1 is 0.9999999999999............

@ Undefined

with this I am in total agreement with...
...am still considering the rest of your post...

 

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Sorry Undefined but I can not agree with this approach to poster Tach. This fora is littered with smart and yet somewhat "crazy " people and you, I and Tach and others are good examples IMO...

Tach's response of "NO" even though unexplained has some foundation. IMO

Firstly,
When dealing with these damn infinitesimals extreme care HAS to be used, as you have already noted..

The issue of adding a zero to an infinite string when multiplying by ten has some merit but has also a need for rigorous debate as I feel at first glance that this is not quite right.
However the point that you make that, presumptions of validity when dealing with such "contextual axioms" vs "isolated axioms" need to be carefully scrutinized has huge merit. IMO
edit, and I might add , adding a zero at the end when multiplying by ten is only another proposed axiom or rule

Secondly,
To warn other posters about another member is denigrating/dis-empowering other members ability to decide who is a troll and who is not and serves you very little. It also dis-empowers the moderator from making a proper decision based on current posting behavior and not past posting behavior.
Thirdly,
I wanted to thank you for your long post as I see much merit in it...


just my honest opinion.. ok

 

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The point is that it is very clear that:
10 x 0.999... - 0.999... = 9 x 0.999...

Look:
10 x 0.999... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...
9 x 0.999... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...

Agreed?

Thus proving that 1.999... = 2

Thus proving that 8.999... = 9

If \(1/\infty\) has any value, then that value is zero. Or do you think that Achilles really can't catch the tortoise?

 

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No, we move the decimal point one place to the right.

For example, the decimal expansion of \(10\pi\) is 31.4159... and it does not end in 0 (it does not end at all).

 

So, the complete answer, is "no, you don't know what you are talking about": \(10*0.(9)=9.(9)\)

 

no we haven't.
You keep repeating that we have agreed and I keep repeating that we haven't agreed...

again:
9.999... - 0.999... = 9
yet
9 x 0.999... = 8.999....

for
9 x 0.999... to equal 9
an infinitesimal needs to be added.

so
9 x 0.999... + 1/infinity = 9

 

@ Pete,
Compare the two following identities

yours
10 x 0.999... - 0.999... = 9 x 0.999...

and mine
10 x 0.999... - 0.999... = 9 x 0.999... +1/infinity

which one is correct?

 

Let me help you:

10*0.(9)-0.(9)=9.(9)-0.(9)=9

 

gosh Tach .... not sure how that helps ...assuming I read it right...that is
It appears you are using a different context....

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In post 83, you said you had no problem with that identity (10 x 0.999... - 0.999... = 9 x 0.999... )

Look:
10 x 0.999... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...
9 x 0.999... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...

Agreed?

 

No, I am just using basic math.

 

The resolution to all of this is really simple: either 1 = 0.999... or, it doesn't.
Mathematics doesn't allow any other solution, it doesn't have any theorems that are almost true, or nearly false.

0.999... + 0.999... = 1.999..., or it doesn't. QQ appears to believe there is room for some 'infinitesimal' number, but 0.9 + 0.9 = 1.8, and 0.09 + 0.09 = 0.18 and so on. Adding two identical numbers which have an infinitely repeating decimal doesn't change the rules of addition.

 

Hi Pete.

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With respect I must disagree with your unexplained "moving the decimal point one place to the right".

I would humbly suggest that fundamentally that is the consequence of extending the number string itself (irrespective of parsing to begin with) such that the leading 9 in 0.999... is FORCED to move to the left across the decimal point 'parsing' notation used for the string overall.

In short, there is no basis for just 'moving the decimal point', since the decimal point is a NOTATION element, not an effective part of the number string/operation of multiplication by 10 itself (which brings the ending '0" to the string per se as part of the "10" number expression).

Moving a 'parsing' symbol is not a mathematical operation; but rather it is a consequence of the math operation which extended the string to force the leading nine over into the 'unitary' place IN that notation format of the actual number elements involved.

That is just one of the straightforward fundamental things which is being 'lost' because of the abstract manipulation of symbols being confused with the numbers/quantities they help 'format' rather than those symbols being fundamental numbers themselves. Just as QQ is saying about the use of Algebraic notations/symbols trivially because they are mixing context of instance and value attached to the numbers/entities involved in the mixed context algebraic 'proofs' he and you are still discussing.

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Agrees [generally] However when using infinities there could be an issue...but finding the reasoning in the various theorems is not going to be easy...

it is true that
0.9999999999999999999999999999999999999999999999999999
plus
0.9999999999999999999999999999999999999999999999999999
equals
1.9999999999999999999999999999999999999999999999999998

however this presumes an ending to an infinite string of digits....the paradox that the use of infinity creates becomes evident [ Zeno's paradox]
so in my humble opinion... extreme care is needed when discussing these things.

 

See what Pete did? He replied properly and politely, and not just your usual 'contrariness' "No" without any supporting comment. Now that you have seen Pete's answer, and my reply to him, you can in future reply in a proper manner or not at all (unless your intention is to add more 'noise' to the thread?).

As for my reply to yours now, please read my reply to Pete. Thanks.

 

Looking at it that way, moving the decimal point is a consequence of multiplying by ten.
Adding a zero to the end of an integer multiplied by ten is a similar consequence.

"Extending the number string" is not multiplication by ten, it's not a math operation at all. It's a parsing operation in exactly the same way as shifting the decimal point.

 

@undefined
Oh my goodness [ chuckle] I wasn't wrong when I wrote the fora has some smart but crazy posters...

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how can an infinite string be extended another digit? hee hee ... so therefore the end digit when extended must be zero... [love it! ha]

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might be totally wrong but I dig that sort of adventure....

just multiply 0.999... by ten and we have proof of zero [chuckle]

 

I may have agreed in post 83 but have since spent many post showing that I actually disagree

For the record:
I DO NOT Currently agree with the equation as being sound
10 x 0.999... - 0.999... = 9 x 0.999...

I do however currently agree that this equation is sound

10 x 0.999... - 0.999... = 9 x 0.999... +1/infinity

Can you at least agree that I do not agree?

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He just proved you wrong. So did I.

 

Thanks for your kind and fair appreciation of my contribution to the on-topic discourse, QQ.

Regarding the treatment/comment about the Tach troll post, I should perhaps say that it was Admin who gave me and others excellent advice to ignore that troll whenever he makes his usual "anti-discussion" tactics for which has been banned more than once recently in the hopes that he will learn the lesson NOT to make such obviously vacuous and unargued replies to 'isolated excepts' from context etc etc which have all too often in the past ruined very good threads/discussions (including some of the mods' own threads/discussions!). So I was only following excellent admin advice re such troll posts, and spreading the advice to others who may have missed it from the admin/mods about such trolls/tactics as Tach tried on again there! If he has learned anything it was not evident from that vacuous troll post of his despite being banned for such behaviour recently.

Anyhow, Let's see if the message has got through to him. It does seems the message to behave properly may be starting to get through (evidence his latest attempt at reply....even though it merely asserts some axiomatic 'treatment' without discussing the reasons for why it is fundamentally valid or not as the case may be in light of what you and I have posted that indicates otherwise...IN MY opinion, as argued, of course.

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