1 is 0.9999999999999............
- Thread starter chinglu
- Start date
A Banana For Billy T for Sincere Effort and Simlistic Approach
Huh? One( 1 ) is not a value? I think your incorrect on that assessment also BT. infinite value of 0.999... is more like to not fit a description of a "value" more than 1 not being value.
Please direct me the the websites that clearly show that 1 is not a value.
Primary concern on any group or forum is NOT to learn mathematics. That is an aside benifit, although mathematics can become very complicated and it fairly obvious that few agree on what is the correct math for this, that or another issue. Just follow this thread to see how much dissagreement on what is correct mathematical procedure to follow, that there is.
So here again, irrespective of any base, a finite, or a finite value, or a finite number value etc.....
will never ever equal a infinite , or infinite value, or a infinite number value etc......
Finite never ever equals infinite is really the basis for the non-crux/non-connundrum of this and associated threads BT.
There is only one or two ways to attempt to rationally equate or associate a finite with an infinite, and that is as follows;
We begin with a finite whole ergo a sum-total whole potential, and say that, we eternally, micro-infinitely subdivide the finite whole.
This scenario invokes or involves a process, over time.
A decimal is subdivisonal processing of a finite whole and that subdivison has two possible ways of occuring;
1) finite set,
..10, 8, 4 and 2....
2) infinite set.
...9, 7, 5 and 3....
For some weird reason, that, I do not understand yet, the numbers 9, 7,6 and 3 invoke/induce involve and infinite process i.e. if we had cal. with infinite spaces, then the cal. would never stop subdivisional process.
And the numbers 10, 8, 4 and 2 invoke/induce a finite process.
Seems pretty simple to me. Infinite is infinite in any base. Finite is finite in any base. So please do not come back and tell me that we can not have and infinite in bases other than 10, or,
infinite value in bases other than 1o, or,
infinite number value in bases other than 10.
Thx for attempt to offer the a "base 10" as a simple answer to why finite 1.0 = infinite 0.999...
I'm sorry BT, you get a banana for sincere effort, and simplicity over RPenner who cannot reduce his complex "mapping" math to a rational, logical, common sense, and relatively simple, explanatory guide. Big Dummies Guide.
r6
Huh? One( 1 ) is not a value? I think your incorrect on that assessment also BT. infinite value of 0.999... is more like to not fit a description of a "value" more than 1 not being value.
Please direct me the the websites that clearly show that 1 is not a value.
Primary concern on any group or forum is NOT to learn mathematics. That is an aside benifit, although mathematics can become very complicated and it fairly obvious that few agree on what is the correct math for this, that or another issue. Just follow this thread to see how much dissagreement on what is correct mathematical procedure to follow, that there is.
So here again, irrespective of any base, a finite, or a finite value, or a finite number value etc.....
will never ever equal a infinite , or infinite value, or a infinite number value etc......
Finite never ever equals infinite is really the basis for the non-crux/non-connundrum of this and associated threads BT.
There is only one or two ways to attempt to rationally equate or associate a finite with an infinite, and that is as follows;
We begin with a finite whole ergo a sum-total whole potential, and say that, we eternally, micro-infinitely subdivide the finite whole.
This scenario invokes or involves a process, over time.
A decimal is subdivisonal processing of a finite whole and that subdivison has two possible ways of occuring;
1) finite set,
..10, 8, 4 and 2....
2) infinite set.
...9, 7, 5 and 3....
For some weird reason, that, I do not understand yet, the numbers 9, 7,6 and 3 invoke/induce involve and infinite process i.e. if we had cal. with infinite spaces, then the cal. would never stop subdivisional process.
And the numbers 10, 8, 4 and 2 invoke/induce a finite process.
Seems pretty simple to me. Infinite is infinite in any base. Finite is finite in any base. So please do not come back and tell me that we can not have and infinite in bases other than 10, or,
infinite value in bases other than 1o, or,
infinite number value in bases other than 10.
Thx for attempt to offer the a "base 10" as a simple answer to why finite 1.0 = infinite 0.999...
I'm sorry BT, you get a banana for sincere effort, and simplicity over RPenner who cannot reduce his complex "mapping" math to a rational, logical, common sense, and relatively simple, explanatory guide. Big Dummies Guide.
r6
Billy T said: "No 0.999... is an infinitely long (in base 10) way to express the FINITE value 1, or 0.999... which are only two different names, not values, for one finite and rational same point on the number line."
" 0.999... Is an infinitely long (in base 10) way to express the FINITE value 1
Or 0.999... , which are
only two different names, not values,
for one finite and rational same point on the number line."
I.e. they are different names, not different values for the same point or
For the same finite
And
Rational value or
Point on the number line.
That closed tautology is probably the ONLY realm of knowledge where there is 100% agreement by the informed. Truth there is not an opinion, but proven.
Why your opinions on math are worthless.
BTW, typically there are many different proofs for any math truth. The Pythagorium theorem, I think has more than 30 different ones, one even added by US president James Garfiled! His is quite simple, easy to understand, but does use a little simple algebra too. It and several other using geometrical figures are illustrated here: http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html
Instead of repeatedly posting your nonsense here, displaying your math ignorance, Why not OFF LINE, show that 2 + 2 = 5 during the fools moon.
Reading simple English is difficult for many idiots. I'll divide my sentence into parts for you:
" 0.999... Is an infinitely long (in base 10) way to express the FINITE value 1
Or 0.999... , which are
only two different names, not values,
for one finite and rational same point on the number line."
I.e. they are different names, not different values for the same point or
For the same finite
And
Rational value or
Point on the number line.
No, only idiots and the ill educated in math disagree on what is true in math.
That closed tautology is probably the ONLY realm of knowledge where there is 100% agreement by the informed. Truth there is not an opinion, but proven.
Why your opinions on math are worthless.
BTW, typically there are many different proofs for any math truth. The Pythagorium theorem, I think has more than 30 different ones, one even added by US president James Garfiled! His is quite simple, easy to understand, but does use a little simple algebra too. It and several other using geometrical figures are illustrated here: http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html
Instead of repeatedly posting your nonsense here, displaying your math ignorance, Why not OFF LINE, show that 2 + 2 = 5 during the fools moon.
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What's to prove?
Please excuse the formatting of this first step if the columns don't quite line up.
$$
\hspace{37 pt}0.111\overline{1} \\
\frac{1}{9}= 9 \overline{\big) 1.0000...}\\
\hspace{37 pt}-9 \\
\hspace{42pt}\overline{\hspace{7 pt} 1}0 \\
\hspace{43 pt} -9 \\
\hspace{47 pt} \overline{\hspace{7}1}0 \\
\hspace{49 pt} -9 \\
\hspace{56 pt}\overline{\hspace{7}1}0 \\
\hspace{56 pt} -9 \\
\hspace{65 pt} \overline{\hspace{7}1...}
$$
$$ \frac{1}{9}= 0.111\overline{1}$$
$$ \frac{1}{9}=1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...$$
$$ 9 \times \frac{1}{9}= 9 \times (1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...)$$
$$ 9 \times \frac{1}{9}= 9 \times 10^{-1} + 9 \times 10^{-2} + 9 \times 10^{-3} + 9 \times 10^{-4} + ... + 9 \times 10^{-(n-1)} + 9 \times 10^{-n} + 9 \times 10^{-(n+1)}+ ...$$
$$ \frac{9}{9}= 0.999\overline{9}$$
Hi Trippy, everyone.
While I do appreciate, Trippy, your succinctness (and your humour) of those "9/9" responses to chinglu, I would again point out my earlier generally directed observations regarding starting an argument from a facile 'construction'; or even a UNITARY state and not from a fractional state. Because naturally if one starts from 1, all arguments will lead back to 1. Circuitous. Not a way to 'proofs' at all of the fractional case of 'becoming one'. It is a case of now having to 'prove' the definition of .999...=1, rather than just effectively 'restating' it tacitly by using a UNITARY statement like 9/9, or 8/8, -----1/1 (not to mention what is 0/0 = ??? in that same line of UNITARY (or like/like) 'construction'?).
This was not intended to offend anyone. Only putting again my longstanding 'take' so far as it applies to such kinds of responses/assumptions. This was just a general timely observation/reminder, Trippy, everyone, that as far as I can observe the discourse/arguments on these things, it may not be enough to dispel the confusions/dissension with responses involving such facile 'constructions' which may lead to inevitably circuitous, and sometimes very 'peculiar' (as in 0/0 ?) things?
You not liking it does not invalidate it.
See my response to Chinglu and try and follow it. It's not that complicated.
Hi Billy T.
Only two brief observations regarding your replies to rr6, as follows....
PLease read my earlier observations/reminder (see post #1533) about relying on like/like (ie, 1/1 etc) 'facile constructions', as well as about relying on unitary states (such as 1; and again, 9/9 etc like/like) circuitous-inevitable starting states whenever attempting to make 'proof' arguments. They do not really go anywhere except back where you started from, which is the facile 1/1 construct and the unitary (not fractional) starting state via the circuitous and self-selecting-logic flow which those 'starting points' inevitably lead to.
Agreed! They are two different names agreed upon by CONVENTION. No argument from me there.
However, when it comes down to analyzing the 'construction' of the 'string' involved, then that invokes all sorts of deeper issues about why that 'other name' should be used at all?
If the decimal system results in 'infinitely long STRING which can only ever be 'determined' to '= 1' via invocation of LIMITS arguments, then it is obviously NOT prima facie '+', but 'contrived' '=' so that we can use that 'other' name for '1'.
It's like using the name "observable universe' (finite) and 'UN-observable universe' (potentially extending infinitely) and then making a statement like "It must end somewhere, so we will use BOTH 'names' because we use a LIMITING ASSUMPTION argument to MAKE THEM EQUAL NAMES, even though they are technically two DIFFERENT things (ie, 'finite observable' and 'potentially infinitely un-observable)?
Anyhow, as long as you and rr6 understand that it is mere LABELING and NAMING convention 'equality' rather than some purely mathematical 'equivalency' of finite and infinite STRING representation/operation, then there is nothing more to be said about that 1= .999... naming/representation 'convention equivalency' only.
Just came in to post that 'lose end' observation/reminder. Haven't much time to linger longer. Will log out again very soon, and resume 'read-only' mode again for a while. Cheers till then, and do all enjoy your various discussions, Billy T, rr6, everyone!
Only two brief observations regarding your replies to rr6, as follows....
PLease read my earlier observations/reminder (see post #1533) about relying on like/like (ie, 1/1 etc) 'facile constructions', as well as about relying on unitary states (such as 1; and again, 9/9 etc like/like) circuitous-inevitable starting states whenever attempting to make 'proof' arguments. They do not really go anywhere except back where you started from, which is the facile 1/1 construct and the unitary (not fractional) starting state via the circuitous and self-selecting-logic flow which those 'starting points' inevitably lead to.
Agreed! They are two different names agreed upon by CONVENTION. No argument from me there.
However, when it comes down to analyzing the 'construction' of the 'string' involved, then that invokes all sorts of deeper issues about why that 'other name' should be used at all?
If the decimal system results in 'infinitely long STRING which can only ever be 'determined' to '= 1' via invocation of LIMITS arguments, then it is obviously NOT prima facie '+', but 'contrived' '=' so that we can use that 'other' name for '1'.
It's like using the name "observable universe' (finite) and 'UN-observable universe' (potentially extending infinitely) and then making a statement like "It must end somewhere, so we will use BOTH 'names' because we use a LIMITING ASSUMPTION argument to MAKE THEM EQUAL NAMES, even though they are technically two DIFFERENT things (ie, 'finite observable' and 'potentially infinitely un-observable)?
Anyhow, as long as you and rr6 understand that it is mere LABELING and NAMING convention 'equality' rather than some purely mathematical 'equivalency' of finite and infinite STRING representation/operation, then there is nothing more to be said about that 1= .999... naming/representation 'convention equivalency' only.
Just came in to post that 'lose end' observation/reminder. Haven't much time to linger longer. Will log out again very soon, and resume 'read-only' mode again for a while. Cheers till then, and do all enjoy your various discussions, Billy T, rr6, everyone!
There's nothing facile about it.
The fact that it goes back to the start point is, to some extent the point - they wouldn't do that if 0.99(9) had any value other than 1.
Do you get it yet? They demonstrate that 0.99(9) behaves as if it was the multiplicative identity, thus proving that it must have the same value.
If 0.999(9) had a value other than that of the multiplicative identity, then what I posted would lead to a contradiction. It doesn't, therefore the statement is true.
I explained why. It is not about my 'not liking it'. It is about the possible circuity and logical self-referencing which makes the 'proofs' mere 'facile construction-deconstruction' exercises which does not lead to any real 'answers' to the issues raised.
And regarding your following 'proof' in response to chinglu....
...I note that you merely trivially RE-FORMATTED the DIRECTLY EVOLVED 'long-division result' STRING, ie,...
from:
$$ \frac{1}{9}= 0.111\overline{1}$$
to:
$$ \frac{1}{9}=1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...$$
...and then just as trivially used that facile RE-arranged format ONLY in further facile 'formatting based' treatments, so as to RE-INTRODUCE a '9' UNIT TIMES factor which merely effectively re-inserts the 9/9 unitary factor in order to further re-arrange the same re-formatted result into a UNITARY trivial reformatting of the argument based on the like/like construction you want to introduce circuitously.
These are the very sort of trivial/facile 'proof' arguments 'formatting/circuitous' constructions/treatments which my observations/reminders caution about. While they are 'correct' as 'formatting treatments', they are not really any answer/proofs to the issues raised which go beyond the 'naming/labeling/formatting conventions aspects.
Please see my post to Billy T above, thanks.
Thanks again for your polite and interesting discussions, Trippy, Billy T, everyone!
BT Thinks 2 + 2 = 5---Ok Please Explain Dude
"two differrent names, not values"..duhh, sorry BT, you are the one having problem understanding your own english words, not me.
1 is a value BT and you are incorrect on this and that finite value 1.0 = infinite 0.999...and this is just more logical common sense you do are in denial of because of your ego not able to ackowledege your error, now in two places.
Again, your the one who has trouble understanding english, infinite, or infinite value or infinite number value is infinite irrespective of what base is being used. This is common sense which you yet to show much of.
Ditto the above 1 is a is value. Common sense dude.
finite 1.0 does not equal infinite 0.999...and never will. Common sense.
Yours and others( "mapping" ) repeated, mathematically illusionary, mental masturbation will never have finite 1.0 = infinite 0.999... and this rational, logical common sense, that you others are in denial of.
As I stated previously, we begin with a finite and your case it is finite number line, and that can be micro-infinitely subdivide ergo multiplication of the decimal places via divsion of the number line. Fuller was big keen our finite Universe doing a micro-infinite-- ergo eternal ---multiplication-by-division processs/procedure.
I don't buy into that i.e. I think there are quantum gravitonic limits--- that may vary( see Loop Quanty Gravity in theory ---ans specifically limits related to macro-finite Universe of occupied space.
Perhaps, you all concerned that I "don't want to learn math" yet I repeat, get out of your denial and just look how much controversy there is amongest those in this thread regarding, who is doing the correct math, who is not, ergo who is your alledge "idiots" and who is not.
I know enough math to know your and others in the camp of finite 1.0 = infinite 0.999... are incorrect, and have yet to offer us a rational, logical, common sense and relatively simple explanatory guide in those regards.
PLease, when you do have i,t spell it out for me, and I will disscuss it one line at a time.
Origin posted 1/3 and 1.9 * this or that and gave no explanation, then he revised with some latex condoms and no explanation.
Whose mathematical proof is correct, yours, his, RPenner, Undefined, Arfa-brane etc......yeah, you certainly don't elaborate on their givens with rational, logical common sense and relatively simple explanatory guide. Why? Cause do not agree with them, they do not agree with your etc......
That closed tautology is probably the ONLY realm of knowledge where there is 100% agreement by the informed. Truth there is not an opinion, but proven.
Why your opinions on math are worthless.
Yeah, that was Feynmans speciality to arrive at same resultant with some many differrent mathematical procedure. Fine read and address my latter comment above, use others given proofs and meet my challeng with given criteria. You have not and will not, just as they have not and will not.
I will not be holding my breath on that one either.
Great!, Let see you post it here and I will be the judge of how simple it is.
My guess is that none of them will prove that finite 1.0 = infinite 0.999... and will go check after this reply to you.
Huh? Dude you the one who thinks a finite value is equal to and infinite value. Your ego blocks you to truth.
Huh? BT what are you talking about now. Stating falsehoods in no way brings you any closer to truth.
I can explain to you how synergy works i.e. 1 + 1 = 4 and 3 + 3 = 12 but your given "2 + 2 = 5" is something you do not explain and I rather doubt you ever will, unless your talking about one woman and one man can make 3 children so not it is family of 5.
Maybe that is what your going on about. Who knows what your talking about in those regards? my guess is nobody.
r6
"two differrent names, not values"..duhh, sorry BT, you are the one having problem understanding your own english words, not me.
1 is a value BT and you are incorrect on this and that finite value 1.0 = infinite 0.999...and this is just more logical common sense you do are in denial of because of your ego not able to ackowledege your error, now in two places.
Again, your the one who has trouble understanding english, infinite, or infinite value or infinite number value is infinite irrespective of what base is being used. This is common sense which you yet to show much of.
Ditto the above 1 is a is value. Common sense dude.
finite 1.0 does not equal infinite 0.999...and never will. Common sense.
Yours and others( "mapping" ) repeated, mathematically illusionary, mental masturbation will never have finite 1.0 = infinite 0.999... and this rational, logical common sense, that you others are in denial of.
As I stated previously, we begin with a finite and your case it is finite number line, and that can be micro-infinitely subdivide ergo multiplication of the decimal places via divsion of the number line. Fuller was big keen our finite Universe doing a micro-infinite-- ergo eternal ---multiplication-by-division processs/procedure.
I don't buy into that i.e. I think there are quantum gravitonic limits--- that may vary( see Loop Quanty Gravity in theory ---ans specifically limits related to macro-finite Universe of occupied space.
Perhaps, you all concerned that I "don't want to learn math" yet I repeat, get out of your denial and just look how much controversy there is amongest those in this thread regarding, who is doing the correct math, who is not, ergo who is your alledge "idiots" and who is not.
I know enough math to know your and others in the camp of finite 1.0 = infinite 0.999... are incorrect, and have yet to offer us a rational, logical, common sense and relatively simple explanatory guide in those regards.
PLease, when you do have i,t spell it out for me, and I will disscuss it one line at a time.
Origin posted 1/3 and 1.9 * this or that and gave no explanation, then he revised with some latex condoms and no explanation.
Whose mathematical proof is correct, yours, his, RPenner, Undefined, Arfa-brane etc......yeah, you certainly don't elaborate on their givens with rational, logical common sense and relatively simple explanatory guide. Why? Cause do not agree with them, they do not agree with your etc......
That closed tautology is probably the ONLY realm of knowledge where there is 100% agreement by the informed. Truth there is not an opinion, but proven.
Why your opinions on math are worthless.
Yeah, that was Feynmans speciality to arrive at same resultant with some many differrent mathematical procedure. Fine read and address my latter comment above, use others given proofs and meet my challeng with given criteria. You have not and will not, just as they have not and will not.
I will not be holding my breath on that one either.
Great!, Let see you post it here and I will be the judge of how simple it is.
My guess is that none of them will prove that finite 1.0 = infinite 0.999... and will go check after this reply to you.
Huh? Dude you the one who thinks a finite value is equal to and infinite value. Your ego blocks you to truth.
Huh? BT what are you talking about now. Stating falsehoods in no way brings you any closer to truth.
I can explain to you how synergy works i.e. 1 + 1 = 4 and 3 + 3 = 12 but your given "2 + 2 = 5" is something you do not explain and I rather doubt you ever will, unless your talking about one woman and one man can make 3 children so not it is family of 5.
Maybe that is what your going on about. Who knows what your talking about in those regards? my guess is nobody.
r6
me said:
Undefined said:
Well if numbers are "more fundamental" than any representation of them, then any representation has 'structure'.
In base 10, each digit is multiplied by some power of 10, and the powers decrease by 1 from left to right. In base 2, it's powers of 2 decreasing by 1 from left to right. In base b it's powers of b doing likewise.
Changing a number from one base to another preserves this structure, which is an ordering we impose on digits.
This is true for any number system that 'represents' numbers, and so therefore any (pairwise) operation, such as addition, multiplication, etc, preserves the same left-to-right structure.
But then, the idea that numbers are more fundamental than their representation leaves the question: "what is a number that isn't represented?"
So do numbers exist if we don't write them or get some machine to? Does it matter?
Have you heard of the Redundancy Ministry of Redundancy?
Not so. If one assumes, as Billy T has just confirmed, that the .999... is just a formatting/naming 'equivalency', then there is no route to 'proving' that equivalency is also a mathematical one.
Which makes all such purported 'proofs' based on .999... somehow involving an equivalency to '1' in mathematical manipulations moot.
Which makes all such 'proofs' which use unitary 'constructions' like that even less meaningful/valid, circuitous and irrelevant/uninformative as to the other issues raised.
And the fact that (as Billy T confirms) it is a NAMING convention which ASSUMES the result is 'equivalent', then of course all such trivial arguments/proofs 'recover' the UNITARY ASSUMPTION it was all based on from the start of that circuitous 'treatment' in trivial ways.
You seem to have this idea that the representation of a number is "just formatting". Is there some other way to show two numbers are equivalent, and does it not require that either number is represented or mentioned?Undefined said:
So ultimately, all you can say is: "two numbers are equivalent, but we can't tell you which ones because that's just formatting" (??)
Yes, of course. That is the whole point of designing number systems which can (hopefully) preserve all the essentials (information/extrapolation etc) one needs to 'read off' from such a formatting/representation whatever type it may be and whatever it may be more suited for in practice than another representation. No argument from me there!
It remains that the fundamental mathematical operations they REPRESENT in their own specific formatting way is always the common starting point for all expressions of that fundamental operation and result. The operation/result comes first, fundamentally, and we then can use all sorts of formatting and representational or even GRAPHING 'shortcuts' and 'algorithms to make the outputs more quickly and read them off more conveniently etc etc.
In this context/aspect per se, that was the point of my earlier observations regarding the difference between the fundamental 'mathematical operation' and the convenient 'moving the format 'decimal point' expression re-arrangement of the 'string' in question etc.
Cheers!
Undefined Narrows the Playing Field---Almmos There---Line-of-Demarcation<<) | (<<
I think what your trying to say it, that for practical purposes, finite 1.0 = infinite 0.999... can be contrived, to be considered to be, equal, even tho, they are not.
Again you appear to using the word "convention" to reference 'for all practical purposes' and that is really not differrent that rounding higher or lower, of and decimal for practical purposes of the circumstances--- needed or not precission ---involve.
Technically-- absolute truth ---is that I and any others who believe as I do, are in the correct camp. Common sense evaluation that does not require years of mathematics. Years of mathematics tends to allow for some to arrogantly play their matematically illusionary, mental masturbations and act arrogantly as the non-idiot, more superior individual, intellectually and moral above the "retards"/"idots"/"ingorant" etc....what shameful behaviour
finite 1.0 = infinite 0.999....NOT and that is common sense. imho
R6
I think what your trying to say it, that for practical purposes, finite 1.0 = infinite 0.999... can be contrived, to be considered to be, equal, even tho, they are not.
Again you appear to using the word "convention" to reference 'for all practical purposes' and that is really not differrent that rounding higher or lower, of and decimal for practical purposes of the circumstances--- needed or not precission ---involve.
Technically-- absolute truth ---is that I and any others who believe as I do, are in the correct camp. Common sense evaluation that does not require years of mathematics. Years of mathematics tends to allow for some to arrogantly play their matematically illusionary, mental masturbations and act arrogantly as the non-idiot, more superior individual, intellectually and moral above the "retards"/"idots"/"ingorant" etc....what shameful behaviour
finite 1.0 = infinite 0.999....NOT and that is common sense. imho
R6
Math is not done by "common sense" nor by popular vote, nor by opinion, BUT BY PROOFS.
Several different proofs that 1 =0.999... have been presented. Some are even simpler than mine, but not as rigorous as require multiplying an infinitely long decimal expression to conclude that 1 and 0.999... are identical values, both rational and finite as they are the same , identical, number, only two different names for that same number. My proof, without need of multiplying an infinitely long decimal string, nor any limiting procedure that some valid proofs do use, and every step is logically derived from stated definitions, so it is quite rigorous.
In contrast you give ONLY your opinion and no supporting proof; further more you can not find any error in the proof I gave and I even number the steps for you to tell what step you did not think was valid.
SUMMARY: YOU HAVE ZERO UNDERSTANTING OF MATH (and very poor comprehension of English)
I will admit you now show a slight improvement in you written text: I. e. no longer write "infinite value 0.9999..." but still think that in same old false claim (only you ignorantly assert)* "that finite 1 can not equal infinite 0.999..." That is of course false as 0.999... is not infinite in value, only has an infinitely long decimal expression that has the value of very finite 1. (Much like 1/3 =0.3333.... has an infinitely long decimal expression for finite, rational fraction 1/3.)
* Still persisting in the ignorant claim that the value must be infinite if the decimal expression of it is infinitely long.
You don't even realize / understand that 1/4 = 0.25 is also infinitely long decimal expression given more correctly (no assumption about less significant decimal place / locations being zero needed as that is explicitly so stated.) as 1/4 = 0.25000000000000000000 ....
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Not at all, arfa. A mathematical quantity 'equivalency' is an 'equivalency' irrespective of what we choose to represent different 'instances' of the values involved.
The issues that seem to plague this 1= .999... situation is the FORMATTING system which the decimal system introduces when trying to represent a never ending decimal FRACTIONAL 'result' as an infinite string. That's it, really.
Billy T has put his finger right on the causes of these cross-purpose discussions. The labeling/naming conventions get in the way of the real issues of mathematical equivalency per se IRRESPECTIVE of the way we can/do 'represent' certain fractional results.
That's all, arfa. I make observations on these discussions from no more than that is already been agreed about the number system/formatting representations. The discussions as to actual mathematical equivalency, and how one 'gets there from here' mathematically without introducing the LIMITS argument is still not quite settled as far as I have observed. Good luck getting substantive non-trivial agreement/proofs on that aspect!
There's nothing facile or circuitous about the proof. It demonstrates that 0.99(9) has the same properties as the multiplicative identity.
And regarding your following 'proof' in response to chinglu....
I trivially reformated nothing, I wrote out explicitly what 0.111(1) recurring is and extended it to the generic case to prove that the statements 9*0.111(1) = 0.999(9) is true.
Let me lead you through it by the hand, step by step, seeing as how you're obviously not comprehending it on your own....and then just as trivially used that facile RE-arranged format ONLY in further facile 'formatting based' treatments, so as to RE-INTRODUCE a '9' UNIT TIMES factor which merely effectively re-inserts the 9/9 unitary factor in order to further re-arrange the same re-formatted result into a UNITARY trivial reformatting of the argument based on the like/like construction you want to introduce circuitously.
Starting point (after proving it by stepping through the long division - I stated long ago that this should be done):
$$ \frac{1}{9}= 0.111\overline{1}$$
Expand the decimal representation of 1/9 to show the decimal powers (I mentioned long ago that people needed to keep in mind what the numbers actually represent):
$$ \frac{1}{9}=1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...$$
Multiply both sides of the equation by 9
$$ 9 \times \frac{1}{9}= 9 \times (1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...)$$
Expand the brackets
$$ 9 \times \frac{1}{9}= 9 \times 10^{-1} + 9 \times 10^{-2} + 9 \times 10^{-3} + 9 \times 10^{-4} + ... + 9 \times 10^{-(n-1)} + 9 \times 10^{-n} + 9 \times 10^{-(n+1)}+ ...$$
Simplify.
$$ \frac{9}{9}= 0.999\overline{9}$$
There is nothing trivial, facile, or circuitous about this, nor does it rely on simply reformatting numbers.
So far, the only facile thing in this conversations has been your responses.
Look, do you think that 1/9 can give a result other than 0.111(1)?
Do you think that 1x9 can ever equal anything other than 9?
Hi Trippy. I am running out of time again, so I will be as brief as I must.
It's not your treatment's USAGE benefits I am pointing to; it is the logical NEED for it above and beyond the fundamental operations involved which give the initial long-division result/string.
See? I am only pointing out that all your 'overlay' for 'generic' case is merely re-formatting to more generic cases which lose sight of the fundamentals involved. Yes, they are convenient usages, and I already stated that your use of the 'powers' representation/treatment merely replicates the 'decimal location' involved in each part of the string. No biggie. No great 'revelation' or 'insight' gained from that 'overlay' of new formatting/manipulation. It does not actually do anything different except introduce more assumptive/formatting overlays which hide the simple operations which they are even MORE ABSTRACTLY referring to.
See? It always gets back to whether or not the UNITARY 'starting point' is used as part of the 'proof/treatment'....or if the convenient treatment you use gets us back to such a unity when YOU introduce the '9' to effectively RECONSTRUCT the facile/trivial unitary 'like/like 9/9 factor which, no surprise, in your treatment as designed WILL lead back to the 'result' YOU designed the abstraction/formatting system/treatment for. Circuitous. Since it does not 'prove' anything outside its own pre-determined result designed into your trivial treatments/overlays.
Look, Trippy, I have to log out soon. If we cannot get beyond the obvious misunderstandings between us in this aspect, then I humbly suggest we leave it between us as "agree to disagree"....for th reasons we each have respectively posted. Ok?
PS: Really, I have to log out very soon.
That is why I did not use any limiting procedure (or any multiplying of infinitely long decimal strings)* in my proof given and discussed in its step by step logical development from well defined stating point, the line segment with ends at 0 and 1. (length scale = unity)... The discussions as to actual mathematical equivalency, and how one 'gets there from here' mathematically without introducing the LIMITS argument is still not quite settled as far as I have observed. Good luck getting substantive non-trivial agreement/p proofs on that aspect!
See it all here: http://www.sciforums.com/showthread...ophy-of-Math&p=3140402&viewfull=1#post3140402
* This is a little bit difficult to prove valid as the common multiply algorithmic starts to operate on the "right most end" of the number being multiplied and "carries some value to the next to end place" if the first final product is more than the base, as it usually is for small bases like binary system..
For example consider multiplying 0.659 by 3. We first note 3x9 = 27 and write down in the third decimal place a 7. Then we do 3x5 =15, add the "carried 2" and write down another 7 to have ?.?77 and 1 to "carry." Then we do 3x6 =18, add the carried 1 to get 1.977 but what do we do when there is no "right most end" - This multiply algorithm fails. We need to develop a new one AND PROVE IT IS VALID.
Because I could not give any GENERALLY valid algorithm for multiplying an infinitely long (endless) decimal string MUCH LESS PROVE IT VALID, I avoided ALL MULTIPLAICATIONS.
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Yes, I noted that already. Thanks. It then comes again back to the use of unity 1/1, 9/9 etc like/like facile constructions as part of the pre-determined logic flow which, no suprise, lead back to the unity 1/1, 9/9 etc starting assumption because the 'proof' is circuitous, and no new insight/revelation 'from outside' that 1/1 staring/designed treatment is given so far. Thanks again, Billy T, for putting the whole double-issue (format usages etc being different from actual mathematics operations etc) that is causing all these cross-purpose exchanges.
PS: As foreshadowed above, am logging out now. 'Read you round', Billy T, everyone! Thanks again.
Cheers.