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Jeff Root
2010-Oct-26, 03:17 AM
"The Peano axioms give a formal theory of the natural numbers.
The axioms are:
There is a natural number 0.
Every natural number a has a natural number successor, denoted
by S(a). Intuitively, S(a) is a+1. ...
Are zero, one, and addition defined? I don't see a need to define
them, myself, but that might be because I can't imagine any way
to define them.

-- Jeff, in Minneapolis

astromark
2010-Oct-26, 03:39 AM
The point for me has been lost... I will modify my stubborn stance just a tad...

It must be said... that some of us are not so foolish as to be expected to be taken so seriously.

I am one of those that still 'see' some room for discussion regarding the true value of any number.

That when ~ is used it tends to loosen the values grip on precision of values.

That I am now changing my stance and, accepting the point. That when 0.9999~ is used they might just as well have said 1.

So now tell me you most learned folk... why did they not just say 1. Umm... and I am just joking...

Jeff Root
2010-Oct-26, 04:08 AM
Mark,

What is your understanding of the function or purpose of
the tide (~) as you have used it? I am wondering if you
think of it as meaning "approximately". That is what I have
always understood the tilde to be used for, though always
before a number, not after. Obviously, if we interpreted the
tilde after a number as meaning "approximately", then
0.9999~ would not necessarily equal 1. It *might* equal 1,
depending on just how approximate the value of 0.9999~
is, but it certainly wouldn't have to equal 1.

I have always understood the ellipsis (...) after a decimal
number to indicate that the decimal expansion continues
without end, so that is what I've used in this thread, and
wonder where the odd use of the tilde came from.

-- Jeff, in Minneapolis

HenrikOlsen
2010-Oct-26, 06:25 AM
I read the tilde to mean that the last digit is repeated endlessly, while the ellipsis means it goes on but without specifying how, at least that's the distinction I know, and it's the unending series of 9's I've been arguing equals 1 the entire time.

I think the normal way to indicate this symbolically is by a line over the repeating digits, which has the advantage of being able to express longer repeats such as in 1/7=0.142857... continuing with unending repeats of 142857, but that can't be done in html so the tilde is used by some instead for the single repeating digit case, exactly to distinguish it from the case where the following digits aren't specified.

agingjb
2010-Oct-26, 07:52 AM
Yes, one reason why I always use the word "recurring" is that either the ellipsis or the tilde can be interpreted as not necessarily an infinite sequence - but then I suppose the "recurring" could, at need, be so interpreted. The notation I would use is a dot over a 9, but sadly I cannot find a way to do this.

Ken G
2010-Oct-26, 08:08 AM
Are zero, one, and addition defined? I don't see a need to define
them, myself, but that might be because I can't imagine any way
to define them.Zero just gets the natural numbers started, there's no need for addition yet or the concept of the identity transformation under addition, and there's certainly no need for zero to carry a connotation of "nothing" (or indeed any connotation at all, that's the point). The Wiki just inserted the "intuitively" phrase to help guide understanding, it really doesn't belong there. To have "natural numbers", we don't need addition, we only need a concept of "successor" which then gives a syntactic type of ordering. That's the guts of the natural numbers-- a staircase, with a beginning but no end.

Ken G
2010-Oct-26, 08:13 AM
I am one of those that still 'see' some room for discussion regarding the true value of any number. And this is just a classic example of what I'm talking about-- you are thinking about the "true value" of a number, but this is not what formal mathematics does. In formal mathematics, numbers don't have "true values", they just have definitions and syntactic relations that lead to proofs, there's no "truth" there at all, not in the colloquial sense of the term. All of that "truth" business gets added when we cross over the mysterious bridge from the formal mathematics to real-world notions, and real-world usages of the outputs of formal mathematics. Indeed, as I pointed out for 2+2=4, often the real-world application precedes the pure-math derivations by quite a large time.

Ivan Viehoff
2010-Oct-26, 08:58 AM
But is 2 something that is capable of interpretation in the real world? Ergo, you have just stated that they are different, but can be made the same in pure mathematics. Note that is just precisely what I said above.
It is also precisely what I said in an earlier post. I went on to say that interpretations in which 1.999... was not considered to be a real number were probably not very useful or interesting, and it seemed unlikely that the questioner had such an interpretation in mind. There are several different interpretations of the number 2, also, 2 as an element of different mathematical structures, precisely 2, 2.0 to an accuracy of one decimal place, but these are not the distinctions that are interest to this thread.

As usual, Ken, we agree on the significant points, and it is only at the angels on pinheads level we may differ.

astromark
2010-Oct-26, 09:52 AM
Mark,

What is your understanding of the function or purpose of
the tide (~) as you have used it? I am wondering if you
think of it as meaning "approximately". That is what I have
always understood the tilde to be used for, though always
before a number, not after. Obviously, if we interpreted the
tilde after a number as meaning "approximately", then
0.9999~ would not necessarily equal 1. It *might* equal 1,
depending on just how approximate the value of 0.9999~
is, but it certainly wouldn't have to equal 1.

I have always understood the ellipsis (...) after a decimal
number to indicate that the decimal expansion continues
without end, so that is what I've used in this thread, and
wonder where the odd use of the tilde came from.

-- Jeff, in Minneapolis

OK ... now I feel embarrassed as this might be why I am so sure that my thinking of parts of whole numbers,

expressed as fractions less than the whole can not ever be said to be the whole number.... 0.9999~ is not 1.

BUT... that little ~ which I have been using as a method of expressing a continuing sequence. 0.99999999999

Is expressed as 0.9999~. I can not find a icon for infinitum... this seems to be above my ability to alter...

The little circle on its side with a / through it...I can not have one...( yet. )

But I am equally receptive of the maths., that tells me I am wrong... wrong is not usually a problem for me...

That 0.9999~(infanitum) is equal to 1. Fine. :clap:

pzkpfw
2010-Oct-26, 09:59 AM
My understanding, is that in the context of this thread, the ~ and ... are being used by various posters to mean the same thing.

i.e. that 0.999~ = 0.999... ( = 1 )

It may not be correct, as such, but that's how they seemed to be being used.

Has anyone used them in a way that made them distinct?

Ken G
2010-Oct-26, 01:52 PM
It is also precisely what I said in an earlier post. I went on to say that interpretations in which 1.999... was not considered to be a real number were probably not very useful or interesting, and it seemed unlikely that the questioner had such an interpretation in mind.That would speak to the issue of answering the OP. Note that what I was saying was explanatory in regard to the disputes that were going on in the 80+ pages of the thread. In summary, the problem is not that posters on here are not smart enough to do math right, the problem is that "doing math" actually means something different to many people than what it is supposed to mean. That is, many people think "doing math" is "arguing quantitative truths in the real world." That is not correct, although it is often how the concept is taught, ergo the whole problem.

For example, the usual way that reflexivity, symmetry, and transitivity are taught is that these are properties that equality (which, they imagine, means something else along the lines of "is the same thing as") happens to demonstrate. But what is actually correct is that those properties define equality, that equality in pure math doesn't mean anything else, such that any sense of "is the same thing as" stems solely from those syntactic structural properties, and that this structure is just a building block that goes into building up other more complex syntactic structures like real numbers.

There are several different interpretations of the number 2, also, 2 as an element of different mathematical structures, precisely 2, 2.0 to an accuracy of one decimal place, but these are not the distinctions that are interest to this thread.They are of interest if some of the posters are following those false leads. In that case, instead of telling them their argument is wrong, it is more important to tell them why these false leads exist-- because of a lack of understanding of the philosophy of mathematics.

As usual, Ken, we agree on the significant points, and it is only at the angels on pinheads level we may differ.Yes, I don't think there is any disagreement here, but the way you framed your post made it sound like you were taking issue. Perhaps I just read that in.

a1call
2010-Oct-26, 03:06 PM
Yes, that is correct.

I hereby apologize to Grapes for accusing him of making up rules as he goes along:

I also apologize to agingjb for accusing him of stating an oxymoron when he pointed out my error:

Somehow I had forgotten that integers can not have infinite digits and in the back of my head I had resolved that as just another conflict of definitions involving infinity.

Without integers with infinite digits, I am not able to deduce that since the f(m) (finite series) can not be an integer, then neither could f(n) (infinite series).

However, I am not ready to relinquish the "!=" camp quite just yet. I will post an objection later when I get a chance in a separate post to distinguish it from my stands to date from which I hereby formally resign.

I expect that to be my last argument regarding this thread.

Ken G
2010-Oct-26, 04:24 PM
Thank you for your graceful resignation, and when you post your new objection, be sure to make sure you can frame it in the language of formal abstract and syntactic mathematics, not in the language of what seems true in real-world modes of thinking. I will grant you right now there are many ways to argue the != camp from the latter perspective, indeed I've even offered a few. If the OP is framed as a question about formal mathematics and the structure of "real numbers", the != camp is just an error, but if opened up to more intuitive levels of meaning around the = symbol, that's a different issue.

Ivan Viehoff
2010-Oct-26, 04:57 PM
Yes, I don't think there is any disagreement here, but the way you framed your post made it sound like you were taking issue. Perhaps I just read that in.
I was disagreeing with my misinterpretation of what you said. Not the first time I've done that!

grapes
2010-Oct-26, 06:18 PM
I hereby apologize to Grapes for accusing him of making up rules as he goes along:
http://www.bautforum.com/showthread.php/14593-Do-you-think-0.9999999-1-that-is-infinite-9s.?p=1805751#post1805751Gratefully accepted. With the caveat that sometimes I have to.

Ken G
2010-Oct-26, 06:59 PM
I was disagreeing with my misinterpretation of what you said. Not the first time I've done that!
We've all been there! Sometimes I think I misinterpret things subconsciously just so I can disagree with them.

DonM435
2010-Oct-26, 08:29 PM
You're both right.

Ken G
2010-Oct-26, 10:15 PM
Are you sure you are not misinterpreting what we are saying so you can agree with us?

a1call
2010-Oct-26, 11:12 PM
The once mighty "!=" camp is a shell of its former self. These days it is manned by a fish and a banana.

Using the plural form fish could imply many individual fish(es) of the same species while fishes could imply many individual fish(es) of differing species. At present, there is no name for words that have identical singular and plural forms. (http://en.wikipedia.org/wiki/English_plural)

Does that mean fish(es) are fruits?
Probably, but that's an argument for another thread. For now it's time to attend to the suspense that has been building up regarding my latest objection to the "=" camp. Well the wait is finally over. here it goes:

There must be a definition for decimals fractions somewhere (http://en.wikipedia.org/wiki/Decimals#Decimal_fractions). But we all know what it is even if we can't quite put it into words. Whatever it is,

*- What makes the expression 0.111... valid when the expression 111.../1000... is not? How is the decimal fractions defined to avoid invalidation when it contains the concept of infinity? I am not debating the convergence of any sums here, just the semantics of the expression.

Ken G
2010-Oct-27, 12:25 AM
*- What makes the expression 0.111... valid when the expression 111.../1000... is not?It is because 0.1111... means something more than just "tack on another digit indefinitely", it is an infinite sum, and a convergent one at that. 111.... is not an infinite sum. I don't think one can avoid the convergence of the sums, though, because that is how real numbers are defined. So 0.111... exists within a set of numbers called the reals, and 111... does not exist in that set, nor in the set of integers. You are welcome to define a set in which 111.... is defined as a member, and track the ramifications of such a set, that's just the kind of thing mathematicians do-- but it's not the reals or the integers. For example, all integers are required to have successors-- what is the successor to 111... ?

grapes
2010-Oct-27, 12:31 AM
*- What makes the expression 0.111... valid when the expression 111.../1000... is not? How is the decimal fractions defined to avoid invalidation when it contains the concept of infinity? I am not debating the convergence of any sums here, just the semantics of the expression.Yahbut the decimal fraction is heavily involved with the convergence of sums--and it's as simple as that.

The other expression 111.../1000... appears to be indicating two quantities that are infinite, but infinity is not a number. Besides, it leads to ambiguity--is there anything there that indicates that the quantity is 0.111... rather than 1.111...?

a1call
2010-Oct-27, 12:56 AM
I agree that the invalidation-through-theory-of-numbers not withstanding the 0.111... indicates there are 1 less digits in the numerator than denominator,while the 111.../1000... does not. But, that is not the invalidating point for the expression 111.../1000....

To clarify the objection/question,

*- Let definition/axiom A be the definition of decimal fractional representations of integers in the theory of numbers.

*- Then it would follow that 0.111 = 111/1000

*- Then it would also follow 0.111... != 111.../1000... (and not just because of the missing info regarding number of digits in the numerator and denominator, I would assume)
**- Is the mentioned invalidation a matter of pure deduction (use of infinity as a number) or does the formal definition of decimal-fractional-representations have sort of an amendment that specifically prohibits direct conversion of the repeating decimals.
***- The resolution might be as simple as referencing a formal definition of what decimal fractions represent (Which I haven't been able to find).

Ken G
2010-Oct-27, 01:31 AM
**- Is the mentioned invalidation a matter of pure deduction (use of infinity as a number) or does the formal definition of decimal-fractional-representations have sort of an amendment that specifically prohibits direct conversion of the repeating decimals.I would say the problem is not that 0.1111... doesn't equal 111.../100..., it's that the latter expression has no meaning-- it does not represent a quotient of members of some meaningful set. It looks like you are inventing expressions that you think might/should have meaning, but you have not given them definitions. The definition of a number must come before the number can be used. What is the definition of 111.../100... as a number or quotient of numbers?

***- The resolution might be as simple as referencing a formal definition of what decimal fractions represent (Which I haven't been able to find).They represent an infinite convergent sum, the one you expect.

2010-Oct-27, 03:56 AM
I have to admit that I am shocked by the number of people here who do not understand that .99999~ is equal to 1. There are so many ways to explain this simple concept. Another way is that 1 divided by 3 gives .33333~ and that .99999~ divided by 3 also gives .33333~. The reason why they both give the same answer when divided by the same non-zero number is because they are the same number.

I am also equally shocked that this thread is not in the ATM section of the forum.

HenrikOlsen
2010-Oct-27, 04:27 AM
Does that mean fish(es) are fruits?
They are most closely related to bananas which also tend occur in large groups, in bananas called clusters, in fish schools.

Sorry about the derail, it was too tempting.

a1call
2010-Oct-27, 04:35 AM
Here is a work in progress draft of how I would define decimal fractions.

Decimal fractions are a subset of all fractions in base 10 where the denominator is a power of 10. These fractions are represented by writing the numerator and encoding it with a (decimal) point. The decimal point is placed
x = log(denominator) digits to the left of the right most digit of the numerator. If x > number-of-digits-in-the-denominator then the missing digits to the left of the numerator are represented by 0s.

Are there any errors?
Corrections are welcome and appreciated.
ETA-I:

Historically, any number that did not represent a whole was called a "fraction". The numbers that we now call "decimals" were originally called "decimal fractions"; the numbers we now call "fractions" were called "vulgar fractions", the word "vulgar" meaning "commonplace". (http://en.m.wikipedia.org/wiki/Fraction_(mathematics)?wasRedirected=true)
ETA-I-a: Undoubtedly that is one of the confusing factors

ETA-II:

a1call
2010-Oct-27, 04:39 AM
They are most closely related to bananas which also tend occur in large groups, in bananas called clusters, in fish schools.

Sorry about the derail, it was too tempting.
If anyone should apologize it's me. But I hope I have done enough of that already. I think a bit of derailing is a positive in OTB. There are strict sections for those who do not appreciate any derailing what so ever.

astromark
2010-Oct-27, 05:09 AM
I have to admit that I am shocked by the number of people here who do not understand that .99999~ is equal to 1. There are so many ways to explain this simple concept. Another way is that 1 divided by 3 gives .33333~ and that .99999~ divided by 3 also gives .33333~. The reason why they both give the same answer when divided by the same non-zero number is because they are the same number.

I am also equally shocked that this thread is not in the ATM section of the forum.

No, it can not be ATM because we are seeking the mainstream answer. There is no argument.

My cheap as chips calculator proves what I have been told is correct... Enter 0.999999 and divide it by 3.

and yes sure enough you get 0.3333333 and then do it with number 1. by 3... and yes you get 0.3333333

So that is it. You me or anyone else dare not argue with my cheap as chips calculator.... But, I still want to.... :)

grapes
2010-Oct-27, 01:30 PM
No, it can not be ATM because we are seeking the mainstream answer. There is no argument.

My cheap as chips calculator proves what I have been told is correct... Enter 0.999999 and divide it by 3.

and yes sure enough you get 0.3333333 and then do it with number 1. by 3... and yes you get 0.3333333

So that is it. You me or anyone else dare not argue with my cheap as chips calculator.... But, I still want to.... :)Try this. Calculators keep "guard digits" that are not visible to us, normally.

So, divide 0.999999 by 3 and THEN multiply the answer by 3.

Now, divide 1. by 3 and THEN multiply the answer by 3.

Are they the same? Not on MS Calculator

Boratssister
2010-Oct-27, 05:57 PM
Should I start reading this thread from the start before I commit myself to voting on this public vote?

85 pages and counting, wow!

2010-Oct-27, 06:44 PM
No, it can not be ATM because we are seeking the mainstream answer. There is no argument.

Actually, the mainstream answer (.99999~=1) is already well established and there is no need to seek it. It has already been proven in many ways and countless times in math classrooms all over the world, and is accepted by all who understand it. This certainly qualifies as mainstream.

For this reason, this thread is no less deserving of being in the ATM section than threads that question the validity of Relativity or Quantum Mechanics.

On another note, I posed a question to you in post #2183 on page 73 and explained why it is relevant to this thread. I noticed that you did not answer it so here it is again:

Do you believe that .0000000~ is equal to 0 or do you not?

Kaptain K
2010-Oct-27, 06:51 PM
Should I start reading this thread from the start before I commit myself to voting on this public vote?

85 pages and counting, wow!

I voted before reading any answers (way back near the begininng, when there were only a few). 85 pages haven't changed my view!

grapes
2010-Oct-27, 06:54 PM
Actually, the mainstream answer (.99999~=1) is already well established and there is no need to seek it. It has already been proven in many ways and countless times in math classrooms all over the world, and is accepted by all who understand it. This certainly qualifies as mainstream.

For this reason, this thread is no less deserving of being in the ATM section than threads that question the validity of Relativity or Quantum Mechanics.For that reason, maybe. But as Mark points out, we are all seeking the mainstream answer, in mathematics. Sometimes our assumptions are different, and that changes our answers, but that is also a mainstream approach to mathematics. All sorts of mathematicians introduce negations of commonly-held axioms to produce new and maybe interesting mathematics. We just have to determine the (sometimes unspoken, or unappreciated) assumptions, and the problem is resolved, in the mainstream of mathematics.

For that reason, I do not consider this ATM. Just an opportunity to flex our mathematical skills.

Ken G
2010-Oct-27, 09:26 PM
For this reason, this thread is no less deserving of being in the ATM section than threads that question the validity of Relativity or Quantum Mechanics.
That's not actually true, and that's why I have tried so hard to point out the real mainstream place where this debate can exist-- which is if the OP is not interpreted as a question that is intended to be reserved for formal mathematics, but is allowed to generalize to a real-world understanding of the concept of =. So if the thread is interpreted as an exploration of when 1.999... = 2, versus when 1.999... is something different from 2, then the entire thread is completely mainstream. What made it ATM is that so many people thought they were giving formal mathematical arguments that 1.999... != 2, when in fact they were using "horse sense," or alternatively, as grapes put it, framing a different kind of "formal" mathematics with new axioms that aren't really formalized and probably won't work, but worth exploring.
Do you believe that .0000000~ is equal to 0 or do you not?Yes, this is very much the same issue.

astromark
2010-Oct-27, 10:00 PM
Try this. Calculators keep "guard digits" that are not visible to us, normally.

So, divide 0.999999 by 3 and THEN multiply the answer by 3.

Now, divide 1. by 3 and THEN multiply the answer by 3.

Are they the same? Not on MS Calculator

And you should have used 33333~ not 3. Your calculator is not wrong. Its a different question.

Ken G
2010-Oct-27, 10:05 PM
Try this. Calculators keep "guard digits" that are not visible to us, normally.

So, divide 0.999999 by 3 and THEN multiply the answer by 3.

Now, divide 1. by 3 and THEN multiply the answer by 3.

Are they the same? Not on MS Calculatorwolfram-alpha gets both those right: (x*3.)/3. = x for any x.

astromark
2010-Oct-27, 10:12 PM
On another note, I posed a question to you in post #2183 on page 73 and explained why it is relevant to this thread. I noticed that you did not answer it so here it is again:

Do you believe that .0000000~ is equal to 0 or do you not?[/QUOTE]...

A double negative negative... Yes or no. I have little idea what point you are failing to make. YES. 0.0000~ does equal 0.

But what has this got to do with this.

A infinitely minute fraction of less than one. Does never equal one...

but the fraction is unattainable... it being 'infinitum' expressed... ~. Repeated sequence.

So the statement that 1. = 0.99999~ is true.

Jeff Root
2010-Oct-27, 11:22 PM
Try this. Calculators keep "guard digits" that are not visible to us, normally.

So, divide 0.999999 by 3 and THEN multiply the answer by 3.

Now, divide 1. by 3 and THEN multiply the answer by 3.

Are they the same? Not on MS Calculator
calculator, I get the same result for the two sequences.

-- Jeff, in Minneapolis

grapes
2010-Oct-28, 03:49 AM
And you should have used 33333~ not 3. I dunno. :)
Your calculator is not wrong. Its a different question.Yeah, i knew the MS Calculator wasn't wrong. I expected the answers to be different.

calculator, I get the same result for the two sequences.Interesting!

I remember an early calculator my grandparents were given, it sometimes had 7 x 9 = 105

2010-Oct-28, 03:54 AM
A double negative negative... Yes or no. I have little idea what point you are failing to make. YES. 0.0000~ does equal 0.

But what has this got to do with this.

Thanks for the answer. Here is why it has everything to do with it. I will use basic algebra and simple arithmetic by saying x=.99999~ and then solving for x. No calculator is needed.

x=.99999~ (given)

10x=9.99999~ (simply multiply by 10)

Subtract the first equation from the second so that 10x-x=9x which is equal to 9.00000~ (Notice that all of the 9's right of the decimal point cancel each other out)

This means that 9x=9.00000~ (Simple subtraction)

Therefore 9.00000~=9.0 which is equal to 9

This means that 9x=9 and that x=1 (Simply divide both sides by 9)

The only way to disagree with this result is to disagree that .00000~=0 (Which you did not).

Now, do you agree that .99999~=1 (The only way to not do so at this point is to contradict yourself)

a1call
2010-Oct-28, 04:06 AM
The only way to disagree with this result is to disagree that .00000~=0 (Which you did not).

Now, do you agree that .99999~=1 or are you willing to contradict yourself?
Cool proof.
What if someone (or some fish) disagrees that .99999~ (or 0.999...)is even a valid expression?
Wouldn't that be another way of disagreeing?

Ken G
2010-Oct-28, 05:27 AM
What if someone (or some fish) disagrees that .99999~ (or 0.999...)is even a valid expression?
Wouldn't that be another way of disagreeing?Only if they use some criterion to decide what is "valid" that is different from the definition of a real number in formal mathematics. That's just what I mean about using a "real-world" idea of what is "valid", rather than the syntactic relationships that follow directly from the rules of formal mathematics.

a1call
2010-Oct-28, 05:49 AM
Only if they use some criterion to decide what is "valid" that is different from the definition of a real number in formal mathematics.
What if they have no issue with the definition of a real number or the fact that the sum of any particular series converges to 1. But rather that the decimal-fractional-expression-of-repeating-decimals is not a valid representation of sums of such series. What if their objection is purely semantic. They don't object expressing a sum using sigmas, but object to expressing it using decimal-fractions since decimal-fractions are just a semantically different way of writing regular fractions. As such if the expressions of regular fractions would be invalid, then so would expressions of decimal fractions?

Jeff Root
2010-Oct-28, 06:04 AM
a1call,

I don't understand what you are saying, but I *can* imagine
the objection that the decimal part of 0.999... might not be
equal to the decimal part of 10 x 0.999... Even though they
both have infinitely many nines, isn't there one more nine
to the right of the decimal point in the first number? :)

-- Jeff, in M inneapolis

a1call
2010-Oct-28, 06:19 AM
I thought my objection was clear, but perhaps it would be clearer to say:
*- assuming decimal fractions such as 0.9 are defined simply as a different way of expressing the notion 9/10. Then since there is no valid expression of the notion 1/9 as a fraction with a denominator which is a power of 10, then there can be no valid decimal-fractional expression of the same notion namely 0.111...
ETA: undefined is a more appropriate word than invalid.

Strange
2010-Oct-28, 08:48 AM
I thought my objection was clear, but perhaps it would be clearer to say:
*- assuming decimal fractions such as 0.9 are defined simply as a different way of expressing the notion 9/10. Then since there is no valid expression of the notion 1/9 as a fraction with a denominator which is a power of 10, then there can be no valid decimal-fractional expression of the same notion namely 0.111...
ETA: undefined is a more appropriate word than invalid.

But earlier you said this hypothetical person didn't object to the use of sigma (and, presumably, an infinite series). The decimal exapansion is just a syntactically different way of writing the series:
0.111~ = 1/10 + 1/100 + 1/1000 + ~

It is not semantically different. And I don't see how it could be seen as semantically different.

grapes
2010-Oct-28, 01:46 PM
I meant to comment on this when it first appeared, but I think it's even more appropriate now.
Here is a work in progress draft of how I would define decimal fractions.

Decimal fractions are a subset of all fractions in base 10 where the denominator is a power of 10. These fractions are represented by writing the numerator and encoding it with a (decimal) point. The decimal point is placed
x = log(denominator) digits to the left of the right most digit of the numerator. If x > number-of-digits-in-the-denominator then the missing digits to the left of the numerator are represented by 0s.

Are there any errors?
Corrections are welcome and appreciated.Decimal fractions (http://en.wikipedia.org/wiki/Decimal#Decimal_fractions) are not a subset, they are the fractions that can be expressed as a fraction with denominator a power of ten. But there are other fractions that can be expressed as a decimal, e.g., 1/3 = 0.333..., which is sometimes called a decimal fraction.

Ken G
2010-Oct-28, 02:08 PM
What if they have no issue with the definition of a real number or the fact that the sum of any particular series converges to 1. But rather that the decimal-fractional-expression-of-repeating-decimals is not a valid representation of sums of such series. As Strange said, you can't "object" to a definition. The decimal-fractional-expression-of-repeating-decimals is notation for the sum of that series-- you can not like a certain notation, but you can't say it is "invalid." It's just notation, like objecting to the alphabet.

What if their objection is purely semantic. If their objection is semantic, it means that have entered the real-world place where meaning is possible. But everything in mathematics is just syntactic labels.

a1call
2010-Oct-28, 02:28 PM
I meant to comment on this when it first appeared, but I think it's even more appropriate now.Decimal fractions (http://en.wikipedia.org/wiki/Decimal#Decimal_fractions) are not a subset, they are the fractions that can be expressed as a fraction with denominator a power of ten. But there are other fractions that can be expressed as a decimal, e.g., 1/3 = 0.333..., which is sometimes called a decimal fraction.
There lies my/the whole problem. What exactly is this illusive definition of decimal fractions.
Let's try the step by step approach. I am posting this in a post with a Quote by Grapes but it is addressed to Ken and Strange as well.

*- Are decimal fractions by definition fractions of the form where the denominator is a power of 10?
Yes, or No?

P.S. I do realize that Grapes has implied No, but let's make it clearer.

Strange
2010-Oct-28, 03:02 PM
*- Are decimal fractions by definition fractions of the form where the denominator is a power of 10?
Yes, or No?

Yes. (If you choose to define it that way. It is not a term I have come across before.)

a1call
2010-Oct-28, 03:15 PM
Yes. (If you choose to define it that way. It is not a term I have come across before.)

Let's choose to define it that way, just to see where it's going to get us. Even if the real answer might be No, we might get an insight into why threads/discussions like this never converge.

Next question:

*- Is there a fractional expression a/b of the notion 1/9, where a and b are integers and b is a power of 10?
Yes, or No?

Boratssister
2010-Oct-28, 03:31 PM
After very little deliberation the answer is categorically- no..
1 IS A NUMBER and 0.999-is a PIECE OF STRING. If I had A COMPUTER that could calculate 0.999-forever THEN YOU WOULD SEE THAT 0.999- does not even get close TO 1 . Now LET IT be........

grapes
2010-Oct-28, 04:00 PM
*- Are decimal fractions by definition fractions of the form where the denominator is a power of 10?
Yes, or No?

P.S. I do realize that Grapes has implied No, but let's make it clearer.I thought I meant to imply Yes :)

I included a wikipedia link to that effect. But obviously if you are expressing a fraction as a decimal (i.e., 1/3 = 0.333...) there is a tendency to call that a decimal fraction.

HenrikOlsen
2010-Oct-28, 04:00 PM
Computers only work with approximations, not with mathematics.
Experience with computers is lousy for giving intuition about mathematics.
The question was about mathematics, so stop treating it as computer science.

agingjb
2010-Oct-28, 04:04 PM
Terminating decimals represent rational numbers of the form k/(2n5m).

a1call
2010-Oct-28, 05:07 PM
Terminating decimals represent rational numbers of the form k/(2n5m).
I would say k/(10n)

Since (2^2)*5=20 which is not a power of 10.

BTW, Did I miss somebody answering my latest yes or no question?

*- Is there a fractional expression a/b of the notion 1/9, where a and b are integers and b is a power of 10?
Yes, or No?

Ken G
2010-Oct-28, 06:32 PM
*- Is there a fractional expression a/b of the notion 1/9, where a and b are integers and b is a power of 10?
No. Any a/b = 1/9 must have b be a multiple of 9 (specifically, a*9), so 32 will be in its prime factorization. Powers of 10 don't have any 3s in their prime factorizations.

a1call
2010-Oct-28, 06:54 PM
No. Any a/b = 1/9 must have b be a multiple of 9 (specifically, a*9), so 32 will be in its prime factorization. Powers of 10 don't have any 3s in their prime factorizations.

Thank you Ken,

So let's recap:

A- Decimal fractions are fractions of the form a/b for integers a and b where b is a power of 10
B- The notion 1/9 has no expression of the form a/b where b is a power of 10
C- Thus the notion 1/9 can not have a decimal fractional expression
D- 0.111... is decimal fractional expression of the notion 1/9
E- C & D conflict

Strange
2010-Oct-28, 07:03 PM
Thank you Ken,

So let's recap:

A- Decimal fractions are fractions of the form a/b for integers a and b where b is a power of 10
B- The notion 1/9 has no expression of the form a/b where b is a power of 10
C- Thus the notion 1/9 can not have a decimal fractional expression
D- 0.111... is decimal fractional expression of the notion 1/9
E- C & D conflict

But you are using "decimal fraction" to mean two different things. Firstly, as you defined it in A, and secondly to mean ... well, what? Any number less than 1? Or any rational number expressed in decimal form? If you use the latter definition then the answer to the question "can 1/9 be represented as a decimal fraction" is (obviously) yes.

a1call
2010-Oct-28, 07:32 PM
I think you might have just made my point.
Another way of saying it is Bingo.

Strange
2010-Oct-28, 07:34 PM
I think you might have just made my point.
Another way of saying it is Bingo.

Well, you have lost me there then.

a1call
2010-Oct-28, 07:49 PM
The point is if we say A is true then D must be false.
If we say D is true then A must be false.
A & D can not be both true. But there are refences to both statement being true.
This has a lot to do with why debates like this do not resolve.

Ken G
2010-Oct-28, 09:05 PM
But the issue of what we wish to call a decimal fraction is irrelevant to the thread. All that matters is the notation 0.111... is exactly the same thing as the notation 1/9 in formal mathematics-- within the definition of the real numbers, 1/9 and 0.111... are just two different notations for the same thing, because 0.111... is notation for an infinite sum that can be proven (within formal mathematics) to converge to 1/9.

a1call
2010-Oct-28, 09:18 PM
All I heard was:
But, D is true.
It doesn't matter if you say A is true.:)

Strange
2010-Oct-28, 09:34 PM
Thank you Ken,

So let's recap:

A- Decimal fractions are fractions of the form a/b for integers a and b where b is a power of 10
B- The notion 1/9 has no expression of the form a/b where b is a power of 10
C- Thus the notion 1/9 can not have a decimal fractional expression
D- 0.111... is decimal fractional expression of the notion 1/9
E- C & D conflict

OK. Let's try again.

If A defines "decimal fraction" then D is not true.

If D is true, you have a different defintion of "decimal fraction"; A is true but not a definiton; these are only a subset of "decimal fractions".

F for effort.

2010-Oct-28, 10:28 PM
Cool proof.
What if someone (or some fish) disagrees that .99999~ (or 0.999...)is even a valid expression?
Wouldn't that be another way of disagreeing?

What if they have no issue with the definition of a real number or the fact that the sum of any particular series converges to 1. But rather that the decimal-fractional-expression-of-repeating-decimals is not a valid representation of sums of such series.

(Relevant sections above bolded by me) It would still be a self contradiction and here is why:

Astromark agreed that .00000~ is equal to 0. This means that he agrees that .00000~ (a repeating decimal) is a valid expression that is equal to 0.

So shift the goal posts (after my proof in post#2540) by saying that repeating decimals are not valid would be self contradictory.

Computers only work with approximations, not with mathematics.
Experience with computers is lousy for giving intuition about mathematics.
The question was about mathematics, so stop treating it as computer science.

Well said. It seems that a big part of this is due to people treating computer (or calculator) approximations of math as if these approximations are real math.

a1call
2010-Oct-29, 12:00 AM
This means that he agrees that .00000~ (a repeating decimal) is a valid expression that is equal to 0.

Let's make it clear that I was not referring to anyone other than myself. On that note I beg to disagree.
Repeating 0s do not meet the definition of repeating decimals:

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000..." one simply writes "1.585".[1] The decimal is also called a terminating decimal. (http://en.wikipedia.org/wiki/Repeating_decimals)
above bolded by me as well as the original article.

So shift the goal posts (after my proof in post#2540) by saying that repeating decimals are not valid would be self contradictory.

Contradiction of A & D is the whole basis of my objection.

ETA: I do not object to your proof if we consider D valid.

pzkpfw
2010-Oct-29, 12:30 AM
Is there a difference between:

"decimal representation of a fraction"

and

"decimal fraction"

?

a1call
2010-Oct-29, 01:23 AM
Is there a difference between:

"decimal representation of a fraction"

and

"decimal fraction"

?

It's all a matter of definitions.

"Decimal fractions" are obsolete. "Decimals" are supposed to refer to what "Decimal fractions" used to represent.

Historically, any number that did not represent a whole was called a "fraction". The numbers that we now call "decimals" were originally called "decimal fractions"; the numbers we now call "fractions" were called "vulgar fractions", the word "vulgar" meaning "commonplace". (http://en.m.wikipedia.org/wiki/Fraction_(mathematics)?wasRedirected=true)

However, there are a few references to that obsolete term:

If we establish that what grapes offered as a definition of fractional decimals is indeed authenticated mainstream definition then I have no objection. He amended the regular definition with special permission for repeating decimals. Without that amendment repeating decimals would be undefined and a direct as well as indirect basis for numerous objections.

To answer your question, I have not been able to find an un-vague all inclusive definition for decimals
which at some point where referred to as fractional decimals.

From one school of thought decimals are merely a different way of expressing fractions with denominator of powers of 10:

IE expressing 1/100 as 0.01

At this school of thought 0.111... is undefined since no a/b form of the notion where b is a power of 10 exists. IE all repeating decimals as well as infinitely long decimal representation of irrational numbers are undefined.

From another school of thought repeating decimals as well as infinitely long decimal representation of irrational numbers are defined as expression of infinite sums. The two definitions are vague as far as I cans see and not mutually inclusive.

Like I hinted before Google's listing of definitions of decimals (http://www.google.com/search?q=define%3Adecimals) do not include repeating decimals (specifically).

Ken G
2010-Oct-29, 01:55 AM
All I heard was:
But, D is true.
It doesn't matter if you say A is true.
Yes, that's what you should have heard.

HenrikOlsen
2010-Oct-29, 02:01 AM
Thank you Ken,

So let's recap:

A- Decimal fractions are fractions of the form a/b for integers a and b where b is a power of 10
B- The notion 1/9 has no expression of the form a/b where b is a power of 10
C- Thus the notion 1/9 can not have a decimal fractional expression
D- 0.111... is decimal fractional expression of the notion 1/9
E- C & D conflict
D is wrong, as 0.111... is not an expression of the form a/b where b is a power of 10, hence not a decimal fraction as you defined it in B.
Hence no conflict and no gotcha.

Sorry, you're trying to argue about real numbers using concepts from a very limited subset of the rational numbers. All A-definition decimal fractions are rational numbers, but not all rational numbers are decimal fractions, as you explicitly exclude all fractions with numerators with prime factors other that 2 and 5.

Any number expressible as a finite series of digits (what you call decimal fractions) will be a number with the canonical form k/2n5m with integer k, n and m, and where k has no prime factors 2 or 5, i.e. fully reduced. most fractions can't be reduced to this form and are thus not expressible as decimal fractions.

a1call
2010-Oct-29, 02:13 AM
D is wrong,
Again, bingo. I never said that D followed from C. I just mentioned some facts for alphabetic reference. I thought it was just to obvious what followed what. but lets clarify:

From definition A, B follows, then C follows.
A = > B => C

Then there is definition D

2010-Oct-29, 02:13 AM
On that note I beg to disagree.
Repeating 0s do not meet the definition of repeating decimals:

And that is where you got it wrong. If you notice what you quoted, it says that a decimal with repeating 0's is also called a terminating decimal.

The truth is that the set of terminating decimals is a subset of repeating decimals. In other words, all terminating decimals are repeating decimals but not all repeating decimals are terminating decimals. In this case the digit that repeats for the terminating decimal happens to be 0.

There are many ways to express any given integer. For example, the integer 1 can be expressed as the following:

2^0, 3^0, 4^0, etc (^ represents exponent)
cosine of zero degrees
3-2, 6-5, etc
-(i^2)
etc.

What many people here fail to realize is that .99999999~ is also one these many ways.

Jeff Root
2010-Oct-29, 02:18 AM
Do "D is true" and "D is wrong" contradict each other?

Consecutive posts #2570 (Ken) and #2571 (Henrik).

-- Jeff, in Minneapolis

a1call
2010-Oct-29, 02:22 AM
Do "D is true" and "D is wrong" contradict each other?

Consecutive posts #2570 (Ken) and #2571 (Henrik).

-- Jeff, in Minneapolis
Yes, But what I am saying is:
A-is-true and D-is-true contradict each other.

a1call
2010-Oct-29, 02:26 AM
The truth is that the set of terminating decimals is a subset of repeating decimals. In other words, all terminating decimals are repeating decimals but not all repeating decimals are terminating decimals

Not so, scroll down to where it says:

Every rational number is either a terminating or repeating decimal
in Bold

HenrikOlsen
2010-Oct-29, 02:26 AM
Again, bingo. I never said that D followed from C.
<snip>
Then there is definition D
I'm not saying that D doesn't follow from C, I'm saying D is a contradiction all by itself, based on your own definition in A.
In A you defined a decimal fraction, then in D you claim that something which by your own definition in A isn't a decimal fraction is a decimal fraction.

Sorry, you don't get to change definitions of terms mid-proof.

Still no bingo.

Yes, But what I am saying is:
A-is-true and D-is-true contradict each other.
Which isn't a problem since D is false, by definition A.

a1call
2010-Oct-29, 02:31 AM
Henrik, As far as I can see we are stating identical things. Why are we arguing?

See my post regarding the two schools of thought:

a1call
2010-Oct-29, 03:35 AM
Yes, that's what you should have heard.
Well that might just be a resolution. It certainly agrees with the official stand that is the "=" camp.

But until we can clearly state that why the chicken came before egg (sort a speak) the objections will continue well after I am a kilopi:) (If you do the math it would be a few hundred years)

Why do you give preference to the converging sum definition of decimals over the older expression of k/(10^n) definition of decimals?

Keeping in mind that at least on the web, the k/(10^n) seems to be the common definition?

2010-Oct-29, 03:40 AM
Not so, scroll down to where it says:

in Bold

If you want to go that route, that's fine. However, you said that .99999~ is not equal to 1 and that the reason is that it is not a valid expression for a number. So here is a question:

How can a rational number (such as .99999~) not be a number (without creating a self contradiction)?
After all, all repeating decimals are rational numbers.

a1call
2010-Oct-29, 03:56 AM
It can't(without creating a self contradiction). Which is the whole point of objecting based on a contradiction in what seems to be accepted definitions.

Ken G
2010-Oct-29, 04:47 AM
We're getting into a rather pointless side debate here-- what a1call is saying is correct, it just doesn't matter. The A-->C argument he gave contradicts D, regardless of which definitions are taken. So yes, that is correct, but the significance has not yet become clear. What do you see that is relevant in that contradiction, other than that different definitions can contradict?

a1call
2010-Oct-29, 05:17 AM
All I'm saying is that this is one source of disagreement. I wouldn't be surprised if the real Main stream mathematics (the top few authoritative mathematicians rather than Wikipedia or Google-define) have formulated definitions which are all inclusive. The fact that "decimal fractions" is obsolete might be an indication of perhaps, that the k/(10^n) definition is no longer valid. Another resolution would be to amend the k/(10^n) definition by special permissions for repeating decimals as well as decimal expressions of irrationals as Grapes was eluding to.
Since Google and Wikipedia carry major weight, threads like this won't make everyone happy until better job is done in describing what exactly, decimals are.

agingjb
2010-Oct-29, 07:16 AM
Any discussion that simply ignores the modifier "terminating" in classifying decimals is not likely to achieve the aim of describing what decimals are.

And in my view using 10n rather than 2m5n as a characterising denominator is likely to be misleading.

HenrikOlsen
2010-Oct-29, 09:56 AM
Henrik, As far as I can see we are stating identical things. Why are we arguing?
Because either you aren't clear in expressing what you're trying to accomplish or you're trying to prove something which is false in formal mathematics.

D is formulated as if you're trying to make the point that 0.111~ ≠ 1/9, at which it is failing miserably since it isn't saying anything about 0.111~ because it isn't a decimal fraction by your definition in A, so you can't use properties of A-definition decimal fractions to show anything about it.

As mentioned previously, your A-definition decimal fractions are only a very small part of the rational numbers, a part which 0.111~ isn't in.

2010-Oct-29, 01:07 PM
It can't(without creating a self contradiction). Which is the whole point of objecting based on a contradiction in what seems to be accepted definitions.

Here is the issue:
After my proof, you said that repeating decimals do not qualify as numbers. After I said that a terminating decimal is a repeating decimal, you disagreed and quoted a source which said that repeating decimals and terminating decimals are 2 types of rational numbers. However, this contradicted your assertion that a repeating decimal is not a number. When I pointed this out, you responded with this post which is essentially refuting the part of the source which you quoted. In other words, you accepted the part that says that terminating and repeating decimals are two different types of rational numbers and are now rejecting its accepted definition of a number.

What I am trying to say is that being human, we are all susceptible to making logical errors. Do you think that now might be a good time to step back and re-evaluate your position on this issue? :)

a1call
2010-Oct-29, 02:06 PM
A'- My car is black
D'- My car is white

Where is the logical error in conclusion E', based on distinct definitions A' and D'?

grapes
2010-Oct-29, 02:21 PM
Where is the logical error in conclusion E', based on distinct definitions A' and D'?I think we've established a clear definition of what is meant by 0.999... and have also found that there is no uniformly accepted definition of "decimal fraction". If you wish to exploit the ambiguity of the term "decimal fraction", then that's all there is, ambiguity, no conclusion possible.

SeanF
2010-Oct-29, 02:23 PM
Am I missing something?

It seems to me that A1call is claiming that 0.111... does not meet the definition of a "decimal fraction." But it likewise seems to me that 1/9 doesn't meet the given definition, either.

So I'm at a loss as to understand why that would tell us anything at all about whether or not 0.111... and 1/9 are equal to each other. :think:

a1call
2010-Oct-29, 03:20 PM
and have also found that there is no uniformly accepted definition of &quot;decimal fraction&quot;.
That is a bit more diplomatic than I hoped for, but:
That is the only conclusion that I set out to establish. If that is indeed established, then my discussion is over. Perhaps it's best to let this thread sink, until it's resurrected again in the year 2051.999... .
Incidentally, I hear that's the year where the time traveler in the news departed from.:doh:

amazeofdeath
2010-Oct-29, 05:13 PM
I'm not sure what's your point, a1call. SeanF above your post finely summarized the "problem" you have introduced to muddle the waters; the question is not about definitions like you have proposed, it's about real maths. 1/9 is a valid fraction, equal to 0.111... and 9/9 = 1 = 9 * 0.111... = 0.999... No matter of redefining fractions to consist only decimal fraction changes that from the real mathematical point of view.

Ken G
2010-Oct-29, 05:41 PM
I believe what a1call is saying is that since there are different definitions of decimal fractions, theorems that apply for one definition might not apply for another. Using a theorem improperly like that could lead one to imagine that 1.999... != 2, so can be a source of confusion in the thread. That seems to be the entire point at this stage, you'll note the main point, that 1.999... = 2 in pure math, has already been surrendered.

Subduction Zone
2010-Oct-31, 12:50 AM
There have been all sorts of proofs that .999... is equal to one. There are formal mathematical proofs and informal ones. Only by trying to make a new definition of what a "decimal fraction" is can a1call even hope to make a point. He also does not seem to understand what a rational number is. Too many people look at the word irrational and think "crazy" instead of looking closer at the roots of the word. A rational number is one that can be written as a ratio of two integers. In other words it can be written as a/b where a and b are two integers, and b cannot equal zero. An irrational number cannot be described as a ratio. A terminating decimal is a number that can be written k/2^a5^b where k,a and b are integers. A repeating decimal can be written as k/(99...99)(2^a5^b) where a,b, and k are again an integers and the number of 9's is the same as the period of repetition. So for numbers like 1/9 or 1/6 (15/(9*2*5)) the number of 9's is one, 2/11 (18/99) the number of 9's is two etc. . There is nothing exceptionally special about a repeating fraction. If you use another base for your number system some fractions that used to repeat will terminate and vice versa. It relates with how the ratio can be written in each base. And in any base .a repeating where a = b - 1 where b is the base of the system (so it would always be written 10 regardless of base) is 1. So in base 2 .1111..... repeating is 1 in base 3 0.22222... repeating is 1 and so on.

jfribrg
2010-Oct-31, 04:00 AM
A terminating decimal is a number that can be written k/2^a5^b where k,a and b are integers.

The general rule is that if every prime factor in the denominator is also a prime factor of the base, then the decimal terminates. Otherwise it repeats. In base 10, the two prime factors are 2 and 5, and therefore, the denominators 2,4,5,8,10,16,20,25,32, etc. repeat. If you use base 12, then 2,3,4,6,8,9,12,16,18,24,27, etc repeat.

grapes
2010-Oct-31, 04:23 AM
If you use base 12, then 2,3,4,6,8,9,12,16,18,24,27, etc repeat.Terminate, but otherwise OK.

SeanF
2010-Oct-31, 04:24 AM
In base 10, the two prime factors are 2 and 5, and therefore, the denominators 2,4,5,8,10,16,20,25,32, etc. repeat.
Terminate, right?

grapes
2010-Oct-31, 04:38 AM
Gluteus maximus sinister

a1call
2010-Oct-31, 05:27 AM
Rather than repeat something over and over again, I made a blog regarding the point that I made. This is because it seems that some, perhaps understandably, are confused:

ETA: To avoid any further confusion and easy reference to the point that I am making, I added the link to the above mentioned blog to my board signature.

I've removed the links. There is enough clamor to get this thread moved to ATM that I think it is only fair that you make any arguments within this thread. You can then link to the part of the thread where you make the argument. I would have changed the link for you, but I can't find within this thread where you make the same point that you make at the blog. [grapes]

grapes
2010-Oct-31, 01:31 PM
and have also found that there is no uniformly accepted definition of &quot;decimal fraction&quot;.
That is a bit more diplomatic than I hoped for, but:
That is the only conclusion that I set out to establish. If that is indeed established, then my discussion is over. Perhaps it's best to let this thread sink, until it's resurrected again in the year 2051.999... I'm sure there'd be plenty of posters who'd be happy with that! :)

Ken G
2010-Nov-01, 01:01 AM
A rational number is one that can be written as a ratio of two integers. In other words it can be written as a/b where a and b are two integers, and b cannot equal zero. An irrational number cannot be described as a ratio. A terminating decimal is a number that can be written k/2^a5^b where k,a and b are integers. A repeating decimal can be written as k/(99...99)(2^a5^b) where a,b, and k are again an integers and the number of 9's is the same as the period of repetition. So for numbers like 1/9 or 1/6 (15/(9*2*5)) the number of 9's is one, 2/11 (18/99) the number of 9's is two etc. . There is nothing exceptionally special about a repeating fraction. If you use another base for your number system some fractions that used to repeat will terminate and vice versa. It relates with how the ratio can be written in each base. And in any base .a repeating where a = b - 1 where b is the base of the system (so it would always be written 10 regardless of base) is 1. So in base 2 .1111..... repeating is 1 in base 3 0.22222... repeating is 1 and so on.Now that's interesting. Sometimes it takes a lot of pages beating a dead horse to shake out a jewel of learning!

jfribrg
2010-Nov-01, 01:14 AM
Terminate, right?

Doh. Yes, I really meant "terminate". I'm glad that folks actually read my posts.

Solfe
2010-Nov-14, 04:54 AM
Ok, four pages later and I am back. I still see a difference between 1 and .9999.... and I am armed with a digital calculator. Digital, of course being pronounced the same way as it is in HHGTTG. ;) It is entirely possible, I have nothing useful to add except I got you to say "dig-i-tal."

I was thinking of the non-number between .9999.... and 1 which cannot be constructed in a reasonable/acceptable way.

What if you took a different approach. Picture a coil constructed like a phone cord (which hasn't been seen in real life since things went digital) where the there were 4 loops of the coil per 1 unit forward. Make it fairly long. Repeat, but this time replace the one with .9999.... Before long, you see a big difference.

Now a different approach. 1/1=1 as does .9999..../.9999.... So, 1/1, 2/2, 3/3 all equal 1.
What happens when you multiply? 2/2*4/4=8/8 or 1. How about 8/8*.9999..../.9999...., Would that work out to be 7.9999..../7.9999.... It is 1 again, but not in the standard way we are used to. What happens when you take 8/8-1/1? Zero. Now what about 8/8-.9999..../.9999....? Again it is zero, but not exactly the way we expect it.

Now as you can see with the rather simplistic way I have worded this, I am not here to win people to my side. I am sort of expecting to see how/why I am wrong because I am likely the student described in the paper.

grapes
2010-Nov-14, 05:11 AM
Before long, you see a big difference.After long. :)
I am sort of expecting to see how/why I am wrong because I am likely the student described in the paper.Except for that, nothing. :)

cjl
2010-Nov-14, 11:46 AM
Ok, four pages later and I am back. I still see a difference between 1 and .9999.... and I am armed with a digital calculator. Digital, of course being pronounced the same way as it is in HHGTTG. ;) It is entirely possible, I have nothing useful to add except I got you to say "dig-i-tal."

I was thinking of the non-number between .9999.... and 1 which cannot be constructed in a reasonable/acceptable way.

What if you took a different approach. Picture a coil constructed like a phone cord (which hasn't been seen in real life since things went digital) where the there were 4 loops of the coil per 1 unit forward. Make it fairly long. Repeat, but this time replace the one with .9999.... Before long, you see a big difference.

The difference you would see would be equal to n*10-h, in which n was the number of iterations of the process, and h was the number of 9s in your number. If you take the limit as the number of 9s goes to infinity, this difference goes to zero, regardless of iteration count.

Now a different approach. 1/1=1 as does .9999..../.9999.... So, 1/1, 2/2, 3/3 all equal 1.
What happens when you multiply? 2/2*4/4=8/8 or 1. How about 8/8*.9999..../.9999...., Would that work out to be 7.9999..../7.9999.... It is 1 again, but not in the standard way we are used to. What happens when you take 8/8-1/1? Zero. Now what about 8/8-.9999..../.9999....? Again it is zero, but not exactly the way we expect it.

Now as you can see with the rather simplistic way I have worded this, I am not here to win people to my side. I am sort of expecting to see how/why I am wrong because I am likely the student described in the paper.

In all of these cases, you get the correct answer, and I really don't see what you mean when you state "not exactly the way we expect it". I don't see anything particularly strange or unusual about your conclusions.

Solfe
2010-Nov-14, 03:46 PM
Cjl,

Late at night, answers other than integers are not exactly what I was expecting? Actually that really was my point and I was going to go into this whole idea about how numbers are "constructed", but in reading the previous posts about constructing numbers between 1 and .9999 I already conceded that point.

I will now disappear into the tiny space between 1 and .9999....

caveman1917
2010-Nov-15, 01:13 AM
Late at night, answers other than integers are not exactly what I was expecting?

This may be some source of confusion. They are integers. You're perhaps not used to the notation, but whichever symbol we would like to use to represent a number doesn't change that number. Should we like to represent "1" as say a little square, does that mean 1 suddenly isn't an integer anymore?

Disinfo Agent
2010-Nov-24, 06:42 PM
(Edit: Further, I thought "lim" meant "nearly, but never quite, gets there". Am I wrong here too?)Consider the sequence

0, 1, 0, 1/2, 0, 1/4, 0, 1/8...

and find its limit. Is the limit ever attained?

Disinfo Agent
2010-Nov-25, 11:18 AM

I would form it as .3333~ ≈ 1/3 for the main reason that not matter how long I calcualted 1/3, I'd be pushing up daisies before I could actualy prove or disprove matmatically that .3333~ is actualy equal to 1/3.There are mathematical techniques for handling such situations, that don't take a lifetime. It's true that some people never learn them in their lifetime, but that is not what we are talking about here. I'm curious about what techniques you mean. Could you exemplify?

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. (http://en.wikipedia.org/wiki/Rational_number)

Since 0.999... is a rational number we will "express" it as:

999.../1000...But of course 999... and 1000... are not integers. That's silly.

The equality 0.999... = 1 is not just a matter of convention ('syntax', to use Ken G's term).* Rather, it is an inescapable consequence of the assumptions we usually make about the real numbers and their representations in decimal form. Sure, you can come up with nonstandard structures and interpretations where the two sides of the equation are not equal -- but there's good reason why such structures and interpretations are not standard.

* Although I'll concede that choosing to even assign a meaning to expressions like '0.999...' is a matter of convention. However, the viable choice here is not between the standard meaning and another meaning; it's between the standard meaning and no meaning whatsoever.

forrest noble
2010-Nov-28, 11:04 PM
My opinion concerning how the poll was set up is that unlimited .9's is not the same thing as the "limit" of infinite .9's.

forrest noble
2010-Nov-28, 11:23 PM
pzkpfw,

Further, I thought "lim" meant "nearly, but never quite, gets there". Am I wrong here too?

Using "lim" as in limit is a mathematical term concerning the limit line of an asymptotic function or numerical sequence. If there is a numerical solution it is an exact quantity like pi, where pi is the limit of the ratio of the circumference of a circle to its diameter. The limit could also be a simpler function than the original.

grapes
2010-Nov-29, 10:56 AM
I'm curious about what techniques you mean. Could you exemplify?Merely taking limits.
My opinion concerning how the poll was set up is that unlimited .9's is not the same thing as the "limit" of infinite .9's.What's the difference? O yeah, .000...1 :)

If there is a numerical solution it is an exact quantity like pi, where pi is the limit of the ratio of the circumference of a circle to its diameter. Pi is not the limit of the ratio of the circumference of a circle to its diameter--it is exactly equal to the ratio.

Frank T
2010-Dec-21, 10:13 PM
Zero x ∞ = 1 because: ∞ is not a number. You cannot say it's the largest number in the world because infinity is always larger than any number. So we can say 1 x ∞ = ∞, but 2 x ∞ is impossible because we can never get beyond 1 x ∞. Similarly any number except one times ∞ is impossible for the same reason. If ∞ is not a number, what is it? It's a concept. But it's not an infinity of concepts, but one concept: infinity. Therefore all (not any) numbers -including zero- x ∞ = ∞ or 1x∞. In fact, whatever we do, whichever numbers we add or subtract to the formula, we are always left with 1∞. But as ∞ is not a number, but a mere concept, we must delete it from the formula, leaving … ONE. Therefor 0 x ∞ = 1

caveman1917
2010-Dec-22, 02:04 AM
Zero x ∞ = 1 because: ∞ is not a number. You cannot say it's the largest number in the world because infinity is always larger than any number. So we can say 1 x ∞ = ∞, but 2 x ∞ is impossible because we can never get beyond 1 x ∞. Similarly any number except one times ∞ is impossible for the same reason. If ∞ is not a number, what is it? It's a concept. But it's not an infinity of concepts, but one concept: infinity. Therefore all (not any) numbers -including zero- x ∞ = ∞ or 1x∞. In fact, whatever we do, whichever numbers we add or subtract to the formula, we are always left with 1∞. But as ∞ is not a number, but a mere concept, we must delete it from the formula, leaving … ONE. Therefor 0 x ∞ = 1

Welcome to BAUT. But your statement is false. 0 x inf is undefined. We can encounter it as a limit for a function, but then we can get any real number as the answer (it depends on the function).

jfribrg
2010-Dec-22, 02:39 PM
Zero x ∞ = 1 because: ∞ is not a number. You cannot say it's the largest number in the world because infinity is always larger than any number. So we can say 1 x ∞ = ∞, but 2 x ∞ is impossible because we can never get beyond 1 x ∞. Similarly any number except one times ∞ is impossible for the same reason. If ∞ is not a number, what is it? It's a concept. But it's not an infinity of concepts, but one concept: infinity. Therefore all (not any) numbers -including zero- x ∞ = ∞ or 1x∞. In fact, whatever we do, whichever numbers we add or subtract to the formula, we are always left with 1∞. But as ∞ is not a number, but a mere concept, we must delete it from the formula, leaving … ONE. Therefor 0 x ∞ = 1

Welcome to BAUT.

To say that we must delete infinity from the formula means that we need to treat infinity as if it were a variable, but it isn't. If it were, we could prove that 2= 3.
2*∞ = 3*∞

If it were a valid operation, we could cancel the ∞ from both sides, leaving 2=3, which is false.

It helps to think of ∞ as both 1/0 and as set of numbers all of which have a magnitude larger than any given number. Certain operations are permitted, such as multiplying by a number, resulting in the same set as we started with. We can also multiply infinities by declaring that ∞*∞ = ∞, but multiplication by zero is not permitted because it will lead to contradictions similar to what I showed above:
2*0*∞ = 3*0*∞
using your definition that 0*∞ = 1, again we would reach the incorrect conclusion that 2=3.

We have two properties of numbers:
0 * anything = 0
∞ * anything = ∞

These two properties are incompatible if we were allowed to multiply 0 and ∞. If we assign any specific value to 0*∞, then it is not hard to produce contradictions like I did above.

The only way to avoid these contradictions is to declare the operation 0*∞ to be invalid, or in other words, that 0*∞ is indeterminate. If 0*∞ (or it's equivalent form of 0/0) is used in an algebraic manipulation, then any subsequent conclusion you might reach is worthless.

Kaptain K
2010-Dec-22, 07:04 PM
Please note that there is more than one "order" of "infinity".

The number of points on a line (infinity) is (infinitely) greater than the number (infinity) of whole numbers.

Andrew D
2010-Dec-23, 04:48 AM
Ok, I voted no, but on more careful consideration:

http://latex.codecogs.com/gif.latex?.999999R%20=\sum_{n=1}^{\infty%20}\frac{ 9}{10^{n}}=9\sum_{n=1}^{\infty%20}\frac{1}{10^{n}}

We can find the sum of the geometric series, but we have to account for the n=0 term by subtracting it out,

http://latex.codecogs.com/gif.latex?9\sum_{n=1}^{\infty%20}\frac{1}{10^{n}}= 9\sum_{n=0}^{\infty%20}\frac{1}{10^{n}}-\frac{9}{10^{0}}

http://latex.codecogs.com/gif.latex?9\sum_{n=1}^{\infty%20}\frac{1}{10^{n}}= 9\left%20(\sum_{n=0}^{\infty%20}\frac{1}{10^{n}}-{1}\right%20)

for those of you unfamiliar with the maths, the sum of a geometric series is
http://latex.codecogs.com/gif.latex?\sum_{n=0}^{\infty%20}ar^{n}=\frac{a}{1-r}

in this case a=1 and r=1/10, so

http://latex.codecogs.com/gif.latex?9\sum_{n=1}^{\infty%20}\frac{1}{10^{n}}= 9\left(\frac{1}{1-\frac{1}{10}}-{1}\right%20)=9\left(\frac{1}{\frac{9}{10}}-{1}\right%20)=9\left(\frac{10}{9}-{1}\right%20)=9\left(\frac{1}{9}\right%20)=1

Sorry if someone has already used this method, I didn't look through all 88 pages.

Strange
2010-Dec-23, 10:20 AM
Please note that there is more than one "order" of "infinity".

The number of points on a line (infinity) is (infinitely) greater than the number (infinity) of whole numbers.

More than one? An infinite number, surely.

Paul Beardsley
2010-Dec-23, 10:46 AM
It's been mathematically proven several times now. Isn't it time this thread was closed?

Strange
2010-Dec-23, 11:36 AM
It's been mathematically proven several times now. Isn't it time this thread was closed?

Nearly. But not quite ....

Paul Beardsley
2010-Dec-23, 11:38 AM
Nearly. But not quite ....

The fact that I'm laughing does not necessarily mean I thought that was funny!

Jeff Root
2010-Dec-23, 02:13 PM
I tried to think of something like Strange's last comment just
after Frank T posted (post #2612, above), but couldn't think of
anything that would allow me to maintain my integrity. But I
wanted to say the same thing. We're almost there. But not
quite yet. Not quite yet ....

-- Jeff, in Minneapolis

Paul Beardsley
2010-Dec-23, 02:16 PM
Er, I assumed Strange's post was a joke - along the lines of, "We're 99.99999~% finished." Clearly there isn't really anything else to add.

Jeff Root
2010-Dec-23, 04:40 PM
quite done yet. Very close. But not quite yet ...

-- Jeff, in Minneapolis

caveman1917
2010-Dec-24, 12:25 AM
More than one? An infinite number, surely.

Two, in fact. At least as far as one goes by the Continuum hypothesis (http://en.wikipedia.org/wiki/Continuum_hypothesis).

Jeff Root
2010-Dec-24, 02:42 AM
It looks like there are only two known levels of cardinality of
infinities, the two referred to by Kaptain K: One exemplified
by the set of integers or the set of rational numbers; the other
exemplified by the set of real numbers or the points on a line.
If there are others, they have not been discovered.

-- Jeff, infinity

Swift
2010-Dec-24, 03:04 AM
I have hated this thread for at least as long as I can remember. It is 99.9999~% more trouble than it is worth. It is done as far as I'm concerned.

If you have an amazing reason why it should be reopened, report this post and it will be considered (but don't count on it).