What is .99 repeating as a fraction?
Quote: weaselmanUmmm ... no, it doesn't. Where did you read this?
"it" "doesn't" What?
Statement to vague to analyze.....*ERROR*
Does adding two infinitesimals equal another infinitesimal?
And how is that you can add three infinitesimals together, and get a non infinite number?
If infinity goes on forever, you just made it stop.
Quote: YoDiceRoll11"it" "doesn't" What?
Statement to vague to analyze.....*ERROR*
Too much activity in this thread :)
I did not expect three other posts to get inserted between yours and mine.
I mean that the hyperreal theory does not support the idea you are defending, you are wrong about it. If you care to provide a reference to the source that made you believe this, I will probably be able to explain what you misunderstood there.
In short, hyperreal theory does not alter the properties or behaviour of real numbers in any way. It introduces a different kind of entity (called non-standard numbers), that does have some peculiar properties, but what is true about a real number in the "standard" theory, remains true in hyperreal as well ... and vice versa.
This generally holds for any "alternative" theory - they are not really alternatives per se, more like extensions.
Quote: YoDiceRoll11Here's another way to look at this guys. I'd like your input on the following:
Does adding two infinitesimals equal another infinitesimal?
Yes, it does.
But the difference between 1 and 0.999... is not an infinitesimal - it is exactly 0.
Quote: weaselmanToo much activity in this thread :)
I did not expect three other posts to get inserted between yours and mind.
I know tell me about it.
Quote:In short, hyperreal theory does not alter the properties or behaviour of real numbers in any way. It introduces a different kind of entity (called non-standard numbers), that does have some peculiar properties, but what is true about a real number in the "standard" theory, remains true in hyperreal as well ... and vice versa.
This generally holds for any "alternative" theory - they are not really alternatives per se, more like extensions.
You are correct, they are extensions.
Here, this is one such argument that I agree with:
Hyperreal numbers and extensions involving repeating decimals
And like I said from the beginning. You can disagree. Just watch the name calling. ;)
Quote: weaselmanQuote: YoDiceRoll11Here's another way to look at this guys. I'd like your input on the following:
Does adding two infinitesimals equal another infinitesimal?
Yes, it does.
But the difference between 1 and 0.999... is not an infinitesimal - it is exactly 0.
Ok.....
now address the second question:
Quote:And how is that you can add three infinitesimals together, and get a non infinite number?
Quote: YoDiceRoll11
Ok.....
now address the second question:Quote:And how is that you can add three infinitesimals together, and get a non infinite number?
If by "non infinite" you mean "non-infinitesimal", I then the answer is, it is impossible.
Quote: weaselmanQuote: YoDiceRoll11
Ok.....
now address the second question:Quote:And how is that you can add three infinitesimals together, and get a non infinite number?
If by "non infinite" you mean "non-infinitesimal", I then the answer is, it is impossible.
Yes, thank you.
Than what is .333...+.333...+.333....?? 1 right?
I thought you just said it was possible. Now it's impossible?
I usually don't like using the term "agree to disagree", but this is one such instance where I can accept the majority not agreeing with this. I'm ok with that.
Quote: YoDiceRoll11Quote: weaselmanQuote: YoDiceRoll11
Ok.....
now address the second question:Quote:And how is that you can add three infinitesimals together, and get a non infinite number?
If by "non infinite" you mean "non-infinitesimal", I then the answer is, it is impossible.
Yes, thank you.
Than what is .333...+.333...+.333....?? 1 right?
I thought you just said it was possible. Now it's impossible?
Huh? 0.333... is not an infinitesimal (infinitesimal means infinitely small). What are you talking about?
I think I have read all of your posts. Including the ones where you claim that .333... is an irrational number, but I have not seen any explanation of that other than the one I questioned: you stated it is irrational because it cannot be expressed as 1/3 (although it can be). Then you appeared to claim that because it is irrational it cannot equal 1/3. I pointed out that this did not appear to be sound logic, and you admitted that circular reasoning is not the best, but you never justified your thought/belief/conclusion that .333... is irrational.Quote: YoDiceRoll11Really, read my above post.
Quote: DocI think I have read all of your posts. Including the ones where you claim that .333... is an irrational number, but I have not seen any explanation of that other than the one I questioned: you stated it is irrational because it cannot be expressed as 1/3 (although it can be). Then you appeared to claim that because it is irrational it cannot equal 1/3. I pointed out that this did not appear to be sound logic, and you admitted that circular reasoning is not the best, but you never justified your thought/belief/conclusion that .333... is irrational.
Link from above
Check the above link involving hyperreal numbers.
And than show me how you can add three infinite numbers to get a non infinite number.
Quote: weaselmanQuote: YoDiceRoll11Quote: weaselmanQuote: YoDiceRoll11
Ok.....
now address the second question:Quote:And how is that you can add three infinitesimals together, and get a non infinite number?
If by "non infinite" you mean "non-infinitesimal", I then the answer is, it is impossible.
Yes, thank you.
Than what is .333...+.333...+.333....?? 1 right?
I thought you just said it was possible. Now it's impossible?
Huh? 0.333... is not an infinitesimal (infinitesimal means infinitely small). What are you talking about?
Did you even read the link I posted? Whether you define it as infinitely small or infinite, you can't add three infinite numbers and get a real integer.
Quote: boymimboI will maintain that .3333... where the ... is an infinite set of 3s is 1/3 and that 1/3 x 3 = 1, which is the same as .999...
And I will maintain you can't multiply an infinite number and expect a non infinite number. That's like saying multiply pi by 2 and give me a rational, non infinite, integer. It's retarded.
Ok, I'm done on this thread boys. Re-read my posts if you will, check out the link. If MathExtremist would like to add something to the link that I posted, I would like his analysis on that.
Otherwise, have a good day all and thank you for the discussion.
Quote: DocIt appears that, while well behind, YDR11 declared victory and retreated.
I did no such thing. In fact I did the opposite. I said that I had made my point, and you can disagree, in fact I expect you to disagree.
How is that victory? This isn't a win/lose thing. Good discussion all the way around.
Quote: YoDiceRoll11Ok, I'm done on this thread boys.
Quote: DocIt appears that, while well behind, YDR11 declared victory and retreated.
Quote: YoDiceRoll11I did no such thing. In fact I did the opposite. I said that I had made my point, and you can disagree, in fact I expect you to disagree.
How is that victory? This isn't a win/lose thing. Good discussion all the way around.
Sorry, that was just a little joke on my part to test/poke-fun-at whether you were really "done on this thread." I really need to get my warped sense of humor under control.
Quote: YoDiceRoll11Link from above
Check the above link involving hyperreal numbers.
From that article:
"This leaves the limit of a series of finite sums as the only consistent interpretation of that expression." ("that expression" being 0.999...)
Very well. If 0.999... is the limit of the series 0.9 + 0.09 + 0.009 + 0.0009 + ... + 9*0.1^n, n->+inf, then what is the limit?
Are you suggesting the limit isn't 1? There is a big difference between any particular finite summation of terms, which is < 1, and the *limit* of the infinite summation of those terms, which does equal 1. You're throwing out a big chunk of calculus if you dispute that result because such would be rewriting what "limit" means. As it is, your prior reply to me indicated that you do not believe the operation "addition" is valid on any expression of the form 0.111..., so I think what's really going on here is that you don't believe the notation 0.111... is a valid representation of a number, and specifically that 0.111... is not a valid representation of 1/9. I disagree, of course, but I have no desire to argue definitions with anyone.
Carry on.
Quote: YoDiceRoll11
Did you even read the link I posted? Whether you define it as infinitely small or infinite, you can't add three infinite numbers and get a real integer.
I did not read it very attentively (as it appeared way too verbose for the amount of information it was trying to convey), so I could have missed it ... But if it really says this, it is just plain wrong.
First 0.333 ... is not an infinite number (and neither is it infinitely small, infinitesimal ... or anything like that), if you ever see any page calling it that, just leave, as it is a clear indication that the author simply has absolutely no idea what he is talking about. Second, 0.3333... is definitely a real number (and also a rational number), and there is nothing unusual or unexpected in being able to add two real numbers and get an integer.
And, finally, even if it was not real, you certainly can add two (non-standard)hyperreal numbers and get an integer (this is not relevant to the discussion though, because all the numbers we are adding are real to begin with).
If you find it surprising that adding infinite decimal fractions can yield a final one, I urge you to consider such numbers as
0 = Pi + (-Pi), 1 = sin^2(pi/3) + cos^2(pi/3), 2 = 2*sin(pi/4) + (2-sqrt(2)) etc.
All the additive components above would have infinite number of digits if they were represented in a decimal form.
Quote: buzzpaffI am a little lost here. If .999=1, then does 3.999=4 ? Is not, why not ?
Well, .999≠1 and 3.999≠4. It is important to include the ellipses (is this a correct application of this term?).
The topic under discussion is whether .999... = 1, which if true would mean that 3.999... = 4. I agree with both of these. YoDiceRoll11 disagrees and cited an article that he believes supports his position.
My latest interpretation of the conflict is that everyone here who believes as I do defines the expression ".999..." to mean a decimal number with a truly infinite series of 9s, while the article that YDR11 linked to explicitly claims that 0.999... "does not correspond to an expansion with an infinite number of 9s!"
If you are going to base your argument on statements that reject the definition of the term as used by others, then I don't think its even possible to ever reach agreement through discussion. If everyone who currently holds the same position as I do were to suddenly redefine/agree that .999... doesn't involve an infinite series of 9s, then we would draw a completely different conclusion. But we don't.
Quote: YoDiceRoll11And I will maintain you can't multiply an infinite number and expect a non infinite number. That's like saying multiply pi by 2 and give me a rational, non infinite, integer. It's retarded.
PI is an irrational number. That by definition means, you can't get an integer by multiplying it by another rational.
1/3=0.333... by contrast is a rational number, which means ... (you guessed it!) ... multiplying it by a rational will always yield another rational (which sometimes can happen to be an integer).
On the other hand, there is nothing unusual in multiplying two irrational numbers to get an integer - evidently,
Pi*(1/Pi) = 1, despite the fact that both nominator and denominator would take an infinite number of digits to be expressed in a decimal notation.
BTW, if you think, that number 1 has finite number of digits you are mistaken - it is actually 1.000... (infinite number of zeroes). It is simply a convention that we usually omit leading trailing zeroes in decimal fractions. It does not mean they are not there, just that we agreed to not spell them out explicitly.
This is the mathematical proof (I think)
Give a {delta}>0 I can construct a series of nines being .99999999....9 (ends) that is within that value of 1.
In simple terms, give me any positive number (however small, positive and non-zero) and I can get closer than that to "1" (essentially since N nines gives a value of 1 - 10^-N, by working out which N is needed to create 10^-N that is smaller than your value).
Similar logic means 1 + 1/2 + 1/4 ... being "2" (essentially by recognizing the last term is how far out the total so far is).
btw our club tie was "{epsilon}<0" and had a quorum of "one more than the number present".
Quote: YoDiceRoll11Edit: MathExtremist, dude much respect to you btw, I just can't allow what I believe to be cutting a corner by allowing one to add an infinite decimal. It's like saying that you can add pi plus pi and give me a rational number.
It's not at all like saying you can add pi plus pi and give you a rational number. Rationality has nothing to do with it. pi + pi = 2pi, which has a non-periodic decimal representation that begins with 6.283185 and carries on to an infinite number of places. I cannot accurately depict 2pi in decimal notation.
However, I *can* depict any rational number (where rational number = a number expressable via a/b) via the use of a periodic representation.
1/2 = 0.500(0)... = 0.5
1/4 = 0.2500(0)... = 0.25
1/3 = 0.333... Based on the nature of the base-10, i.e. "decimal", numbering scheme, there is no finite numeric representation of the fraction 1/3, so we invent one: the vinculum, the bar that goes over the top of the 3 to indicate it repeats. The ellipses I've been using serve the same purpose, because I can't figure out how to get a vinculum to print here.
However, let me expand your mind a bit and introduce the *ternary* numbering system. Base 3, using only the digits 0, 1, and 2. In ternary, decimal number 10 (ten, the number of fingers you probably have) is written 101.Pp
In ternary you can add 0.1 + 0.1 and get 0.2, which equals 2/3 in base-10.
In ternary, the base-10 value 1/3 is written 0.1. It has a finite representation without resorting to a vinculum or ellipses.
However, the base-10 value 1/4 is written 0.02020202(02)... in ternary. That is, an infinite string of 02 following zero and the decimal point.
Similarly, the base-10 value 1/2 is written 0.111... in ternary. In fact, the base-10 rational numbers 1/2 and 1/9, when expressed in ternary and decimal notation respectively, have exactly the same representation: 0.111...
Now I hope you're not going to argue that the validity of the operation "addition" is somehow dependent on the base of the numbering system you're using, because we all know that's not true. The fact is, decimal notation requires a notational shortcut to depict division by 3 while ternary doesn't because the 10 in base-10 has prime factors of 2 and 5. That means any other prime factor in the denominator will result in a periodic representation. If you wanted to use a base-210 numbering scheme, 1/2, 1/3, 1/5, and 1/7 would all have non-periodic representations. In decimal, only 1/2 and 1/5 do -- 1/3 and 1/7 have periodic representations.
But the representation of a number in a given numbering system doesn't change what it *means*. 1/3 is still 0.333... in decimal and 0.1 in ternary. They're all equal, and they can be added together regardless of which base notation you're using.
Quote: weaselmanPI is an irrational number. That by definition means, you can't get an integer by multiplying it by another rational.
1/3=0.333... by contrast is a rational number, which means ... (you guessed it!) ... multiplying it by a rational will always yield another rational (which sometimes can happen to be an integer).
On the other hand, there is nothing unusual in multiplying two irrational numbers to get an integer - evidently,
Pi*(1/Pi) = 1, despite the fact that both nominator and denominator would take an infinite number of digits to be expressed in a decimal notation.
BTW, if you think, that number 1 has finite number of digits you are mistaken - it is actually 1.000... (infinite number of zeroes). It is simply a convention that we usually omit leading trailing zeroes in decimal fractions. It does not mean they are not there, just that we agreed to not spell them out explicitly.
So weaselman, are you saying that 0.999! = 1.000! (with ! representing an infinite number of repeating digits)?
Quote: Ayecarumba
So weaselman, are you saying that 0.999! = 1.000! (with ! representing an infinite number of repeating digits)?
Yeah, that's what I am saying.
Quote: weaselmanYeah, that's what I am saying.
Given that, I can see where YDR11 is coming from. In my base-10 mind, 0.999! is not equal to 1.000!
Quote: MathExtremistIt's not at all like saying you can add pi plus pi and give you a rational number. Rationality has nothing to do with it. pi + pi = 2pi, which has a non-periodic decimal representation that begins with 6.283185 and carries on to an infinite number of places. I cannot accurately depict 2pi in decimal notation.
However, I *can* depict any rational number (where rational number = a number expressable via a/b) via the use of a periodic representation.
1/2 = 0.500(0)... = 0.5
1/4 = 0.2500(0)... = 0.25
1/3 = 0.333... Based on the nature of the base-10, i.e. "decimal", numbering scheme, there is no finite numeric representation of the fraction 1/3, so we invent one: the vinculum, the bar that goes over the top of the 3 to indicate it repeats. The ellipses I've been using serve the same purpose, because I can't figure out how to get a vinculum to print here.
However, let me expand your mind a bit and introduce the *ternary* numbering system. Base 3, using only the digits 0, 1, and 2. In ternary, decimal number 10 (ten, the number of fingers you probably have) is written 101.
In ternary you can add 0.1 + 0.1 and get 0.2, which equals 2/3 in base-10.
In ternary, the base-10 value 1/3 is written 0.1. It has a finite representation without resorting to a vinculum or ellipses.
However, the base-10 value 1/4 is written 0.02020202(02)... in ternary. That is, an infinite string of 02 following zero and the decimal point.
Similarly, the base-10 value 1/2 is written 0.111... in ternary. In fact, the base-10 rational numbers 1/2 and 1/9, when expressed in ternary and decimal notation respectively, have exactly the same representation: 0.111...
Now I hope you're not going to argue that the validity of the operation "addition" is somehow dependent on the base of the numbering system you're using, because we all know that's not true. The fact is, decimal notation requires a notational shortcut to depict division by 3 while ternary doesn't because the 10 in base-10 has prime factors of 2, 2, and 5. That means any other prime factor in the denominator will result in a periodic representation. If you wanted to use a base-210 numbering scheme, 1/2, 1/3, 1/5, and 1/7 would all have non-periodic representations. In decimal, only 1/2 and 1/5 do -- 1/3 and 1/7 have periodic representations.
But the representation of a number in a given numbering system doesn't change what it *means*. 1/3 is still 0.333... in decimal and 0.1 in ternary. They're all equal, and they can be added together regardless of which base notation you're using.
Great explanation. Mmmm, higher math. I almost got a double major including a math degree but gave up after 3 weeks of Real Analysis.
Do you feel that they differ by something that is greater than zero? How much greater?Quote: AyecarumbaIn my base-10 mind, 0.999! is not equal to 1.000!
I took Real Analysis the last term before starting a job as a process engineer for a synthetic fiber manufacturer. At the end of the course, I commented to my professor that I hadn't yet figured out how the concepts of holes in the rational number line and the lack of holes in the real number line were ever going to help me make better nylon. He conceded that they likely would not.Quote: AyecarumbaI almost got a double major including a math degree but gave up after 3 weeks of Real Analysis.
Quote: MathExtremistso I think what's really going on here is that you don't believe the notation 0.111... is a valid representation of a number, and specifically that 0.111... is not a valid representation of 1/9. I disagree, of course, but I have no desire to argue definitions with anyone.
Carry on.
You are most correct in that assumption. I don't think .111... is equivalent to exactly 1/9 nor is .333... exactly equivalent to 1/3. Thank you for your analysis.
Quote: AyecarumbaGiven that, I can see where YDR11 is coming from. In my base-10 mind, 0.999! is not equal to 1.000!
Exactly my point. If you are going to utilize the definition of infinity for one side of the equation (.999...) you should be applying it to both sides.
MathExtremist is spot on with his earlier posts, especially with base 10 and ternary. It's just that, I don't change the base number system or what a number "means" I challenge the definition and application of infinity when applied to a string of numbers.
Quote: YoDiceRoll11Exactly my point. If you are going to utilize the definition of infinity for one side of the equation (.999...) you should be applying it to both sides.
MathExtremist is spot on with his earlier posts, especially with base 10 and ternary. It's just that, I don't change the base number system or what a number "means" I challenge the definition and application of infinity when applied to a string of numbers.
I think you're getting hung up on syntax. The concept of adding 1/3 to 1/3 and getting 2/3 is validly and equivalently expressed in ternary as 0.1 + 0.1 = 0.2 and decimal as 0.333... + 0.333... = 0.666... It makes no sense to suggest that one is a valid equation yet the other is not. Similarly, 0.1 + 0.1 + 0.1 in ternary is 1.0, while 0.333... + 0.333... + 0.333... in decimal is 0.999...
And since ternary 1.0 = decimal 1.0, then decimal 1.0 = decimal 0.999...
Q.E.D. again.
Edited to add -- unless you actually propose that any periodic notation is necessarily an invalid representation of a rational number, in which case we're back to arguing definitions. I think 0.1 in ternary and 0.333... in decimal represent the same value, just as I think 0.111... in ternary and 0.5 in decimal represent the same value. If you don't, that's a definitional dispute that precludes further analysis.
Quote: YoDiceRoll11Exactly my point. If you are going to utilize the definition of infinity for one side of the equation (.999...) you should be applying it to both sides.
I am applying it to both sides - one side has infinite number of nines, the other has an infinite number of zeroes.
Quote:MathExtremist is spot on with his earlier posts, especially with base 10 and ternary. It's just that, I don't change the base number system or what a number "means" I challenge the definition and application of infinity when applied to a string of numbers.
But that's not what ME said at all. The number that are expressed as finite decimals in one base, are infinite decimals in another. That means that there is nothing inherently "wrong" with an infinite decimal, it is as "good" as a finite one.
0.333... in base-10 is just 0.1 in ternary. Because they both refer to the same number, they must either both be "good" or both "bad". Because, you don't seem to think that there is anything "wrong" with 0.1, you have to admit, that , consequently, there is nothing wrong with 0.333... since it is by construction expressing the same exact number.
Quote: Ayecarumba
In my base-10 mind, 0.999! is not equal to 1.000!
Quote: DocDo you feel that they differ by something that is greater than zero? How much greater?
Yes. The difference cannot be represented since the string is infinite, but there is a difference.
Imagine that you are the librarian in a library with only two books. One is assigned the shelving number 0.9999..to a googolplex of 9999's. The other is assigned shelving number 1.0! (with ! representing an infinite repetition of "0's").
Someone then brings in a copy of "Gambling 102" which happens to have shelving number 0.9! (with ! representing an infinite repetition of "9's").
Where would you put this book?
If "the difference cannot be represented", how do you determine that it is different from 0 = 0.000... ? Or don't you think 0 = 0.000... ? If it's the latter, how many decimal places do you think zero can properly be extended to?Quote: AyecarumbaYes. The difference cannot be represented since the string is infinite, but there is a difference.
If by "googolplex" you mean the very large but finite number 10^(10^100), then there is a space between the shelving numbers for the original two books, so there is an appropriate space for some book. Unfortunately, the "Gambling 102" book with shelving number 0.999... or 0.9! has the exact same shelving number as the book identified as 1.0! , so those two belong in the same place. Perhaps the book noted as 1.0! is also "Gambling 102"; otherwise, two different books have been assigned the same shelving number. Place them in whichever order you like, unless there is some sequencing rule other than shelving number.Quote: AyecarumbaImagine that you are the librarian in a library with only two books. One is assigned the shelving number 0.9999..to a googolplex of 9999's. The other is assigned shelving number 1.0! (with ! representing an infinite repetition of "0's").
Someone then brings in a copy of "Gambling 102" which happens to have shelving number 0.9! (with ! representing an infinite repetition of "9's").
Where would you put this book?
If by "googolplex" you actually meant to suggest an infinite number of 9s, then all three books have the same shelving number -- put them in whatever order you like.
Quote: Ayecarumba
Imagine that you are the librarian in a library with only two books. One is assigned the shelving number 0.9999..to a googolplex of 9999's. The other is assigned shelving number 1.0! (with ! representing an infinite repetition of "0's").
Someone then brings in a copy of "Gambling 102" which happens to have shelving number 0.9! (with ! representing an infinite repetition of "9's").
Where would you put this book?
What if the only book in your library was assigned a number 0.5, and someone brought a book numbered 1/2?
Would you have the same problem then?
Quote: weaselman
Quote:MathExtremist is spot on with his earlier posts, especially with base 10 and ternary. It's just that, I don't change the base number system or what a number "means" I challenge the definition and application of infinity when applied to a string of numbers.
But that's not what ME said at all. The number that are expressed as finite decimals in one base, are infinite decimals in another. That means that there is nothing inherently "wrong" with an infinite decimal, it is as "good" as a finite one.
Using that definition, is fine. Like I said, it all has to do how you view infinity, and it's actual, implied, and practical application. Will you relax about this already?
However, as I said before, I liken 0.999... to the probability that a dart thrown between 0 and 10 would not hit pi. That probability of that is 1, but it could still happen. In other words 1 - 1/infinity = 1.
To me, it just goes to show that when trying to conceptualize infinity apparent paradoxes pop up and the discussion can always break down into silliness. In my opinion you just have to accept that 0.999...=1 on faith if you want to sleep better at night.
1/10 = .1
2/10 = .2
10/10 = 1
( where 10 = 1 three and 0 ones ). Get rid of the nines and you get rid of the cognitive dissonance.
Easy-peasy. Just don't let the fact that we have 10 fingers mess up your head.
Consider 1/3. That is the sum of the series 3/10^n, where n goes from 1 to infinity. In other words, 3/10 + 3/100 + 3/1000 + ...
I hope there's no argument about that.
Now, start to write down the terms of that series in decimal notation:
0.3
0.03
0.003
If you add them all up, you end up with 0.333... Ergo, 0.333... = 1/3.
Now, this only holds true if you accept what "limit" means, because otherwise you don't end up with 1/3 = SUM(3/10^n) for all n >= 1. Here's what Wikipedia has to say about mathematical series:
Quote: Mathematical Series
Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a recurring decimal, as in
we are talking, in fact, just about the series
But since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, it should offend no sensibilities if we make no distinction between 0.111… and 1/9. Less clear is the argument that 9 × 0.111… = 0.999… = 1, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more.
I just stumbled upon the page for 0.999... above, but it's got a lot of good stuff in it.
Edit: having read the page, it seems like the proponents of 0.999... != 1 also dispute that 0.999... (and presumably any other infinite decimal notation) is within the set of real numbers. Here's a good quote:
Quote:Although the real numbers form an extremely useful number system, the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999... = 1 is a convention as well:
"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."
For my part, I'm fine accepting that non-terminating repeating decimals are part of the real number system. If I did not, I would have to accept the nonsensical conclusion that converting a fractional value with a finite representation in a first base (e.g. the ternary representation of 1/3 as 0.1) to a second radix where it does not have a finite representation (e.g. the decimal representation of 1/3 as 0.333...) somehow kicks it out of the set of real numbers.
Quote: weaselmanWhat if the only book in your library was assigned a number 0.5, and someone brought a book numbered 1/2?
Would you have the same problem then?
Hehe, This actually is the core of the disagreement. If a book came into the library numbered, "1/2", it would be apparent that it belongs in a different branch of the system.
Quote: MathExtremistI just stumbled upon the page for 0.999... above, but it's got a lot of good stuff in it.
I didn't read all of it, but agree it is a good mathematical treatment of the concept. It shows several explanations why 0.9...=1. For what it is worth, I've never known anyone with a strong mathematical background to claim otherwise.
Here is the link again.
My favorite line?
Quote: WikipediaWith the rise of the Internet, debates about 0.999... have escaped the classroom and are commonplace on newsgroups and message boards...
Quote: WizardI didn't read all of it, but agree it is a good mathematical treatment of the concept. It shows several explanations why 0.9...=1. For what it is worth, I've never known anyone with a strong mathematical background to claim otherwise.
Here is the link again.
My favorite line?
Quote: wikipedia
With the rise of the Internet, debates about 0.999... have escaped the classroom and are commonplace on newsgroups and message boards...
What I find heartening about that is that people are interested in it. Whether it be this, the airplane/treadmill, whatever; people want to figure it out. The normal state is to want to know.
Quote: MoscaWhat I find heartening about that is that people are interested in it. Whether it be this, the airplane/treadmill, whatever; people want to figure it out. The normal state is to want to know.
I had passionate arguments about this as a kid. Whenever it came up the kids of Daisy Circle could argue about it for hours. I was always the only one to claim is was 1.
In math class I can recall the teacher trying to convince the class it was one, and everybody but me just shaking their heads in disbelief or confusion.
Go figure, I'm so calm now.
For example, 2 X ∞ = 3 X ∞ (now divide both sides by ∞):-
2 = 3
