0.999999999.... is equal to 1
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The voting period for this debate has ended.
after 6 votes the winner is...
PotBelliedGeek
Voting Style:  Open  Point System:  7 Point  
Started:  12/10/2013  Category:  Science  
Updated:  3 years ago  Status:  Post Voting Period  
Viewed:  807 times  Debate No:  42072 
Debate Rounds (2)
Comments (5)
Votes (6)
I am arguing that 0.9 repeating is equal to one. My opponent must argue against this. The opponent may use the first round to accept, and present his argument.


0.99999... = 1
(The ... means repeating) The proof for this is quite simple. If there is no difference between two numbers, then logically they are the same number. So, 1  0.999999.... = 0.000000... Hence, they are the same number. A second proof: 1/3 = 0.333333... 0.333333... x 3 = 0.99999... And (1/3) x 3 = 1 Therefore 0.99999... = 1 A third proof: Z = 0.999999... 10Z = 9.99999... 10Z  Z = 9.99999...  0.99999.. 9Z = 9 Z = 1 Therefore 0.99999... = 1 This is counterintuitive, but it is true. It has been proven mathematically (1) (2) If there is no difference between two numbers, they are logically, and mathematically the same number. 1. http://en.m.wikipedia.org...... 2. http://www.purplemath.com... I wish I had more than 2000 characters to address this issue, but I will make due. 1. "1  0.999999.... = 0.000000... Hence, they are the same number" Here, my opponent makes a simple mistake. 10.9 repeating does not equal zero. 10.9...=0.1... 2. 1/3 = 0.333333... 0.333333... x 3 = 0.99999... And (1/3) x 3 = 1 Therefore 0.99999... = 1 Here my opponent makes the assumption that 0.3... is equal to 1/3. My opponent fails to realise that this is an approximation, and is off by an infinitely small amount. Consider Zeno's Paradox[1]. 3.Z = 0.999999... 10Z = 9.99999... 10Z  Z = 9.99999...  0.99999.. 9Z = 9 Z = 1 Therefore 0.99999... = 1 Here my opponent makes a simple error in algebra. In order to multiply 0.9... by any number, one must first establish a constant number of Nines after the decimal. An infinite decimal cannot be multiplied by ten, because this simply makes the number of Nines after the decimal, which is infinity, ten times larger[2]. And so in order to multiply the number by ten one establishes a set number of nines after the decimal and the result is 9.999.....0. This value is not what my opponent included in his proof. and so: Z = 0.999999... 10Z = 9.99999... In this step my opponent added nine to the left side of the equation and multiplied by ten on the right. He has not done the same thing to both sides and therefore his algebra is off balance. Had my opponent done his algebra correctly he would have gotten as follows: z=0.999999... 10z=0.999999...0 This error threw off the algebra subsequent to it and produced the faulty result. Sources: 1. Stanford University http://plato.stanford.edu... 2. University of Utah http://www.math.utah.edu... 
6 votes have been placed for this debate. Showing 1 through 6 records.
Vote Placed by TheUser 3 years ago
The_Tom  PotBelliedGeek  Tied  

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Reasons for voting decision: 0.999999999 is not equal to 1. Con did good in proving that by using reliable sources and more convincing arguments.
Vote Placed by SeventhProfessor 3 years ago
The_Tom  PotBelliedGeek  Tied  

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Reasons for voting decision: The result could have been different ad there been more rounds, but PBG won this one.
Vote Placed by Yraelz 3 years ago
The_Tom  PotBelliedGeek  Tied  

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Reasons for voting decision: Zeno's paradox and Con's final argument regarding the 10x win this debate for him. The two numbers are different insofar as one is irrational and the other is rational. The addition of three 1/3rds is still an irrational approximation, just commonly rounded.
Vote Placed by noprisu 3 years ago
The_Tom  PotBelliedGeek  Tied  

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Reasons for voting decision: Math? Not sure what else to say besides a good reference to Zeno's Paradox.
Vote Placed by Nyx999 3 years ago
The_Tom  PotBelliedGeek  Tied  

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Reasons for voting decision: Intelligent debate. I like how PotBelliedGeek brought up Zeno's Paradox. Pro's sources were purplemath and wikipedia, and Con's sources were the University of Utah and Stanford University, so Con also had the better sources. (Wikipedia discredits even the best debater the slightest bit.)
Vote Placed by Mikal 3 years ago
The_Tom  PotBelliedGeek  Tied  

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Reasons for voting decision: Both cases ban be logically proven to be true using semantics. Con had a hard case but what was noticeable was that his sources were far more credible.
Hmmmmmmmmmmmmmm
Close, but no cigar m8!
Philosophically and theoretically it can get infinitely close, but never quite make it.
Though in Engineering, nobody is going to argue about 0.999999999999999 of an inch if you make it an inch.
But 0.999999999999999 if a light year, might still leave you a big leap to make near the other end