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

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Why bother finding an authority if you can't understand what was written? It was very short.

Did you notice he said "a few proofs" (out of many) and "in set theory" (a topic you don't understand)? Basically he is saying that the mapping between all possible infinite decimal representations to real numbers is surjective (at least one decimal representation exists that maps into any given real number) but not injective (all decimal representation point to a different real number) and therefore the mapping is not bijective which would mean proofs about decimal representations would directly imply proofs about the real numbers. For example, if you wanted to find a solution for a decimal representation which solves "x = 0.9 + x/10" both 1.000... and 0.999... obviously solve it so there are two solutions in decimal representations and so the solution to a linear equation is not unique, but in real numbers there is only a unique solution, 1. So proves of existence aren't the proofs that are complicated, but proofs of uniqueness obviously are, as are proofs that count numbers of solutions.

That's not the redundancy he is talking about.

What rule do you think he is changing?

He is saying if you want to prove "The number of real number solutions to ψ is n" you don't need to know "The number of decimal representations that map to real numbers that solve ψ" but rather "The number of decimal representations that map to distinct real numbers that solve ψ" for which you can choose one of "The number of decimal representations (exclusing those that end in an infinite unbroken string of 0's) that map to real numbers that solve ψ" or "The number of decimal representations (exclusing those that end in an infinite unbroken string of 9's) that map to real numbers that solve ψ".

Making fanciful claims doesn't make your argument appear more cogent. You have simply failed to understand your choice of topic, Jim Loy's incredibly short article. You should have stopped at the first word you didn't understand which in the quoted section was probably "proofs" and almost certainly a word at or before "set theory."

 

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Posters here are ignoring issues relating to context.

Code:

If somebody asks me what is 10 * .999,
I will reply 9.99 & not consider it necessary to add a zero.

If somebody asks me what is 10 * .999 - .0000000001, I will add a few zeros to the product before doing the subtraction.

BTW: The Posts leading to the above remarks are a bit off the basic topic of this Thread.

Now back to the basic issue of this Thread.

I agree that when viewed as a geometric series, .99999 . . . . . . can be proven to have a least upper bound of one, making it reasonable to assert that the sum of the series is one. Note that some pedantic texts express this in a weasel worded manner similar to the following.

The proof I object to uses the notions of ordinary arithmetic. It is as follows.

Code:

x =  .99999[b] . . . . . . [/b]
10*x = 9.99999[b] . . . . . . [/b]
10*x - x = 9

Ergo: 9x = 9 ----->>> x = 1

My objection to this proof is the following.

Code:

x = .999 
10x = 9.99
10x - x = 8.991
9x = 8.991 —>>> x = .999

x = .99999
10x = 9.9999
10x - x = 8.99991
9x = 8.99991 —>>> x = .99999

The above proof can be extended to a string of thousands, millions, a google, a googleplex of nines.

By what concept of elementary arithmetic can it be asserted that there is a magic number of nines called infinitely many, it is valid to assert that the final conclusion is: x = 1?

 

Try this elementary arithmetic: (the (09)[sup]*[/sup] notation means repeat (09) infinitely)

100 = 9 x 11 + 1.
=> 100/11 = 9 + 1/11 = 9 + 0.(09)[sup]*[/sup] = 9.(09)[sup]*[/sup].
=> 10/11 = 9/10 + 1/110 = 0.9 + 0.0(09)[sup]*[/sup] = 0.9(09)[sup]*[/sup].
=> 10/11 + 1/11 = 0.9(09)[sup]*[/sup] + 0.(09)[sup]*[/sup] = 0.(9)[sup]*[/sup] (or 0.999...).
But 10/11 + 1/11 = 11/11 =1.

If you divide 100 by 11, using long division, you should see that you get a repeating decimal. What IS a repeating decimal? How many places are there in this "repeating" decimal?
Unfortunately for some, this number is decided by the finite number of times you can be bothered repeating the division method on part of the actual result.
What is the actual result? It's an infinite sequence of numerals in base 10. What's so bloody hard about it?

 

The 'decimal powers' represent decimal PLACES where ZERO is the symbol for EMPTY PLACE (ie, NON-action 'place-holder') which leaves the OCCUPIED PLACES to determine the string value overall.

So your use of 'decimal powers' merely invokes '0' in another way, that's all. Trivial.

And so it does not answer the point made about the 'introduced 0' being the EFFECTIVE cause of shifting the string, and NOT 'shifting the decimal POINT symbol. The active part of the process is the number place symbols, not the decimal point symbol itself which merely delineates when the number symbols move/transition across the unitary+ and the fractional 'legs' of the string places.

No, adding ANY numeric symbol is what shifts a string. That is the point. The '0" in the decimal system is merely that, in the decimal system. There are ZERO PLACEHOLDER symbols in BINARY strings.

The question is when a string is infinitely completed, then NO numerical string MOVEMENTS can occur, since it is FROZEN in that infinity state of all numbers being there already completely.


Anyhow, just to remind of some of the context: this latest side-discussion about decimal system 'shifting' was about which is the more fundamental/real 'operation/action': shifting the decimal point (a non-number symbol) OR shifting the numerical symbols (ie, '0' or whatever) to EFFECT the 'calculation'?

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Whatever the context, the more fundamental/real operation/action is the shifting of string CAUSED by number inclusion INTO that string (wherever that inclusion occurs in/on the string). By merely saying something like 'we shift the decimal point', it does NOT actually refer to a true fundamental/real process/action, but rather reflects mere FORMATTING AFTER the fundamental/real operation/action involved.

Thanks for your replies, Trippy. Cheers.

 

Yes, I could have easily re-written 22 as 22.0, and merely extracted the zero and let the "numbers shift one place to the right" (note, NOT "shifted the decimal point to the left"....there is a fundamental/real difference: please see my above post to Trippy), but then I would not have been able to explain the actual actions involved fundamentally (ie, that long division PROCESS involving numbers and not just depending on 'short cut'.

After I explained that process, I then REVERSED it using exactly that 'short cut' in the multiplication process. So NOW that the importance/fundamentality of the '0' caused 'shifting' of number string (not decimal point per se), it is obvious to you and all that the short cut in dividing by '10' WOULD have been to re-write the 22 as 22.0 and then remove the zero and let the string shift to the right across the decimal point.

Note again: In reality/action, it is the string that 'transitions across the decimal point'......(NOT the decimal point that 'shifts' to the left, as invalidly used by some as part of the 'argument/action' in their invalid 'proofs').

Thanks for your responses, Monimonika.

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a bit like shifting the decimal point in Pi...
go on add a zero or two to the end of a pi sequence and see what happens... [ chuckle]
Why would an infinite series 0.999... be treated any differently to Pi?

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196
4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609?????????????????????????????? ok.. lets just throw in a zero

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Still at it, Tach? Declaring 'victory' etc when the discussion has not yet completed to conclusion according to the actual discussion in the context/points involved? Are you ever going to stop with your 'chearleading', and 'insulting' and 'arrogant preconclusionary attitude' etc trolling empty posts, Tach? You're lucky I haven't reported that post. Please do better. You can if you really try, I know you can. Good luck!

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Like you said, mate, if the string never determines to a finite value, you are left with some 'limits' loophole for saying anything at all about the 'end' of such an infinite string. Yes?

And if you agree that ZERO is an 'infinitesimal and not 'nothing', then adding a last zero is merely reflecting whatever VALUE that last effective 'step' to resolution has in reality which transitions the string from unreal/unending abstraction to a real/determinable concrete value that brings it back from 'infinity' unknown state. Hence the valid hypothesis that 'nothing' does NOT exist in reality energy-space.

Zero as 'nothing' only exists as an abstraction of 'placeholder' in numeric system.....OR as a real BALANCE/TRANSITION/SUPERPOSITION state/location in REAL energy-space terms between one direction/system of dynamical phenomena sets and another.

Hence the hypothesis/axiom that '0' is an infinitesimal NON-nothing 'real value/point'.....and NOT '0' as currently treated axiomatically as the NOTHING which leads to all the undetermined/undefined etc absurdities and excuses.

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It looks like you're trying really hard to find something "wrong" with 0.9(9)* = 1.0(0)*.
Now you're thinking this end of an infinite string has something wrong with it. And it does: it doesn't exist.

There is no last step, instead you have the (n + 1)th step. Oh dear.

This time what fails to appear, like the end of an infinite string, is any connection between numbers and that "hypothesis".

Look, 0 is a number. It's a number that has to "exist" mathematically, in order that certain rules of addition and multiplication are closed.

What is absurd is the mental gymnastics on display by people who want to say "there must be something wrong", in the face of many examples that have nothing, that's zero, wrong with them.

 

Sorry, I'm not going to wade through past posts in this thread to search for whether that part in code is supposed to be a quote from you, if you're just implying agreement with it, or is something else entirely. At the very least you should include post #s or links so I can read the "context" for myself.

Either way, you're the one who brought up the whole "issue" with appending zero and whatnot when you replied to me regarding clarification of your post #901:

Now let's see what that has to do with what you state you object to:

The answer is... ABSOLUTELY NOTHING! This so-called "elementary arithmetic" of appending a zero when multiplying by 10 does not show up even in the code that you, Dinosaur, yourself wrote as a proof that can "be extended to a string of thousands, millions, a google, a googleplex of nines" and that supposedly also uses the same "elementary arithmetic". You seem to constantly pick up and drop what constitutes as "elementary arithmetic" whenever you please.

I will repeat what I said in my post #944:

Of course, instead of clarifying as I had requested, you just introduced an unrelated "elementary arithmetic" concept (post #1330), then declared it off-topic (post #1344), then simply repeated what you had in post #901 without specifying what the heck is the difference between the first proof and the second proofs (post #1344).

I have to ask, is the concept of:

9x = 9
9x/9 = 9/9
x = 1

not a part of your "elementary arithmetic"? Are you really that obtuse?

Or is it the part where it goes from:

10*x = 9.99999 . . . . . .
to
10*x - x = 9

? If so, then just say that to me instead of trying to imply it with unrelated garbage. If not, you are one very, very confused individual.

 

That is the part he doesn't get. He doesn't understand the notion 10*0.(9)=9.(9)

 

Better But Too Complex To Grasp For Most Humans

A vaguely recall "proof" but if his "proof" is not understood by me, then it will not appear as "proof" too me.

Ok, cool, three words that I have never heard of in my life--- some 50 years ---and may never hear of again.

Yeah this the key point--- called mapping ---where Column X with n becomes a in Column Y with a etc....again this at first sight and with minimal knowledge of what will appear to most humans, as complex branch mathematics, that, is changing X-n( frog ) into Y-a( dog ) i.e. to me all of this still leads to what is obvious to most humans who would view the original statements again;

0.999...infinite... has not #1 in it and the real/natural whatever we some want to call it, where there #1 has to exist, if we want to have our given irrational and infinite decimal equal 1. 0.999...infinite...does not equal 1. Frog( 0.999... ) does not equal dog( 1.000 ).

Ok great( thx ) an actual process, that maybe some humans will be able to get grasp/understand, to have a mathematical process of converting( mapping ) a frog to become a dog. So let me see if I can figure out what your symbols mean.

X = some number( n ).

0.9 + x divided by 10.

Then you dont say equal( = ) you just state both 1 and .999 so you maybe just forgot the =.

To this x reads as n so, 14 + ( 14/10 ) = 15.4 and not "1" or 0.999 and certainly not 0.999...infinite.....

So please RP, give us a simple--- big dummies guide ---and clear explanation, as to how x( of is n ) or frog, is equal to dog?

I'm glad for the link to complex mathematic that I-- and most of humanity ---will never understand much less ever use, but please skip all of the complicated mathematics of mapping and explain to it us as tho were 8 14 year old, that a finite rational frog( 1 ) is equal to and infinite irrational dog( 0.999...infinite....

The one that says finite rational frog( 1 ) is equal to an infinite irrational dog( 0.999.... ).

It appears to me, that most humans will never grasp your and lovejoys concepts as explained, and that you do not understand the simplicity of why it seems like nonsense to me, to say that a finite rational frog( 1 ) equals and irrational infinite dog( 0.9999...). Do you understand why me if not we would think you are off your rocker or maybe should be put into a rocker...

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...just kidding with you.

Sorry I have no idea what the pitch fork symbol is supppose to mean. "rea'" solutions fake solutions, magically mathemiatcally solution etc...make no differrence to me--- if not many others RP. I and others need to see some simple relatively simple explanation of how a rationally finite frog( 1 ) is equal to irrationally infinite dog( 0.9999.... )?

Making fanciful complex/complicated does not make you appear more cogent to me.

You have failed to give us a relatively simple, rational logical and understandable to the average human, within this one email message with ingecti, surgjectives etc....thx again, but no banana for you. I dont think you are able to understand why I perhaps others think your in error and have yet to sifficiently explain why your not irrational and should be in rocking chair in old folks home.

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sincerely RP.

Come back with something Eartly to most humans if you ever want to convince us that a frog is equal to a dog, other than saying their both biologicals. And in some ways perhaps you "mapping" is just a fancy way of saying that, yeah their equal to each because their both biologicals, at that level. Idunno because I cannot really grasp the fancy( compliecated ) mathematics. Sorry. No offense just from another planet and we need someone who can speak both Earth language and extra-terrestiral langugage.

r6

 

Hi arfa brane, everyone.

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To help forestall any further cross-purpose exchanges, i refer you to the following post of mine to rpenner (in QQ's philosophy of maths thread) where I outline the overall thrust and issue about the infinitesimal and treatments based on concepts like infinity etc. I essentially point out that the infinitesimal may be in the same category of IMAGINARY NUMBER but with a real basis in physical reality because the 'o' (nothing) is an abstraction whereas the '0' as infinitesimal SOMETHING (an 'imaginary', if you will) DOES have a NON-zero (but infinitesimal) value BUT is not like any other number (ie, hence in the category of 'imaginary infinitesimal').

Anyhow, here is the post quoted below for your convenience. I trust that the issues will become clearer as to 'reasonableness' and 'intent/approach' for making the axiomatic assumption/notion of 'point' be NO LONGER treated as "NO DIMENSION" status, and instead be given axiomatically a NON-NOTHING value status that is 'imaginary' and 'exists' at the 'end' of all 'infinity-based' strings/concepts as the final cut/step of effectiveness/transition etc, as already explained...

Enjoy the unexpected 'imaginary' twists and turns in the journey towards the consistent completion of the maths and physics, folks! Keep discussing your own takes on it all! Cheers all.

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The original point, in case you weren't reading closely (again), was if the concept of "appending a zero when multiplying by 10" was REQUIRED

then wouldn't dividing by 10 also REQUIRE that a zero be removed?

Dinosaur decided to skip out on answering that question, and you decided to answer with your explanation of the long division process.

I then pointed out what I thought were inconsistencies in your explanation of how both "append a zero when multiplying by 10" and "remove a zero when dividing by 10" could work out. And you have totally lost me as to what the heck you are trying to (not) say:

The above implies that you DID NOT take the hypocrite's route of adding an extra zero in order to have something to remove when dividing by 10. This is fine.

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Yeah, you spelled that out for me quite clearly in your previous reply to me:

Yes, you already stated in your first sentence that you did not take that route. So...?

What does that have to do with the "removing a zero when dividing by 10" thing I was asking about? You know you have not yet addressed the contradiction I pointed out, right?

22/10 = 2.2 [removal of the "0" from the "20" part of "22" during long division]
22.0/10 = 2.20 [removal of the "0" from the "20" part of "22.0" during long division]

Combining the above REQUIRED (Undefined's) method of "removing a zero from the original value when dividing by 10" with the REQUIRED (also Undefined's) "appending a zero to the original value when multiplying by 10" results in the following contradiction:

2.2*10 = 22.0
22.0/10 = 2.20

So, 2.2*10/10 = 2.2 * 1 = 2.20 [...wait, what?]

2.20*10 = 22.00
22.00/10 = 2.200

So, 2.20*10/10 = 2.20 * 1 = 2.2 * 1 * 1 = 2.200 [...huh? Why is another extra "0" notation REQUIRED here?]

2.200*10 = 22.000
22.000/10 = 2.2000

So, 2.200*10/10 = 2.200 * 1 = 2.2 * 1 * 1 * 1 = 2.2000 [...oh come on, this is getting ridiculous!]

Undefined, you know that your method of "removing a zero when dividing by 10" does not work with the supposedly REQUIRED "append zero when multiplying by 10", right? Are you going to respond?

And then there was nothing. No conclusion whatsoever from the wordy rambler (that's you, Undefined) who obviously has no reading comprehension skills, logic skills, nor math skills.

Edited to add: And don't start yammering about infinities, imaginary numbers, what others said about decimal places, etc. I am working only with finite strings of numbers here and you are currently running away from answering a VERY SIMPLE contradiction I pointed out within your post. Face it or just admit you have no idea what you were trying to do in the first place.

 

Hi Monimonika.

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Sorry if I implied I was only dealing with your aspect in all this. I am keeping to the wider context, as explained in my posts already, here and elsewhere.

As far as THIS particular side-discussion/aspect about 'appending a zero' (or any other actual number symbol) is concerned, the whole point arose when someone used the "shift the decimal point" to the left (or right, as the case may be) AS IF THAT WAS some valid numerical action/process in reality. Which is not, since it is the behaviour of the NUMBERS themselves which occupy OR NOT the decimal PLACES which 'causes' the shifting of the number string (and NOT the decimal point which is NOT a number and does not 'shift' anywhere).

That was the overall context of my posts about the behavior of numbers as the fundamental action, and not just the decimal notation point itself. Do you understand now where this aspect/issue (ie, 'add a zero' to the string to make the string move along/transition etc) came from? And why that is the fundamental 'action' and not the 'move the decimal point' route/arguments?

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Anyhow, it doesn't matter what number, whether a 9 or a '0' is appended, that is the point. Unless 'something' is changed IN THE NUMBER STRING ITSELF, there ins NO 'movement' of anything, and that especially includes the infinite string 0.999... which is currently assumed to be mathematically FROZEN ALREADY with all the 9s already all there.

My point has been about what can make that infinite string 'transition' from never ending fractionality 'infinite string' to become unity and above (ie, move across the decimal point, not the decimal point moving) value string?

That is where an 'imaginary number' called the 'inifinitesimal' (ie, a non-nothing zero symbol) comes in.

That is, an imaginary infinitesimal (what we call SOMETHING-not-NOTHING '0' in this context) MAY OCCUPY ALREADY the 'last possible EFFECTIVE STEP position in the infinite string' such that its necessary inclusion there effectively TRUNCATES that infinite string and brings it into the next state, that of UNITY. The decimal point/notation is immaterial. The whole issue revolves about gettinfg the infinitely FRACTIONAL string to become a FINITE UNITY string.

And so far the only real/fundamental way to solve the axiomatic impasse and JUSTIFY the use of the LIMITS argument is to recognize that when we include the 'infinity' symbol in our limits argument, we are effectively acknowledging that an IMAGINARY (ie, INFINITESIMAL last effective step) NUMBER is already a part of that 'infinity' invoked, and represents the last REAL non-zero (superposition zero point, if you will) step which IS the 'limiting/transition state' change in reality as well as in maths!

I hope you didn't get offended by my trying to keep to the wider context/issue as the thread posts will show is what we are all heading for in the end! Cheers and thanks for your contributions, Monimonika. Most useful and interesting and, I assure you, most appreciated.

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Much shorter Undefined: I did not actually read Monimonika's posts and just replied to them because I imagined they had something to do with whatever word salad I am making at this time. Hopefully my extra-wordiness will miraculously make me look intelligent. Thank you, [insert various names here]!

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You just demonstrated that you don't know what imaginary numbers are. Why you insist in demonstrating your ignorance while pretending to be "debating" higher order concepts is a mystery. You aren't impressing anyone, more exactly, you are producing a very bad impression.