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The Potter
Feat of Clay
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 Posted: Tue Feb 12, 2013 12:21 am Post subject: 41 |
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A long time ago, I would have attempted to argue for them being not equal to each other. Fortunately I came to understand the position mathematicians hold. Converging infinite sequences have a solution.
I do not completely understand infinity still. I also never have to work with math where it shows its ugly double head.
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The Potter
Feat of Clay
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| Posted: Tue Feb 12, 2013 12:23 am Post subject: 42 |
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And thank you MNOWAX for cutting away this garbage from the unsolved proof thread. It was your fault it started.
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LordKinbote
Daedalian Member
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| Posted: Tue Feb 12, 2013 2:12 am Post subject: 43 |
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| Death Mage wrote: |
Yes. Now you're learning. Infinity - infinity != 0 |
Who's subtracting infinity minus infinity? I see 9 - 9 an infinite number of times. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 2:37 am Post subject: 44 |
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Cat.
That's how you spell "cat". |
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LordKinbote
Daedalian Member
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| Posted: Tue Feb 12, 2013 3:06 am Post subject: 45 |
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Can I ask a serious question?
How egotistical do you have to be to declare "All mathematicians are wrong?"
I mean, there are really only three options here.
1) DM is wrong, and thinks so extremely highly of his own intelligence that he is unwilling to contemplate a world in which he is wrong.
2) DM is smarter than every one of the thousands of mathematicians who have contemplated this very problem. I mean, this isn't a global warming issue, where you can twist the evidence one way or the other, but that most people agree is one way. The matter is dead and buried.
3) DM is trolling us all. |
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Zag
Unintentionally offensive old coot
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| Posted: Tue Feb 12, 2013 3:35 am Post subject: 46 |
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| The Potter wrote: |
| And thank you MNOWAX for cutting away this garbage from the unsolved proof thread. It was your fault it started. |
'Twas I, actually; and you're welcome. |
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Zag
Unintentionally offensive old coot
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| Posted: Tue Feb 12, 2013 3:39 am Post subject: 47 |
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| Death Mage wrote: |
| Zag, your quote is right, but you're applying it to the wrong person. I'm not the one trying to apply infinity as a rational number. |
Well, I respect your right to stand alone, with all of the world's serious mathematicians standing against you.
Do you understand Xeno's paradox? Do you realize that, if 0.9999 doesn't equal 1, then Achilles can never catch the tortoise, and you can never walk all the way to the front door of your house? And yet, you do it every day. Perhaps there's a reason you're standing alone. |
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Death Mage
Raving Lunatic
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| Posted: Tue Feb 12, 2013 5:22 am Post subject: 48 |
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Still waiting for anyone to give a proof that doesn't rely on a rounding error/flaw in the decimal system, infinity - infinity, or a formula for estimations - not exact amounts.
Or anyone to answer the question of a representation of the largest possible value for x that fits the definition x < 1. Or an answer to 1-(1/infinity).
You can keep making the same tired assertions all you like, but they'll be just as wrong as the first time you said them. Trying to ad hominem me isn't going to make your position any stronger either.
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MNOWAX
0.999... of a Troll
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| Posted: Tue Feb 12, 2013 5:27 am Post subject: 49 |
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| Zag wrote: |
| The Potter wrote: |
| And thank you MNOWAX for cutting away this garbage from the unsolved proof thread. It was your fault it started. |
'Twas I, actually; and you're welcome. |
yeah I was going to say it wasn't me, but eventually I would have done it if Zag didn't.
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jadesmar
Bad Puppy
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| Posted: Tue Feb 12, 2013 5:30 am Post subject: 50 |
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| LordKinbote wrote: |
Can I ask a serious question?
How egotistical do you have to be to declare "All mathematicians are wrong?"
I mean, there are really only three options here.
1) DM is wrong, and thinks so extremely highly of his own intelligence that he is unwilling to contemplate a world in which he is wrong.
2) DM is smarter than every one of the thousands of mathematicians who have contemplated this very problem. I mean, this isn't a global warming issue, where you can twist the evidence one way or the other, but that most people agree is one way. The matter is dead and buried.
3) DM is trolling us all. |
Appeal to authority fallacy. This really doesn't count as a mathematical proof. |
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LordKinbote
Daedalian Member
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| Posted: Tue Feb 12, 2013 5:40 am Post subject: 51 |
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| jadesmar wrote: |
| Appeal to authority fallacy. This really doesn't count as a mathematical proof. |
No, you're right, of course it doesn't, and I realize that my exasperation reeks of the appeal to authority fallacy.
But I'm not using it in place of a mathematical proof. I'm using it when, in the presence of accepted, flawless mathematical proof, someone still says "Nope, I'm right, you're wrong."
In other words, I'm not saying "Thousands of experts believe it is so, therefore DM is wrong." I'm saying "It can be proven so, therefore DM is wrong, but it's annoying that he's declaring himself correct over not just a bunch of schmoes on the internet, but everyone in the field of mathematics." |
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LordKinbote
Daedalian Member
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| Posted: Tue Feb 12, 2013 5:49 am Post subject: 52 |
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| Death Mage wrote: |
| Still waiting for anyone to give a proof that doesn't rely on a rounding error/flaw in the decimal system, infinity - infinity, or a formula for estimations - not exact amounts. |
Can you please point out exactly where infinity - infinity is occurring?
9.999... - 0.999..., as I stated earlier, is not the same thing as infinity - infinity. The mere fact that we're using phrases like "infinity - infinity" as if "infinity" is a number at all shows that we're not really discussing this accurately. Hell, I don't even like using "infinity" as a noun when referring to anything mathematical because it obscures its meaning. Doing something an infinite number of times, or heading in the infinite direction feels more mathematically honest, since infinity is more a direction than a number. |
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jadesmar
Bad Puppy
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| Posted: Tue Feb 12, 2013 5:53 am Post subject: 53 |
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| Amb wrote: |
| 0.9999 stops at the planck length. After that, the digits are imaginary and cease to be useful. |
Can you tell me your interpretation of Zeno's paradox considering a discrete number line? With and without a discrete timeline would be preferred.
I think that the Plank length argument is identical to the 0.99999.. does not equal 1 argument. The number line is either continuous or discrete. If discrete, then there is strictly a maximum number greater than zero and less than one. |
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Jack_Ian
Big Endian
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| Posted: Tue Feb 12, 2013 10:58 am Post subject: 54 |
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| referee wrote: |
Assume they're different
1 != .999...
multiply by 10
10 != 9.999...
If we subtract the first line from the second
10 - 1 != 9.999... - .999...
9 != 9
Contradiction! |
If 1 != .999… then you can't subtract 1 from one side of an equation and 0.999… from the other and make assumptions about the answer.
e.g.
1 != 2
add 0 to both => 1 != 2
Subtract the first line from the second => 0 != 0 |
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Jack_Ian
Big Endian
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| Posted: Tue Feb 12, 2013 11:09 am Post subject: 55 |
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| LordKinbote wrote: |
Can I ask a serious question?
How egotistical do you have to be to declare "All mathematicians are wrong?" |
Well they were wrong about 1 being a prime number.  |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Tue Feb 12, 2013 12:02 pm Post subject: 56 |
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| Death Mage wrote: |
| Or anyone to answer the question of a representation of the largest possible value for x that fits the definition x < 1. |
You've already gotten two proofs that there isn't such a number as long as < respects addition and division. That proof doesn't exploit decimal representation, since it doesn't use decimal representation.
The fact that there is no largest number, is a feature not a bug. Some sets don't have a maximum element. (There is no largest rational number < 1, for example.) You haven't answered my question on why they should.
We're at the point where I'd like to know what you want a number system to be. That is:
1. What operations can you do with numbers?
2. What rules are there for when you can do those operations?
My answers are that for 1. I can try to compute a+b,a-b,a*b,a/b and get a number and I can determine if a<b is true or false. For 2. the only rule is that division by 0 is forbidden (well, and there's a bunch of rules that describe how the different operations play nicely with each other).
Last edited by Thok on Tue Feb 12, 2013 1:48 pm; edited 1 time in total |
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LordKinbote
Daedalian Member
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| Posted: Tue Feb 12, 2013 1:07 pm Post subject: 57 |
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| Jack_Ian wrote: |
| Well they were wrong about 1 being a prime number. |
They weren't wrong about 1 being a prime number. Their manufactured definition of primes required 1 to be excluded from too many theorems, so it was more convenient to leave 1 out of the definition than to keep excluding it from things where 1 was an exception.
And I suppose that we could think of the "1 = .999..." problem in a similar way, where it's untrue but only if you change the accepted definition of infinity and infinitesimality (?) but it would cause more things to break than it would fix. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 1:50 pm Post subject: 58 |
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| Death Mage wrote: |
| Still waiting for anyone to give a proof that doesn't rely on a rounding error/flaw in the decimal system, infinity - infinity, or a formula for estimations - not exact amounts. |
We're all waiting for an explanation of where you think there was a "rounding error" or what "flaw in the decimal system" there is. Nobody is rounding. Rounding introduces error. We round dollar amounts to the nearest penny, because we don't want to deal with fractions of pennies. There is no rounding here, hence no rounding error. You said earlier "you forgot to round". One doesn't round when one wants correct and accurate answers. One rounds to throw away some unwanted precision.
Nobody used infinity-infinity. 9.999... - 0.999... is not the sort of infinity-infinity problem that produces nonzero results. We're subtracting 9*10 -n from 9*10 -n , for all n>0, and the value is always zero (and n is always finite), and never involves an infinity - infinity. That leaves that 9.999... - 0.999... = 9.000... which is 9
And no formulas for estimations were used. The formulas for infinite sums are not for estimations, but for exact amounts. 1/2 + 1/4 + 1/8 + 1/16 + ... ... that infinite sum has an exact value, and it is 1. We calculate the sum of the infinite series as the limit of longer and longer finite sums, and we get an exact answer, not an estimate.
| Death Mage wrote: |
| Or anyone to answer the question of a representation of the largest possible value for x that fits the definition x < 1. |
There is no such number, and it's simple to prove that, as we did. Let X be the largest real number less than 1. (X+1)/2 is the greater than X and less than 1, contradicting that X is the largest real number less than 1. Given any two real numbers, one can always take their average to find a number midway between them. You said "no such number" is "not a valid answer", but it is trivial to come up with all sorts of supposed properties such that there is no number having such properties. Like, what is the smallest number X>0 such that 2*X=3*X. No such number. Etc.
| Death Mage wrote: |
| Or an answer to 1-(1/infinity). |
"infinity" is not a number. You may as well ask what 1-(1/beer) is. |
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jadesmar
Bad Puppy
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| Posted: Tue Feb 12, 2013 2:09 pm Post subject: 59 |
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| extro...* wrote: |
| Death Mage wrote: |
| Or anyone to answer the question of a representation of the largest possible value for x that fits the definition x < 1. |
There is no such number, and it's simple to prove that, as we did. Let X be the largest real number less than 1. (X+1)/2 is the greater than X and less than 1, contradicting that X is the largest real number less than 1. Given any two real numbers, one can always take their average to find a number midway between them. |
If the number line is discrete as claimed by DM and Amb, there is no guarantee that (X+1)/2 is a number. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 2:36 pm Post subject: 60 |
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| jadesmar wrote: |
| extro...* wrote: |
| Death Mage wrote: |
| Or anyone to answer the question of a representation of the largest possible value for x that fits the definition x < 1. |
There is no such number, and it's simple to prove that, as we did. Let X be the largest real number less than 1. (X+1)/2 is the greater than X and less than 1, contradicting that X is the largest real number less than 1. Given any two real numbers, one can always take their average to find a number midway between them. |
If the number line is discrete as claimed by DM and Amb, there is no guarantee that (X+1)/2 is a number. |
It isn't discrete. If it's discrete, we're dealing with integers (integer multiples of the smallest discrete unit). Let's just put everything to the left of the decimal point. No fractions. 1/3? No such number. |
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jadesmar
Bad Puppy
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| Posted: Tue Feb 12, 2013 3:01 pm Post subject: 61 |
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| extro...* wrote: |
| It isn't discrete. If it's discrete, we're dealing with integers (integer multiples of the smallest discrete unit). Let's just put everything to the left of the decimal point. No fractions. 1/3? No such number. |
Right, that's the argument as I see it. And, you have correctly outlined your side of it. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 3:36 pm Post subject: 62 |
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| jadesmar wrote: |
| If the number line is discrete as claimed by DM and Amb, there is no guarantee that (X+1)/2 is a number. |
I also don't think DM is claiming it's discrete.
Let X
1
= 0.5;
Now repeatedly do the following: Let X
i
= (X
i-1
+ 1) / 2
I don't think DM denies you can repeat that N times, for any finite value of N. Each time you get an X value closer to 1, still less than 1, but you never get an X value closest to 1, for which there is no closer value. What you get is X
N
= 1 - (1/2 N ), and that's true for every positive integer N. It's true for ALL positive integers, and that's the only place infinity comes in, but he doesn't deny an exception, some large but finite N for which it isn't true. He just claims that, in addition to the continuous rational number line, and in addition to the reals, there are these other "numbers", not rational, not real, and which you can't do math with like you can with other numbers. Numbers like his hypothetical "largest X<1", which is larger than ALL values of 1 - (1/2 N ) for a positive integer N. What makes them numbers, he hasn't explained. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 3:39 pm Post subject: 63 |
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| DM doesn't deny that for any two rational numbers A and B, that (A+B)/2 is a rational number between them. He's not denying the rational numbers are continuous. |
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jadesmar
Bad Puppy
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| Posted: Tue Feb 12, 2013 3:47 pm Post subject: 64 |
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| extro...* wrote: |
| Numbers like his hypothetical "largest X<1", which is larger than ALL values of 1 - (1/2 N ) for a positive integer N. What makes them numbers, he hasn't explained. |
Intuitively I feel that the existence of a largest x < 1 leads necessarily, perhaps by definition, to accepting a discrete number line, but I have no proof of that. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 5:32 pm Post subject: 65 |
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| jadesmar wrote: |
| extro...* wrote: |
| Numbers like his hypothetical "largest X<1", which is larger than ALL values of 1 - (1/2 N ) for a positive integer N. What makes them numbers, he hasn't explained. |
Intuitively I feel that the existence of a largest x < 1 leads necessarily, perhaps by definition, to accepting a discrete number line, but I have no proof of that. |
It makes no sense. If we accept a discrete number line in order to jam this alleged less-than-one value of 0.999... onto that line, then we have to reject 0.9...9 for some finite number of 9s. Or, this 0.999... is, like beer and infinity, not a number one can do arithmetic with. Of course, DM could argue "beer is not a number" and "largest x < 1 is not a number" are not valid statements. They're numbers, just not numbers you can do arithmetic with. |
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extro...*
Guest
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| Posted: Tue Feb 12, 2013 5:55 pm Post subject: 66 |
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| The funny thing is DM rejects that there's no such number as the largest rational (or real) number X<1, yet when asked what 1-X equals, the answer seems to be it's not a number. If 1-X=0, X=1. If 1-X is a number >0, then we can divide it by 2, add that to X, and have a number >X, <1. |
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referee
June 21st, 2004 Member
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| Posted: Tue Feb 12, 2013 7:15 pm Post subject: 67 |
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| Jack_Ian wrote: |
| referee wrote: |
Assume they're different
1 != .999...
multiply by 10
10 != 9.999...
If we subtract the first line from the second
10 - 1 != 9.999... - .999...
9 != 9
Contradiction! |
If 1 != .999… then you can't subtract 1 from one side of an equation and 0.999… from the other and make assumptions about the answer.
e.g.
1 != 2
add 0 to both => 1 != 2
Subtract the first line from the second => 0 != 0 |
If you read the whole thing, it explains that in this case you can do it, because you are still working with multiples of the first line,
Assume X != Y
Times 10, 10 X != 10 Y
Subtract, 9 X ! = 9 Y.
But in this case, 9 X = 9, and 9 Y = 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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The Potter
Feat of Clay
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| Posted: Tue Feb 12, 2013 8:12 pm Post subject: 68 |
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Assume X != Y
Times 10, 10 X != 10 Y
Subtract, 9 X != 9 Y. ------on the left side you are subtracting X. On the right side you are subtracting Y. You specifically said X != Y. Therefore you just subtracted different amounts from each side.
The idea of working with multiples of the line would only be valid if X=Y.
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Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy. |
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The Potter
Feat of Clay
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| Posted: Tue Feb 12, 2013 8:20 pm Post subject: 69 |
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Bad Math
1 ≠ 2 (starting inequality)
1 ≠ 4 (square line 1)
2 ≠ 4 (2* line 1)
1 ≠ 0 (line 3 - line 2)
2 ≠ 2 (line 1 + line 4)
I love how wrong this is 
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referee
June 21st, 2004 Member
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| Posted: Tue Feb 12, 2013 8:25 pm Post subject: 70 |
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X != Y |X-Y| = d > 0
10 X != 10 Y |10X - 10Y| = 10d > 0
9 X != 9 Y |9X - 9Y| = 10d - d = 9d > 0
It still works, no matter how many bad math examples you give.
The problem with your example is with the addition/subtraction. Line 4 in your last example, making Line 5 wrong.
_________________
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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The Potter
Feat of Clay
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| Posted: Tue Feb 12, 2013 8:46 pm Post subject: 71 |
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You can get to your third line by multiplying by 9 but not by "subtracting line 1 from line 2". This operation, as you noticed in my example, is invalid.
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Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy. |
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referee
June 21st, 2004 Member
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| Posted: Tue Feb 12, 2013 8:48 pm Post subject: 72 |
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Are you even trying? Multiplying by 10 and subtracting is EXACTLY the same thing as multiplying by 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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The Potter
Feat of Clay
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| Posted: Tue Feb 12, 2013 9:22 pm Post subject: 73 |
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The end result may be the same but subtracting an inequality from another inequality doesn't prove what you want.
_________________
Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy. |
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Chuck
Daedalian Member
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| Posted: Tue Feb 12, 2013 10:33 pm Post subject: 74 |
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if 1 ≠ 0.999... then 1 = 0.999... + d, d ≠ 0
multiply both sides by 10
10 = 9.999... + 10d
subtract first equation from the second
9 = 9 + 9d
since d ≠ 0, 9 ≠ 9 |
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Death Mage
Raving Lunatic
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| Posted: Tue Feb 12, 2013 11:10 pm Post subject: 75 |
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You're still trying infinity - infinity.
.999... - .999... != 0. It is undefined.
_________________
* 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: Tue Feb 12, 2013 11:49 pm Post subject: 76 |
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| Death Mage wrote: |
You're still trying infinity - infinity.
.999... - .999... != 0. It is undefined. |
.999... is infinity? And less than 1? I'll say it again: There's no such number. |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Feb 13, 2013 12:23 am Post subject: 77 |
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| Death Mage wrote: |
You're still trying infinity - infinity.
.999... - .999... != 0. It is undefined. |
For what numbers is subtraction defined then? Is pi-pi also disallowed since pi has infinitely many digits? Is 1.000....-1.0000... disallowed, since there are infinitely many zeroes?
(The .999... = 1 people have a consistent rule for when subtraction is defined. It's unclear that you do.) |
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The Potter
Feat of Clay
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| Posted: Wed Feb 13, 2013 12:26 am Post subject: 78 |
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DM:
Is this an acceptable way of writing 0.999... for you?
_________________
Artwork | Fractals | Don't ignore your dreams; don't work too much; say what you think; cultivate friendships; be happy. |
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extro...*
Guest
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| Posted: Wed Feb 13, 2013 12:30 am Post subject: 79 |
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| Thok wrote: |
| Death Mage wrote: |
You're still trying infinity - infinity.
.999... - .999... != 0. It is undefined. |
For what numbers is subtraction defined then? Is pi-pi also disallowed since pi has infinitely many digits? Is 1.000....-1.0000... disallowed, since there are infinitely many zeroes?
(The .999... = 1 people have a consistent rule for when subtraction is defined. It's unclear that you do.) |
Furthermore, DM says 0.999... is the largest number less than 1. That it is a number (unlike infinity). But you can't subtract this number from itself. |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Wed Feb 13, 2013 12:54 am Post subject: 80 |
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| extro...* wrote: |
| Furthermore, DM says 0.999... is the largest number less than 1. That it is a number (unlike infinity). But you can't subtract this number from itself. |
That also means his number line has a hole in it. I could make his number line bigger by defining a new number x and declaring that .999... < x <1.00..., and then extending addition as desired. |
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