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 [quote="extropalopakettle"][quote="LordKinbote"][quote="jadesmar"] Obviously the bit that claims 1/3=0.333... Is just an approximation.[/quote] And it's off by...how much, exactly?[/quote] I would imagine taking 1/3 - 0.333... would, according to DM's theory, present the same problem as taking 1 - 0.999... The problem being, that while 0.999... is the largest number less than 1, and 0.333... is the largest number less than 1/3, when you try to do subtraction or other operations with these numbers, they cease to be numbers and transform into infinite strings (though somehow remaining comparable to ordinary numbers with "less than" and "greater than").[/quote]
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esme*
Posted: Fri Feb 15, 2013 9:24 am    Post subject: 1

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.

Ok, my self-control is wavering, so let me add to the conversation that the above is a valid equation in the 2-adics, since -1 is simply the unique Banach limit of this series.

Also, it is perfectly useful and meaningful to calculate with diverging series to get correct results for ordinary numbers in the end.

Since Euler, mathematicians have formalized these ideas, so phycisists had to pick up the slack on illegal calculations that give correct results.
Jedo the Jedi
Posted: Thu Feb 14, 2013 4:51 pm    Post subject: 0

extropalopakettle wrote:
(I carefully avoided attempting to divide the number 1-0.999... by anything to obtain X, as then it would use its superpowers to transform into an infinite string to protect itself from division)
Quote:
I'm picturing some xkcd style comic where 0.999... transforms back and forth between number and infinite string whenever threatened by mathematical operations.

Oh my goodness, these two lines caused me so much laughter. I'm still laughing as I type this. (I may be a bit stressed with school.)
extropalopakettle
Posted: Thu Feb 14, 2013 1:11 pm    Post subject: -1

I'm picturing some xkcd style comic where 0.999... transforms back and forth between number and infinite string whenever threatened by mathematical operations.
LordKinbote
Posted: Thu Feb 14, 2013 12:59 pm    Post subject: -2

extropalopakettle wrote:
Otherwise:

Let X = 1/3 - 0.333...

Then:
3X = 1 - 0.999 ...

0.999... + 3X = 1

Then:

0.999 ... < 0.999... + X < 0.999... + 3X = 1

Which contradicts that 0.999... is the largest number less than 1.

(I carefully avoided attempting to divide the number 1-0.999... by anything to obtain X, as then it would use its superpowers to transform into an infinite string to protect itself from division)

I'm glad you posted this because I was about to post pretty much the same thing in less formal terms: "If .333... is off by the smallest possible amount, then wouldn't .999 be off by the smallest possible amount which just happens to be three times larger than that other amount?"
extropalopakettle
Posted: Thu Feb 14, 2013 9:46 am    Post subject: -3

Otherwise:

Let X = 1/3 - 0.333...

Then:
3X = 1 - 0.999 ...

0.999... + 3X = 1

Then:

0.999 ... < 0.999... + X < 0.999... + 3X = 1

Which contradicts that 0.999... is the largest number less than 1.

(I carefully avoided attempting to divide the number 1-0.999... by anything to obtain X, as then it would use its superpowers to transform into an infinite string to protect itself from division)
extropalopakettle
Posted: Thu Feb 14, 2013 9:34 am    Post subject: -4

LordKinbote wrote:

Obviously the bit that claims
1/3=0.333... Is just an approximation.

And it's off by...how much, exactly?

I would imagine taking 1/3 - 0.333... would, according to DM's theory, present the same problem as taking 1 - 0.999...

The problem being, that while 0.999... is the largest number less than 1, and 0.333... is the largest number less than 1/3, when you try to do subtraction or other operations with these numbers, they cease to be numbers and transform into infinite strings (though somehow remaining comparable to ordinary numbers with "less than" and "greater than").
referee
Posted: Thu Feb 14, 2013 5:01 am    Post subject: -5

MNOWAX wrote:
Do the long division.

Understatement of History.
MNOWAX
Posted: Thu Feb 14, 2013 4:49 am    Post subject: -6

Obviously the bit that claims
1/3=0.333... Is just an approximation.

but...it's not. Go ahead, divide one into three. Do the long division. I'l wait. You'll eventually get bored too.
LordKinbote
Posted: Thu Feb 14, 2013 4:07 am    Post subject: -7

Obviously the bit that claims
1/3=0.333... Is just an approximation.

And it's off by...how much, exactly?
Posted: Thu Feb 14, 2013 3:34 am    Post subject: -8

MNOWAX wrote:
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.

Obviously the bit that claims
1/3=0.333... Is just an approximation.
MNOWAX
Posted: Thu Feb 14, 2013 3:24 am    Post subject: -9

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.
Amb
Posted: Thu Feb 14, 2013 3:12 am    Post subject: -10

Thank you, that explains it
referee
Posted: Thu Feb 14, 2013 3:06 am    Post subject: -11

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.
Thok
Posted: Thu Feb 14, 2013 2:54 am    Post subject: -12

Amb wrote:
Quote:

In base 3, one third can be represented as either 0.1 or 0.0222...

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....
The Potter
Posted: Thu Feb 14, 2013 2:53 am    Post subject: -13

Death Mage:

Is this an acceptable way of writing 0.999999999... ?
Amb
Posted: Thu Feb 14, 2013 2:51 am    Post subject: -14

Quote:

In base 3, one third can be represented as either 0.1 or 0.0222...

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)
extro...*
Posted: Thu Feb 14, 2013 2:44 am    Post subject: -15

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

Quote:
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:

Thok
Posted: Thu Feb 14, 2013 2:43 am    Post subject: -16

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.)
raekuul
Posted: Thu Feb 14, 2013 2:38 am    Post subject: -17

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?
Thok
Posted: Thu Feb 14, 2013 2:23 am    Post subject: -18

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...
LordKinbote
Posted: Thu Feb 14, 2013 2:19 am    Post subject: -19

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.
extro...*
Posted: Thu Feb 14, 2013 1:55 am    Post subject: -20

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.
Death Mage
Posted: Thu Feb 14, 2013 1:47 am    Post subject: -21

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.
Thok
Posted: Thu Feb 14, 2013 1:43 am    Post subject: -22

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.)
LordKinbote
Posted: Thu Feb 14, 2013 1:39 am    Post subject: -23

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...
extro...*
Posted: Thu Feb 14, 2013 1:38 am    Post subject: -24

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.
Thok
Posted: Thu Feb 14, 2013 1:36 am    Post subject: -25

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.
Death Mage
Posted: Thu Feb 14, 2013 1:15 am    Post subject: -26

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?
extro...*
Posted: Thu Feb 14, 2013 1:05 am    Post subject: -27

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
Amb
Posted: Thu Feb 14, 2013 12:50 am    Post subject: -28

Zag
Posted: Thu Feb 14, 2013 12:31 am    Post subject: -29

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.
LordKinbote
Posted: Thu Feb 14, 2013 12:17 am    Post subject: -30

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.

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.
Thok
Posted: Wed Feb 13, 2013 11:27 pm    Post subject: -31

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.
bgg1996
Posted: Wed Feb 13, 2013 10:20 pm    Post subject: -32

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.
Jedo the Jedi
Posted: Wed Feb 13, 2013 9:15 pm    Post subject: -33

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.
Amb
Posted: Wed Feb 13, 2013 8:36 pm    Post subject: -34

Please do. "0.999... of a Troll"
Zag
Posted: Wed Feb 13, 2013 7:53 pm    Post subject: -35

MNOWAX wrote:
I believe I haven't done enough laughing in this thread.

HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA
HAHAHAHAHAHAHA

MNO wins the GL troll award. I might have to change your title.
referee
Posted: Wed Feb 13, 2013 3:08 pm    Post subject: -36

In other words, what is the last digit of .999... ?
Answer: There is no such digit.
extro...*
Posted: Wed Feb 13, 2013 3:08 pm    Post subject: -37

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.
extro...*
Posted: Wed Feb 13, 2013 2:47 pm    Post subject: -38

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
.
.
.
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