operation on. Thus for statement 2 to be true, statement 1 must be false

[Anonymous No.15289756]

1 - 0.000...0001 = 1 because there is no end to 0.000... to add a 1 to it.

[Anonymous No.15289760]

>>15288984
Then it would have to be smaller than itself, based retard

[Anonymous No.15289995]

>>15289409
if you do simple arithmetic. the "..." thing is a theoretical concept that talks about infinity which doesn't work with simple arithmetic.

it's practically "it tends to 1 BECAUSE I said 0.9..." (I _defined_ it that way) (it can't be wrong if it's already defined that way).

[Anonymous No.15290008]

>>15289760
It's not real number. The same way infinity is not a real number.

[Anonymous No.15290170]

>>15288270
false assumption

[Anonymous No.15290182]

>>15288270
0.999... < 0.sneed... < 1

[Anonymous No.15290306]

>>15288278
Division as a technique is only an approximation. 1/3 isn't correct because a unit quantity can never bhi divided into 3 perfect equal halves. The technique assumes 1 to be 0.9999999.... to give us the quotient.

[Anonymous No.15290311]

>>15290306
>technique
It's a function of multiplication by the inverse.

[Anonymous No.15290397]

>>15290311
So a technique.

[Anonymous No.15290409]

>>15288353
>>15288356
[math] \sum_{i=1}^{n} \frac{9}{10^i} [/math]

[Anonymous No.15290423]

>>15288263

[math]\kappa=0.999999...[/math]

[math]\frac{\kappa}{3}=0.333333...[/math]

[math]\frac{1}{3}=0.333333...[/math]

[math]\frac{\kappa}{3}=\frac{1}{3}[/math]

[math]\kappa=1[/math]

[Anonymous No.15290551]

1 - 0.999… = 0.000…

[Anonymous No.15290555]

>>15288263
Don't be polarised by this shit.

[Anonymous No.15290583]

These threads are still here, ok lets contribute
0,999... does not exist because you cannot divide 10 things into 3 groups evenly.
Numbers in Base10 is only divisible by 1, 2, 5 and 10

[Anonymous No.15290700]

>>15288263
>0.111... = 1/9
>0.xxx... = x/9
>0.999... = 9/9

[Anonymous No.15290798]

>>15288263
also if it was the case, nobody would bother with the notation [math]1^-[/math] or [math]1^+[/math]

[Anonymous No.15290869]

>>15288263
When it comes to real numbers, 0.9999999<onandon> is the number that is in interval 0 to 1<closed>, but 1 is not there because it's closed interval.

Those is the example of two real numbers being right next to each other with no space in between.

[Anonymous No.15290872]

>>15288407
>>15290869
Please explain.

[Anonymous No.15290874]

>>15288407
It doesn't work like that because you'll hit planck lenght, and there will be small step you'll miss.

[Anonymous No.15290886]

>>15290869
There might not be real numbers between 0.999... and 1, but there may be some non-real numbers.

[Anonymous No.15290907]

>>15290886
But they are not the same number but next to each other, right?

[Anonymous No.15290909]

>>15290886
like half a number

[Anonymous No.15290922]

>>15290909
You hit your post number, like a head of a nail.

[Anonymous No.15291950]

Look chaps, its very simple. One day some faggots decided to define 0.999... as being equal to 1.
Just like that. It was a purely arbitrary decision which they devised to paper over the flaws in their logic. This sort of shit got bridges built and rockets launched so nobody really gave a damn.

The best way is to think of it like how some primitive people developed cargo cult beliefs. Something happened which was beneficial, so they developed a whole religion around it. Engineers and even many mathematicians today are very much like that. They dont question the fundamentals, close enough is good enough for them. One day the planes will land to deliver the cargo again. They just have to be patient. Best not to point out the deficiencies in their thinking, they just get angry and upset.

[Anonymous No.15292035]

>>15288278
This is one of the more intuitive ways to make somebody who doesn’t see it get it. Yet there are still people who stubbornly see symbols and stick to their wrong intuition.

[Anonymous No.15292051]

>>15289472
> "we let n tend to infinity" = "n becomes infinite"
I rest my case

[Anonymous No.15293009]

>>15290874
>planck lenght
>pure math

There is no such thing as physical limit in math... by definition,physics =/= maths

[Anonymous No.15294398]

>>15292035
It suffices for 9th grade students, which is obviously your level. At your level you haven't developed the necessary cognitive skills to see the fallacies.

[Anonymous No.15294427]

>>15290306
>unit quantity can never be divided into 3 perfect equal halves
It can if you use a base that has 3 as a factor instead of decimal. Thats one reason the ancients used sexagimal. They liked exact answers.
Also
>halves

[Anonymous No.15294429]

>>15294427
sexagecimal*

[Anonymous No.15294682]

>>15294427
It cannot be. You chop unit into 30, 60, 3600 pieces and each individual piece would be irrational. You have run into the same problem in another manner. Size of each individual piece is a divisional approximation.

[Anonymous No.15295112]

>>15288323
1/3 =/= 0.333...????

[Anonymous No.15295432]

>>15293009
Physics is applied math.

[Anonymous No.15295464]

>>15295432
No it isn't

[Anonymous No.15296187]

>>15288407
I don't get it

[Anonymous No.15296238]

>>15296187
It's a manipulation, if you do 0,145 x 10 you would get 1,450 if you concider it with same decimal accuracy
So 0,999... x 10 would mean that
0,9 x 10 = 9,0
0,09 x 10 = 0,90
and so on, leading to 9,999...(0) for equal accuracy.
Because anything multiplied by 10 has 0 at the end of it

[Anonymous No.15296243]

>>15296238
consider*

[Anonymous No.15296296]

>>15288263
yes
RL is actually analog not like 4chins

[Anonymous No.15296310]

>>15289461
>That the number terminates with a 1 contradicts the idea of infinity itself
It actually does not, only if the shorthand expands into a finite amount of digits. Funnily enough, it ends up being smaller than an infinitesimal. Each of those zeros being a member of an infinite set, any of the other digits are external to that infinite set (and thus have a countably higher cardinality)

It's not typical to allow this for a positional numeral system, and it breaks a lot of nice assumptions we have about infinitesimals, but there's no inconsistency outside of the axioms we've used to encode those assumptions. Probably best to leave the expression undefined.

[Anonymous No.15297387]

>>15288270
0.999... <thrembo < 1

[Anonymous No.15297455]

>>15288263
Solve this:
[eqn](0.99999...)^\infty[/eqn]

[Anonymous No.15297563]

>>15297455
1^inf is indeterminate

[Anonymous No.15297585]

>>15297455
(0,999... x 0,999...)...
Its open

[Anonymous No.15299831]

>>15288270
Why should there be number between them if they're not equal?

[Anonymous No.15299837]

[Anonymous No.15300438]

[Anonymous No.15301529]

>>15299831
If a and b are unequal, either a < b or b < a. Suppose a < b. Then 2a = a + a < a + b < b + b = 2b and a < (a + b)/2 < b. The other case is similar.

[Anonymous No.15302083]

...the year 3476AD
"Daddy, why didn't we ever reach the stars?"
"Because some filthy sodomites insisted that 0.999... = 1"
"But daddy how could they have been so dumb?"
"Dunno, they had too much anal sex I guess, now eat your cockroaches and go to sleep"