0.9... = 1

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Ubuntu

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0.9... = 1
« on: November 06, 2006, 12:59:17 PM »
0.9... = 1

1/3 = 0.3...

+

2/3 = 0.6...
__________

3/3 = 0.9...

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cadmium_blimp

  • 1499
  • funny, you thought I'd convert, didn't you?
0.9... = 1
« Reply #1 on: November 06, 2006, 01:05:11 PM »
Old news.  Ha, a little trivia, though, I was about to tell you that 9.1... does not equal one.  You made a typo.

Quote from: Commander Taggart
Never give up, never surrender!

0.9... = 1
« Reply #2 on: November 06, 2006, 01:13:27 PM »
1 = 1

0.9... = 0.9...
ooyakasha!

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cadmium_blimp

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0.9... = 1
« Reply #3 on: November 06, 2006, 01:39:48 PM »
You haven't gotten very far in your math studies yet, have you Knight?

http://en.wikipedia.org/wiki/0.999...

Quote from: Commander Taggart
Never give up, never surrender!

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Ubuntu

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0.9... = 1
« Reply #4 on: November 06, 2006, 02:24:48 PM »
Quote from: "cadmium_blimp"
You haven't gotten very far in your math studies yet, have you Knight?

http://en.wikipedia.org/wiki/0.999...


Just in case, Knight, you want to barrage us with arguments, they've all been tried before here:

http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html

0.9... = 1
« Reply #5 on: November 06, 2006, 02:34:08 PM »
I didn't barrage you with any arguments, Ubuntu.  Also, I'm above average in mathematics, cadmium_blimp.  What did I say to generate these responses?  Did I err in my comment anywhere?

Quote from: "I"
1 = 1

0.9... = 0.9...


Let me know what is wrong with my reasoning here.  I'm quite confused.
ooyakasha!

0.9... = 1
« Reply #6 on: November 06, 2006, 02:36:17 PM »
lol reminds me of my Alg 2 days, just to piss my teacher off, .999...8 that way it was not 1
he man in black fled across the desert, and the gunslinger followed.

Advocatus Diaboli

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GeoGuy

Re: 0.9... = 1
« Reply #7 on: November 06, 2006, 03:00:41 PM »
Quote from: "Ubuntu"
0.9... = 1

1/3 = 0.3...

+

2/3 = 0.6...
__________

3/3 = 0.9...


The problem here is that since 1/3+2/3=1, and .333...=1/3, .333...+.666... must also equal 1.

0.9... = 1
« Reply #8 on: November 06, 2006, 03:00:55 PM »
that is why you use the infinite geometric serier

9(1/10^1)+9(1/10^2)+9(1/10^3)+.....

=(9*1/10)/(1-1/10)

this of course, equals 1

go Algebra
he man in black fled across the desert, and the gunslinger followed.

Advocatus Diaboli

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Ubuntu

  • 2392
Re: 0.9... = 1
« Reply #9 on: November 06, 2006, 03:29:54 PM »
Quote from: "GeoGuy"
Quote from: "Ubuntu"
0.9... = 1

1/3 = 0.3...

+

2/3 = 0.6...
__________

3/3 = 0.9...


The problem here is that since 1/3+2/3=1, and .333...=1/3, .333...+.666... must also equal 1.


THAT'S THE PROOF.


More! :D






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Ubuntu

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0.9... = 1
« Reply #10 on: November 06, 2006, 03:38:04 PM »
Quote from: "GeoGuy"
But the proof is no good, it's just taking an equality and assuming that two numbers add up to something they don't.


I think you misunderstand. 1/3 + 2/3 = 1 and 1/3 + 2/3 = 0.9... Therefore, 1 = 0.9...

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BOGWarrior89

  • 3793
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0.9... = 1
« Reply #11 on: November 06, 2006, 03:38:28 PM »
Fucktards, that just shows that something is lost in translation from fractions to decimals.

0.9... = 1
« Reply #12 on: November 06, 2006, 03:41:09 PM »
http://en.wikipedia.org/wiki/Invalid_proof

0.9 repeating = 1 is covered on this page

other things covered:

-1 = 1

the above with alternate method

1 < 0

2 = 1

the above with alternate method

xER = 0 (xER = all real numbers)

a = b

0 = 1

the above with alternate method

i^2 = 1 (i is the square root of -1)

4 = 5

Any angle = 0

all triangles = equilateral

infinity = 1/4

;)
RE*
Try not to be -too- much of an idiot. Or I'll rape you verbally.

1 out of 9 members on this forum that can spell properly.

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Ubuntu

  • 2392
0.9... = 1
« Reply #13 on: November 06, 2006, 03:55:30 PM »
Quote from: "mattz1010"
http://en.wikipedia.org/wiki/Invalid_proof

0.9 repeating = 1 is covered on this page


Actually, no, it isn't. Unless it's hidden somewhere, behind the wiki logo perhaps...

Quote from: "BOGWarrior89"
Fucktards, that just shows that something is lost in translation from fractions to decimals.


Ignorance. Consult the Wiki article and Polymathematics Blog for further details.

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BOGWarrior89

  • 3793
  • We are as one.
0.9... = 1
« Reply #14 on: November 06, 2006, 04:03:28 PM »
Quote from: "Ubuntu"
Quote from: "mattz1010"
http://en.wikipedia.org/wiki/Invalid_proof

0.9 repeating = 1 is covered on this page


Actually, no, it isn't. Unless it's hidden somewhere, behind the wiki logo perhaps...

Quote from: "BOGWarrior89"
Fucktards, that just shows that something is lost in translation from fractions to decimals.


Ignorance. Consult the Wiki article and Polymathematics Blog for further details.


Ignorance is relative.

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Ubuntu

  • 2392
0.9... = 1
« Reply #15 on: November 06, 2006, 04:11:37 PM »
Quote from: "BOGWarrior89"
Quote from: "Ubuntu"
Quote from: "mattz1010"
http://en.wikipedia.org/wiki/Invalid_proof

0.9 repeating = 1 is covered on this page


Actually, no, it isn't. Unless it's hidden somewhere, behind the wiki logo perhaps...

Quote from: "BOGWarrior89"
Fucktards, that just shows that something is lost in translation from fractions to decimals.


Ignorance. Consult the Wiki article and Polymathematics Blog for further details.


Ignorance is relative.


Now you are dancing back into the shadows, not even on topic... mathematical proofs, whereas they are certain, are some of the least relative ideas possible.

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BOGWarrior89

  • 3793
  • We are as one.
0.9... = 1
« Reply #16 on: November 06, 2006, 04:22:58 PM »
Quote from: "Ubuntu"
Quote from: "BOGWarrior89"
Quote from: "Ubuntu"
Quote from: "mattz1010"
http://en.wikipedia.org/wiki/Invalid_proof

0.9 repeating = 1 is covered on this page


Actually, no, it isn't. Unless it's hidden somewhere, behind the wiki logo perhaps...

Quote from: "BOGWarrior89"
Fucktards, that just shows that something is lost in translation from fractions to decimals.


Ignorance. Consult the Wiki article and Polymathematics Blog for further details.


Ignorance is relative.


Now you are dancing back into the shadows, not even on topic... mathematical proofs, whereas they are certain, are some of the least relative ideas possible.


A decimal cannot fully represent a fraction.  A fraction is base x, where x is the number in the denominator.  Decimals are base ten.

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Erasmus

  • The Elder Ones
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0.9... = 1
« Reply #17 on: November 06, 2006, 05:00:00 PM »
Quote from: "BOGWarrior89"
A decimal cannot fully represent a fraction.


Sure it can... especially if the decimal is infinite.

Quote
A fraction is base x, where x is the number in the denominator.


This is probably an unconventional use of the word "base".
Why did the chicken cross the Möbius strip?

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EnragedPenguin

  • The Elder Ones
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0.9... = 1
« Reply #18 on: November 06, 2006, 05:02:28 PM »
Quote from: "BOGWarrior89"
A decimal cannot fully represent a fraction.  A fraction is base x, where x is the number in the denominator.  Decimals are base ten.


You don't have to use the fraction proof.

c=0.999...
10c=9.999...
10c-c=9.999... -0.999...
9c=9
c=1

And there you go. 0.999... equals one.
A different world cannot be built by indifferent people.

0.9... = 1
« Reply #19 on: November 06, 2006, 07:16:41 PM »
Well, this topic was boring so I came up with my own mathematic proof.

Multiply two integers, subtract the sum of the product's digits from it, and it results in a number divisable by 9.

I am a GENIUS[/u][/i].
ttp://theflatearthsociety.org/forums/search.php

"Against criticism a man can neither protest nor defend himself; he must act in spite of it, and then it will gradually yield to him." -Johann Wolfgang von Goethe

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BOGWarrior89

  • 3793
  • We are as one.
0.9... = 1
« Reply #20 on: November 06, 2006, 07:41:43 PM »
Quote from: "EnragedPenguin"
Quote from: "BOGWarrior89"
A decimal cannot fully represent a fraction.  A fraction is base x, where x is the number in the denominator.  Decimals are base ten.


You don't have to use the fraction proof.

c=0.999...
10c=9.999...
10c-c=9.999... -0.999...
9c=9
c=1

And there you go. 0.999... equals one.


I'm telling you that the decimal proof is invalid.

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beast

  • 2997
0.9... = 1
« Reply #21 on: November 06, 2006, 07:49:10 PM »
In my maths extended class in grade 8 or 9 I remember the teacher getting me to prove the same thing - that 0.9999... = 1.  I don't know remember what proof I used or even if my proof was right but I'm fairly confident it's true.  - although it's definitely conceivable my teacher would ask me to prove something that is actually false.

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Erasmus

  • The Elder Ones
  • 4242
0.9... = 1
« Reply #22 on: November 07, 2006, 12:30:01 AM »
Quote from: "BOGWarrior89"
I'm telling you that the decimal proof is invalid.


Luckily for him, nobody is going to believe you unless you can explain why it's invalid.

On the other hand, I can explain why it's valid:

Let dec(f) = sum( f(n) * 10^n, n=-infinity..infinity ), where f is a function mapping integers to the natural numbers { 0, ..., 9 }.  In other words, dec is a function that maps a sequence of decimal digits to a real number by assuming that the sequence is the decimal expansion of that number.  For example, suppose f(n) = 4 if n=1, 2 if n=0, and 0 otherwise: then f(n) can be written out as "...00042.000...", and dec(f) = 42.  Obviously, as long as there exists some N such that for n > N, f(n) = 0, dec(f) must converge.

Let s(n) = 9 if n < 0, and 0 otherwise.  EnragedPenguin's proof goes as follows:

Let c = dec(s).
10c = 10 dec(s)
. . . = 10 * sum( s(n) * 10^n, n = infinity...infinity )  
. . . = sum( 10 * s(n) * 10^n, n = -infinity...infinity )
. . . = sum( 10 * s(n) * 10^n, n = -infinity...-1 ) (since for n >= 0, s(n)=0)
. . . = sum( s(n) * 10^(n+1), n = -infinity...-1 )
. . . = s(-1) * 10^(-1+1) + sum(s(n) * 10^(n+1), n = -infinity...-2 )
. . . = s(-1) * 10^(-1+1) + sum(s(n-1) * 10^n, n = -infinity...-1 )
. . . = s(-1) * 10^(-1+1) + sum(s(n) * 10^n, n = -infinity...-1 ) (since for n <= -1, s(n-1) = s(n))
. . . = s(-1) * 10^(-1+1) + dec(s)
. . . = 9 + c
10c - c = (9 + c) - c = 9.
9c = 9  =>  c = 1.

Therefore dec(s) = 1, so we have proved that the number represented by the infinite decimal string s(n) is the number 1.  If you found my proof somewhat clunky, you'll realize why we use representations like "0.999...".  They don't actually discard any information, since they refer to something abstract.

So basically you just have to remember that when you write a number in "decimal form", you are really talking about an infinite series, but simply use a much more commodious notation.
Why did the chicken cross the Möbius strip?

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BOGWarrior89

  • 3793
  • We are as one.
0.9... = 1
« Reply #23 on: November 07, 2006, 09:11:35 AM »
Quote from: "Erasmus"
Quote from: "BOGWarrior89"
I'm telling you that the decimal proof is invalid.


Luckily for him, nobody is going to believe you unless you can explain why it's invalid.

On the other hand, I can explain why it's valid:

Let dec(f) = sum( f(n) * 10^n, n=-infinity..infinity ), where f is a function mapping integers to the natural numbers { 0, ..., 9 }.  In other words, dec is a function that maps a sequence of decimal digits to a real number by assuming that the sequence is the decimal expansion of that number.  For example, suppose f(n) = 4 if n=1, 2 if n=0, and 0 otherwise: then f(n) can be written out as "...00042.000...", and dec(f) = 42.  Obviously, as long as there exists some N such that for n > N, f(n) = 0, dec(f) must converge.

Let s(n) = 9 if n < 0, and 0 otherwise.  EnragedPenguin's proof goes as follows:

Let c = dec(s).
10c = 10 dec(s)
. . . = 10 * sum( s(n) * 10^n, n = infinity...infinity )  
. . . = sum( 10 * s(n) * 10^n, n = -infinity...infinity )
. . . = sum( 10 * s(n) * 10^n, n = -infinity...-1 ) (since for n >= 0, s(n)=0)
. . . = sum( s(n) * 10^(n+1), n = -infinity...-1 )
. . . = s(-1) * 10^(-1+1) + sum(s(n) * 10^(n+1), n = -infinity...-2 )
. . . = s(-1) * 10^(-1+1) + sum(s(n-1) * 10^n, n = -infinity...-1 )
. . . = s(-1) * 10^(-1+1) + sum(s(n) * 10^n, n = -infinity...-1 ) (since for n <= -1, s(n-1) = s(n))
. . . = s(-1) * 10^(-1+1) + dec(s)
. . . = 9 + c
10c - c = (9 + c) - c = 9.
9c = 9  =>  c = 1.

Therefore dec(s) = 1, so we have proved that the number represented by the infinite decimal string s(n) is the number 1.  If you found my proof somewhat clunky, you'll realize why we use representations like "0.999...".  They don't actually discard any information, since they refer to something abstract.

So basically you just have to remember that when you write a number in "decimal form", you are really talking about an infinite series, but simply use a much more commodious notation.


Ok, now try multiplying 0.9999... by a number that isn't base ten (like 2, for instance).

0.9... = 1
« Reply #24 on: November 07, 2006, 09:25:32 AM »
Quote
Also, I'm above average in mathematics, cadmium_blimp.


So?  As am I, and I disagree with you.  Therefore, you are wrong.

1/9 = .11111....
2/9 = .22222....
3/9 = .33333....
...
...
9/9 = .99999....

So think of it as a limit.  What is the limit as the number of 9s goes to infinity? 9/9 = 1.

QED
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

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BOGWarrior89

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0.9... = 1
« Reply #25 on: November 07, 2006, 09:44:47 AM »
Quote from: "fathomak"

1/9 = .11111....
2/9 = .22222....
3/9 = .33333....
...
...
9/9 = .99999....

Wrong; 1/9 = 0.1111... + 1/(9 * infinity).  Multiply by nine, and you get that 1 = .99999... + 1/(infinity), which is equal to 1  (1/(infinity) = 1/10000...).

0.9... = 1
« Reply #26 on: November 07, 2006, 10:15:50 AM »
Quote
Wrong; 1/9 = 0.1111... + 1/(9 * infinity). Multiply by nine, and you get that 1 = .99999... + 1/(infinity), which is equal to 1 (1/(infinity) = 1/10000...).


...uh, no?  1 = .99999.... + 1/(infinity) is not equal to 1(1/(infinity)).  If we subtract .99999.... from both sides, all we've shown is 1 - .99999.... = 1/(infinity) = 0.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

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Nomad

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0.9... = 1
« Reply #27 on: November 07, 2006, 10:25:08 AM »
This thread is amazing.  And I used to think the Flat Earth Theory was silly.
Nomad is a superhero.

8/30 NEVAR FORGET

0.9... = 1
« Reply #28 on: November 07, 2006, 10:26:58 AM »
I think this is a better question:

Which is larger, the set of all positive real integers, or the set of all primes?
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

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BOGWarrior89

  • 3793
  • We are as one.
0.9... = 1
« Reply #29 on: November 07, 2006, 10:54:45 AM »
Quote from: "fathomak"
I think this is a better question:

Which is larger, the set of all positive real integers, or the set of all primes?


I know your hand-wavy explanation to counter my argument, but I'm going to say it anyway: the set of all positive real integers.