Billy T.."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.
You ignorance about math's basic concepts is appalling. But what is worse (as all are ignorant about some things) you have no desire to lean! Only to post your confused nonsense.
Reading simple English is difficult for many idiots. I'll divide my sentence into parts for you:Huh? One( 1 ) is not a value? I think your incorrect on that assessment also BT. ..
No, only idiots and the ill educated in math disagree on what is true in math.... it fairly obvious that few agree on what is the correct math for this, that or another issue.
Can I see your proof?
Please indicate why induction it not needed.
Or please indicate why it is needed.
Thanks
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?
...my very simple proof that 1/1 =1 =0.999... which does not need any multiplication as a result and is built "from the ground up" starting from a definition for the 0 to 1 length line segment, even then defining the meaning of 2, 3, ... 8 & 9 !...
...
" 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.
...
There's nothing facile about it.PLease read my observations/reminder (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.
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.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.
You not liking it does not invalidate it.
See my response to Chinglu and try and follow it. It's not that complicated.
What's to prove?
Please excuse the formatting of this first step if the columns don't quite line up.
$$
\hspace{37 pt}0.111\overline{1} \\
\frac{1}{9}= 9 \overline{\big) 1.0000...}\\
\hspace{37 pt}-9 \\
\hspace{42pt}\overline{\hspace{7 pt} 1}0 \\
\hspace{43 pt} -9 \\
\hspace{47 pt} \overline{\hspace{7}1}0 \\
\hspace{49 pt} -9 \\
\hspace{56 pt}\overline{\hspace{7}1}0 \\
\hspace{56 pt} -9 \\
\hspace{65 pt} \overline{\hspace{7}1...}
$$
$$ \frac{1}{9}= 0.111\overline{1}$$
$$ \frac{1}{9}=1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...$$
$$ 9 \times \frac{1}{9}= 9 \times (1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...)$$
$$ 9 \times \frac{1}{9}= 9 \times 10^{-1} + 9 \times 10^{-2} + 9 \times 10^{-3} + 9 \times 10^{-4} + ... + 9 \times 10^{-(n-1)} + 9 \times 10^{-n} + 9 \times 10^{-(n+1)}+ ...$$
$$ \frac{9}{9}= 0.999\overline{9}$$
Billy TBilly 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 areonly two different names, not values,
1 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.
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.
me said:The writing conventions, or rules, also preserve something
Undefined said:I'm not sure what you are getting at in your last sentence/question.
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.
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: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.
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.
Undefined'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.
Math is not done by "common sense" nor by popular vote, nor by opinion, BUT BY PROOFS.... finite 1.0 does not equal infinite 0.999...and never will. Common sense. ... r6
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?
So ultimately, all you can say is: "two numbers are equivalent, but we can't tell you which ones because that's just formatting" (??)
There's nothing facile or circuitous about the proof. It demonstrates that 0.99(9) has the same properties as the multiplicative identity.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.
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....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:
$$ \frac{1}{9}=1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...$$
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.
There is nothing trivial, facile, or circuitous about this, nor does it rely on simply reformatting numbers.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.
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.
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?
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!
That is why I did not 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)
See it all here: http://www.sciforums.com/showthread...ophy-of-Math&p=3140402&viewfull=1#post3140402