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MNOWAX
0.999... of a Troll
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 Posted: Wed Feb 13, 2013 5:45 am Post subject: 81 |
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I believe I haven't done enough laughing in this thread.
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_________________
The Man The Myth The Legend
MNOWAX |
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Death Mage
Raving Lunatic
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| Posted: Wed Feb 13, 2013 6:48 am Post subject: 82 |
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Thok:
Alright, in .999... - .999... how many times do you have to subtract 9 from 9? What number represents it?
Let's think about this "proof", shall we?
Is it 1 nine?
.9 * 10 = 9
9 - .9 = 8.1
Nope.
Is it 2 nines?
.99 * 10 = 9.9
9.9 - .99 = 8.91
Nope.
Is it 3 nines?
.999 * 10 = 9.99
9.99 - .999 = 8.991
Nope.
I'm seeing a pattern developing here.
But you claim that there IS a number of 9s that, when applied to this formula, somehow manages to come out evenly. What is this magical number of 9s that can be altered as above yet still come out exactly as 0?
_________________
* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™.
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Last edited by Death Mage on Wed Feb 13, 2013 7:01 am; edited 1 time in total |
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referee
June 21st, 2004 Member
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| Posted: Wed Feb 13, 2013 6:58 am Post subject: 83 |
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There is no such number.
_________________
Jan 21st, 2008: The pillaging continues.
Mar 4th, 2008: Rest in Peace, Gary Gygax. May your dice always roll a natural 20 wherever you are.
Be the Ultimate Ninja! Play Billy Vs. SNAKEMAN today! |
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Nsof
Daedalian Member
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| Posted: Wed Feb 13, 2013 8:40 am Post subject: 84 |
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0.999... = 0.9+0.09+0.009+... is (by definition) sum over all i>=1 of 9*10^(-i)
One does not say an infinite sum is equal to something else as the '+' operator is not well defined over infinite summation. Instead the infinite sum can have a limit/converge to something: a number, infinity (or it does not converge at all).
So how do we figure 1 = 0.999... when the right hand side cannot be handled as we usually do with "normal" numbers.
By mathematical definition two numbers are equal if
a) the difference between them can be as small as we want it. AND
b) once the difference is as small as we wanted - it does stays small.
See http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit for the actual mathematical definition of the above.
(BTW, the delta epsilon notation should go in the "Most beautiful mathematical definitions thread".)
Once you accept that definition then to show that 1 and 0.99... satisfy these two conditions is easy and therefore we say they are the same.
You don't have to accept the definition but then you will be left without many branches of mathematics.
| Death Mage wrote: |
| But you claim that there IS a number of 9s that, when applied to this formula, somehow manages to come out evenly. What is this magical number of 9s that can be altered as above yet still come out exactly as 0? |
There no such magic number. The definition above only requires that you can get arbitrarily close (and stay there). As you have seen yourself you can easily show that the two numbers keep getting closer and closer as much as you want. Its also pretty easy to explain why they will not "grow" apart.
_________________
Will sell this place for beer
Last edited by Nsof on Fri Feb 15, 2013 10:00 pm; edited 1 time in total |
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Death Mage
Raving Lunatic
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| Posted: Wed Feb 13, 2013 12:18 pm Post subject: 85 |
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So, you're saying the numbers aren't actual equal, there's just close enough that you don't care if they aren't anymore.
_________________
* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™.
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Feb 13, 2013 12:29 pm Post subject: 86 |
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| Death Mage wrote: |
| Alright, in .999... - .999... how many times do you have to subtract 9 from 9? What number represents it? |
Obviously, you aren't going to have an integer number of 9's. Which makes it silly to try to subtract them one at a time.
Are you familiar with the concept of a parallel subtractor? You can subtract all of the digits at once, and then resolve all of the borrows. Now you don't have to worry about a single subtraction taking infinitely long, since you can just parallelize the work.
(This is the same as the idea of a parallel adder, where you resolve all of the digit sums at once, and then resolve all of the carries.) |
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extro...*
Guest
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| Posted: Wed Feb 13, 2013 1:13 pm Post subject: 87 |
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| Death Mage wrote: |
| Alright, in .999... - .999... how many times do you have to subtract 9 from 9? |
You only need to do it once, see the answer is 0, and know it will be 0 for ALL OF THEM.
| Death Mage wrote: |
| What number represents it? |
No number.
| Death Mage wrote: |
Let's think about this "proof", shall we?
Is it 1 nine?
.9 * 10 = 9
9 - .9 = 8.1
Nope.
Is it 2 nines?
.99 * 10 = 9.9
9.9 - .99 = 8.91
Nope.
Is it 3 nines?
.999 * 10 = 9.99
9.99 - .999 = 8.991
Nope.
I'm seeing a pattern developing here.
But you claim that there IS a number of 9s that, when applied to this formula, somehow manages to come out evenly. What is this magical number of 9s that can be altered as above yet still come out exactly as 0? |
NOBODY claimed "that there IS a number of 9s that, when applied to this formula, somehow manages to come out evenly." ALL OF THEM. And "ALL OF THEM" is not a number. Nobody suggested there was such a number.
YOU suggested that 0.999... is the largest NUMBER less than 1.
YOU suggested that 0.999... - 0.999... != 0. Yes, you can subtract a number from a number. If the difference isn't zero, the numbers must be different. If decimal representations represent different numbers, they must be different representations, and they must differ at SOME PARTICULAR DIGIT. At what digit does 0.999... differ from 0.999...??? |
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Death Mage
Raving Lunatic
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| Posted: Wed Feb 13, 2013 1:34 pm Post subject: 88 |
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You're still not getting it? Let's keep trying then.
Is it 4 nines?
.9999 * 10 = 9.999
9.999 - .9999 = 8.9991
Nope, not 4.
Is it 5 nines?
.99999 * 10 = 9.9999
9.9999 - .99999 = 8.99991
Nope, not 5.
Yet you continue to assert that there is some value of n nines where the above pattern stops holding true. That you can freely move a decimal place, and somehow it'll add a spot that makes it equal to exactly zero.
Is in n nines?
.999(n) * 10 = 9.99(n)
9.99(n) - 9.99(n) = 8.999(n-1)1
Nope, it doesn't work. It never works. No matter how long the string is, you can't manipulate it that way and come out with an exact 0.
_________________
* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™.
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Feb 13, 2013 1:50 pm Post subject: 89 |
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| Death Mage wrote: |
| Yet you continue to assert that there is some value of n nines where the above pattern stops holding true. That you can freely move a decimal place, and somehow it'll add a spot that makes it equal to exactly zero. |
Why does what happen with the subtraction pattern of a finite number of 9's have anything to do with what happens when you have 9's in every single decimal place? And notice the numbers of getting closer and closer to 9 at each step, since the error is .0...090..... for an increasing number of zeroes.
Give me a rule for when you can subtract decimal numbers. You've been giving me reasons why subtraction doesn't meet your intuition, which just shows me the flaws in your intuition. I will keep repeating this question until you answer it. |
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Jedo the Jedi
Paragon in Training
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| Posted: Wed Feb 13, 2013 1:54 pm Post subject: 90 |
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| Wikipedia wrote: |
| In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, or as 0.9, , 0.(9)) denotes a real number that can be shown to be the number one. |
| Quote: |
| In other words, the symbols "0.999..." and "1" represent the same number. |
| Quote: |
| The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum. |
| Quote: |
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
- Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.
- Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".
- Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit. |
| Quote: |
| students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1 |
Apparently, none of you are using the same language, at least that's where I think the problem stems. As the article suggests, DM may never accept that 0.999...=1, but it still might help if you guys had established the same language.
I think the most important quote is the second one, though I may have misunderstood the disagreement.
_________________
Paragon Tally: 18 mafia, 3 SKs (1 twice), 1 cultist, numerous chat scum...and counting. |
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extro...*
Guest
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| Posted: Wed Feb 13, 2013 2:47 pm Post subject: 91 |
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| Death Mage wrote: |
| Yet you continue to assert that there is some value of n nines where the above pattern stops holding true. |
Nobody asserted it, nor assumed it without asserting it. We all know that's true for every finite n, but 0.999... does not have a finite number of digits. You keep saying someone asserted it, and we keep pointing out they didn't.
Lets be clear:
0.9 != 1
0.99 != 1
0.999 != 1
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0.9999999999999999 != 1
For any finite number of 9s, 0.999...999 != 1
0.999... repeats forever. It does not have a finite number of 9s.
0.999... = 1 |
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extro...*
Guest
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| Posted: Wed Feb 13, 2013 3:08 pm Post subject: 92 |
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| Jedo the Jedi wrote: |
Apparently, none of you are using the same language, at least that's where I think the problem stems. As the article suggests, DM may never accept that 0.999...=1, but it still might help if you guys had established the same language.
I think the most important quote is the second one, though I may have misunderstood the disagreement. |
I think all that's been pointed out to him.
For instance, regarding that there are different ways to represent the same value:
| extropalopakettle wrote: |
backfilling, a simple rational numbers like 1/3 can't be represented as finite decimal numbers in the same way 1/2, 1/5 and 1/8 can (0.5, 0.2 and 0.125 respectively). 1/3 can be represented as an infinite repeating decimal, 0.333..., where the ",,," denotes infinite repetition. However, this notation we use allows us to express some numbers in more than one way. While 1/3 can only be expressed as 0.333..., 1/2 can be expressed as 0.5 or as 0.4999...
Which rational numbers require "..." notation depends on the base of the number system being used. In base 3, "one third" is 0.1, while "one half" is 0.1111... (1/3 + 1/9 + 1/27 + 1/81 + ...). |
In base 3, one third can be represented as either 0.1 or 0.0222...
and regarding those who "think of the limit of a sequence as a kind of infinite process rather than a fixed value":
| extropalopakettle wrote: |
| Why is a sum of an infinite series (like 0.9 + 0.09 + 0.009 + 0.0009 + ...) not a "fully accurate representation"? It's fully accurate because we can take it and use a fixed set of rules to know its value. We have unambiguous methods to compute sums of infinite series (when those sums exist). The sum is defined to be the value of the limit. That's what we mean, and it's absolutely precise, not at all inaccurate. |
His biggest problem, I think, is insisting that "the largest real number that's less than one" must denote some number. He insists this number exists, and furthermore that when you subtract it from itself, the result is not zero. |
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referee
June 21st, 2004 Member
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| Posted: Wed Feb 13, 2013 3:08 pm Post subject: 93 |
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In other words, what is the last digit of .999... ?
Answer: There is no such digit.
_________________
Jan 21st, 2008: The pillaging continues.
Mar 4th, 2008: Rest in Peace, Gary Gygax. May your dice always roll a natural 20 wherever you are.
Be the Ultimate Ninja! Play Billy Vs. SNAKEMAN today! |
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Zag
Tired of his old title
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| Posted: Wed Feb 13, 2013 7:53 pm Post subject: 94 |
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| MNOWAX wrote: |
I believe I haven't done enough laughing in this thread.
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MNO wins the GL troll award. I might have to change your title. |
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Amb
Amb the Hitched.
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| Posted: Wed Feb 13, 2013 8:36 pm Post subject: 95 |
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| Please do. "0.999... of a Troll" |
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Jedo the Jedi
Paragon in Training
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| Posted: Wed Feb 13, 2013 9:15 pm Post subject: 96 |
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Fair enough, extro. I just thought I would post what seemed to be the sticking points in concise language (thank you, wikipedia), and then share the section which might explain why DM maintains his position. My goal was to aid communication and understanding as I was able. I figure with everything laid out like that, it is pointless to continue the conversation, but if you guys get off by beating your head against a wall, so be it.
_________________
Paragon Tally: 18 mafia, 3 SKs (1 twice), 1 cultist, numerous chat scum...and counting. |
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bgg1996
BeeGees are awesome!
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| Posted: Wed Feb 13, 2013 10:20 pm Post subject: 97 |
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| extro...* wrote: |
| Death Mage wrote: |
| Alright, in .999... - .999... how many times do you have to subtract 9 from 9? |
You only need to do it once, see the answer is 0, and know it will be 0 for ALL OF THEM. |
Oh cool, really?
I'm gonna use this awesome new proof method you just informed me of.
| Code: |
X = 1+2+4+8+16+32+64+128+...
2X = 2+4+8+16+32+64+128+...
2X - X = 2+4+8+16+32+64+128+... - (1+2+4+8+16+32+64+128+...)
0+2+4+8+16+32+64+128+...
- 1+2+4+8+16+32+64+128+...
= -1+0+0+0+0+0+0+0+0+0+...
X = -1+0+0+0+0+0+0+0+0+0+...
X = -1 |
I always secretly knew I could apply basic algebra to infinite series to get an answer that doesn't make much sense.
_________________
The one member below 18 |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Feb 13, 2013 11:27 pm Post subject: 98 |
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| bgg1996 wrote: |
| I always secretly knew I could apply basic algebra to infinite series to get an answer that doesn't make much sense. |
Actually, there are systems where that makes sense and is correct, namely the 2-adics, where that's the equivalent of 1=.999... Number theorists find that number system quite useful, but it doesn't work well with real analysis.
| Jedo the Jedi wrote: |
| Apparently, none of you are using the same language, at least that's where I think the problem stems. As the article suggests, DM may never accept that 0.999...=1, but it still might help if you guys had established the same language. |
There's a reason I keep asking Death Mage to provide explicit rules for when you can subtract numbers.
Also a proof that pi is rational, ala Death Mage:
3 = 3/1 is rational.
3.1=31/10 is rational.
3.14 =314/100 is rational.
3.141 = 3141/1000 is rational.
Clearly if I include any finite number of digits of pi, I write that number as a rational fraction (with a denominator that's a power of 10).
By the above pattern, pi must be rational, since otherwise when does it stop being rational?
If you don't like the use of pi (because you think it's a rational number), I can change the proof to any other number. I have relatively simple proofs that sort(2) and e are irrational that I can post in the other thread. |
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LordKinbote
Daedalian Member
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| Posted: Thu Feb 14, 2013 12:17 am Post subject: 99 |
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| Death Mage wrote: |
Yet you continue to assert that there is some value of n nines where the above pattern stops holding true. That you can freely move a decimal place, and somehow it'll add a spot that makes it equal to exactly zero.
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What if I told you I could convince you that you're wrong, and all it would take would be for you to spend one night in some hotel? I think they're full, but I bet I could convince management to make room for you. |
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Zag
Tired of his old title
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| Posted: Thu Feb 14, 2013 12:31 am Post subject: 100 |
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| Amb wrote: |
| Please do. "0.999... of a Troll" |
Om my God I wish I had thought of that myself. Sheer genius. Done, as you can see. |
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Amb
Amb the Hitched.
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| Posted: Thu Feb 14, 2013 12:50 am Post subject: 101 |
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extro...*
Guest
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| Posted: Thu Feb 14, 2013 1:05 am Post subject: 102 |
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| bgg1996 wrote: |
| extro...* wrote: |
| Death Mage wrote: |
| Alright, in .999... - .999... how many times do you have to subtract 9 from 9? |
You only need to do it once, see the answer is 0, and know it will be 0 for ALL OF THEM. |
Oh cool, really?
I'm gonna use this awesome new proof method you just informed me of.
| Code: |
X = 1+2+4+8+16+32+64+128+...
2X = 2+4+8+16+32+64+128+...
2X - X = 2+4+8+16+32+64+128+... - (1+2+4+8+16+32+64+128+...)
0+2+4+8+16+32+64+128+...
- 1+2+4+8+16+32+64+128+...
= -1+0+0+0+0+0+0+0+0+0+...
X = -1+0+0+0+0+0+0+0+0+0+...
X = -1 |
I always secretly knew I could apply basic algebra to infinite series to get an answer that doesn't make much sense. |
If you don't know what you're doing, sure. See here for further instruction:
http://www.greylabyrinth.com/discussion/viewtopic.php?p=487153#487153 |
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Death Mage
Raving Lunatic
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| Posted: Thu Feb 14, 2013 1:15 am Post subject: 103 |
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You guys are still having trouble conceptualizing the largest value of X where X is < 1. Yet you seem OK with visualizing the largest value of X. You can obviously understand the concept, but refuse to apply it elsewhere.
You also seem to have trouble by continuing to apply rules of rational numbers to irrational numbers. If you can agree that infinity - infinity != 0, why is it so hard to do the same to an infinite string?
_________________
* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™.
[YOUR AD HERE!] |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Thu Feb 14, 2013 1:36 am Post subject: 104 |
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| Death Mage wrote: |
| You also seem to have trouble by continuing to apply rules of rational numbers to irrational numbers. |
The real numbers are not disjoint from the rationals. The real numbers were developed to fill in the holes in the rational numbers, so there is a uniform way of working with them, rather than defining new symbols like sqrt(2) or pi or e each time a new irrational numbers is useful. Addition, subtraction, multiplication, and division still apply to the real numbers because we want to keep those properties around.
You have yet to show me you have anything resembling a consistent method for when you can apply very basic operations like subtractions to numbers. Until you at least try to do so, I will ignore your complaints that subtraction doesn't work that way, because you won't tell me how subtraction does work.
| Quote: |
| You guys are still having trouble conceptualizing the largest value of X where X is < 1. Yet you seem OK with visualizing the largest value of X. |
Neither a largest value of X nor a largest value of X<1 exist. (Assuming X is a real number.)
| Quote: |
| If you can agree that infinity - infinity != 0, why is it so hard to do the same to an infinite string? |
With infinity-infinity, your trying apply subtraction to two things that aren't numbers and subtraction wasn't built for; it's like asking what is apple - world peace. With the real numbers, the subtraction has been built into the construction. |
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extro...*
Guest
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| Posted: Thu Feb 14, 2013 1:38 am Post subject: 105 |
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| Death Mage wrote: |
| You guys are still having trouble conceptualizing the largest value of X where X is < 1. |
In what sense is it a number? You say you can't subtract X from X. If it were a number you could. Does this number X have a square root? If so, the square root must be greater than X (and still less than 1).
YOU can't conceptualize why it isn't a number.
| Death Mage wrote: |
| Yet you seem OK with visualizing the largest value of X. |
Largest value of X? I have no idea what you mean. Infinity? that's not a number.
| Death Mage wrote: |
| You also seem to have trouble by continuing to apply rules of rational numbers to irrational numbers. |
First, we're not dealing with any irrational numbers here.
Second, many of the same rules apply to irrationals as rationals. Such as, if A<B, then (A+B)/2 is greater than A and less then B. This is true whether rational or irrational.
| Death Mage wrote: |
| If you can agree that infinity - infinity != 0, why is it so hard to do the same to an infinite string? |
Infinity is not a number, that's why you can't subtract it from itself.
On the other hand, you said:
| Death Mage wrote: |
| "There is no such number" is not a valid answer for "what is the largest value of X where X < 1". |
You're claiming "the largest value of X where X < 1" is a number, NOT an infinite string. If "the largest value of X where X < 1" were a number, it would not be a string (finite or infinite) and it would not be "infinity". It would be a number. And you can subtract numbers from numbers. But it's not a number. Not everything is a number. |
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LordKinbote
Daedalian Member
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| Posted: Thu Feb 14, 2013 1:39 am Post subject: 106 |
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| Death Mage wrote: |
You guys are still having trouble conceptualizing the largest value of X where X is < 1. Yet you seem OK with visualizing the largest value of X. You can obviously understand the concept, but refuse to apply it elsewhere.
You also seem to have trouble by continuing to apply rules of rational numbers to irrational numbers. If you can agree that infinity - infinity != 0, why is it so hard to do the same to an infinite string? |
Because they're not analogous. Listen, instead of telling how we Just Don't Understand, answer the question of how 9.999... - .999... is different from 3.14159... - 3.14159... |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Thu Feb 14, 2013 1:43 am Post subject: 107 |
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| For what it's worth, Death Mage's instance that there are largest objects is starting to get close to Russell's paradox. (It's not quite there yet, but it's easy to move the conversation there.) |
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Death Mage
Raving Lunatic
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| Posted: Thu Feb 14, 2013 1:47 am Post subject: 108 |
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| LordKinbote wrote: |
| Listen, instead of telling how we Just Don't Understand, answer the question of how 9.999... - .999... is different from 3.14159... - 3.14159... |
Are you going to tell me that pi*10-pi=9? or 3?
You can see how changing the decimal place changes the numbers in pi, but you can't see it in .999... so you think it doesn't happen.
_________________
* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™.
[YOUR AD HERE!] |
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extro...*
Guest
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| Posted: Thu Feb 14, 2013 1:55 am Post subject: 109 |
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For any real number X (including any rational number X, which need not be said), X-X = 0
For any two real numbers X and Y, if X<Y, then (X{+Y)/2 is greater than X and less than Y.
There is no largest number X less than 1, quite simply because for ANY X less than 1, (X+1)/2 is greater than X and still less than 1. |
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LordKinbote
Daedalian Member
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| Posted: Thu Feb 14, 2013 2:19 am Post subject: 110 |
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| Death Mage wrote: |
Are you going to tell me that pi*10-pi=9? or 3?
You can see how changing the decimal place changes the numbers in pi, but you can't see it in .999... so you think it doesn't happen. |
Okay, I walked into that one. No, I do not think that pi* 10 - pi = 9. It is of course 9*pi. And I submit that it's a number that exists, and if you subtracted all aligned digits infinitely, you'd find it.
I submit this as an exercise for the reader. Better start at the back to get the carries right. |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Thu Feb 14, 2013 2:23 am Post subject: 111 |
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| Death Mage wrote: |
| Are you going to tell me that pi*10-pi=9? or 3? |
Of course not. 10*pi-pi = 28.274333882...
Of course 10*pi-pi doesn't equal 9 because pi doesn't equal 1. In fact, the only decimal strings where 10*x-x=9 are 1.000.... and 0.999... |
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raekuul
Lives under a bridge & tells stories.
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| Posted: Thu Feb 14, 2013 2:38 am Post subject: 112 |
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| Thok wrote: |
| 10*pi-pi = 28.274333882... |
Okay, I'm a little confused. I always thought that the ellipses at the end implied a degree of repetition, which is clearly not the case where pi is involved. Or is 9pi suddenly rational? |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Thu Feb 14, 2013 2:43 am Post subject: 113 |
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| raekuul wrote: |
| Okay, I'm a little confused. I always thought that the ellipses at the end implied a degree of repetition, which is clearly not the case where pi is involved. Or is 9pi suddenly rational? |
I'm sure you've seen somebody write pi = 3.14159... before. That should answer your question.
(Ellipses just mean continuation, not repetition. You overline the last few digits if you need to emphasize that the pattern keeps repeating.) |
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extro...*
Guest
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| Posted: Thu Feb 14, 2013 2:44 am Post subject: 114 |
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| raekuul wrote: |
| Thok wrote: |
| 10*pi-pi = 28.274333882... |
Okay, I'm a little confused. I always thought that the ellipses at the end implied a degree of repetition, which is clearly not the case where pi is involved. Or is 9pi suddenly rational? |
http://en.wikipedia.org/wiki/Ellipsis#In_mathematical_notation
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Normally dots should be used only where the pattern to be followed is clear, the exception being to show the indefinite continuation of an irrational number such as:
 |
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Amb
Amb the Hitched.
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| Posted: Thu Feb 14, 2013 2:51 am Post subject: 115 |
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In base 3, one third can be represented as either 0.1 or 0.0222...
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Extro - mind expaining that to me? How can 1/3 in base three be represented in two ways? I'd have thought in base 3:
1/3 (decimal) = 1/10 (Trinary) = 0.1 (Trinary fraction) |
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The Potter
Feat of Clay
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| Posted: Thu Feb 14, 2013 2:53 am Post subject: 116 |
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Death Mage:
Is this an acceptable way of writing 0.999999999... ?
_________________
Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy. |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Thu Feb 14, 2013 2:54 am Post subject: 117 |
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| Amb wrote: |
| Quote: |
In base 3, one third can be represented as either 0.1 or 0.0222...
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Extro - mind expaining that to me? How can 1/3 in base three be represented in two ways? I'd have thought in base 3:
1/3 (decimal) = 1/10 (Trinary) = 0.1 (Trinary fraction) |
The other way is 2/9+2/27+2/81+2/243+....+2/3^n+...
That infinite series is a geometric series with ratio 1/3, so it converges to 2/9/(1-1/3)=1/3.
More generally, in base n, 0.1
n
= 0.0(n-1)(n-1)(n-1)....
n
by using the same geometric series argument. The base 10 version is 0.01 = 0.00999.... |
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referee
June 21st, 2004 Member
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| Posted: Thu Feb 14, 2013 3:06 am Post subject: 118 |
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Amb: Same reason you can say 1/10 in base ten is .0999... as well as .1
In fact, for any base n, 1/(n-1) = .111...
If we multiply that by (n-1) we get (assuming m is the symbol for n-1) .mmm... which is equal to 1. In our regular base 10, that m is 9.
_________________
Jan 21st, 2008: The pillaging continues.
Mar 4th, 2008: Rest in Peace, Gary Gygax. May your dice always roll a natural 20 wherever you are.
Be the Ultimate Ninja! Play Billy Vs. SNAKEMAN today! |
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Amb
Amb the Hitched.
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| Posted: Thu Feb 14, 2013 3:12 am Post subject: 119 |
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Thank you, that explains it  |
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MNOWAX
0.999... of a Troll
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| Posted: Thu Feb 14, 2013 3:24 am Post subject: 120 |
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I will do one thing to add to this conversation, since I was the .99999... of the troll that I am.
1/3*3=1
1/3=.33333....
.33333...* 3 = 1
.99999...=1
Disprove that.
_________________
The Man The Myth The Legend
MNOWAX |
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