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# 0.999999999.... is equal to 1

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PotBelliedGeek
 Voting Style: Open Point System: 7 Point Started: 12/10/2013 Category: Science Updated: 5 years ago Status: Post Voting Period Viewed: 1,225 times Debate No: 42072
Debate Rounds (2)

 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.Report this 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...Report this Argument 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. 1-0.9 repeating does not equal zero. 1-0.9...=0.1... 2. 1/3 = 0.333333...0.333333... x 3 = 0.99999...And (1/3) x 3 = 1Therefore 0.99999... = 1Here 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.3.Z = 0.999999...10Z = 9.99999...10Z - Z = 9.99999... - 0.99999..9Z = 9Z = 1Therefore 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. 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...0This error threw off the algebra subsequent to it and produced the faulty result.Sources: 1. Stanford Universityhttp://plato.stanford.edu... 2. University of Utah http://www.math.utah.edu...Report this Argument 5 comments have been posted on this debate. Showing 1 through 5 records.
Posted by Sagey 5 years ago
0.999999999 equaling 1.
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
Posted by peach93 5 years ago
0.999999999... is not equal to 1 but it can be rounded to 1
Posted by themohawkninja 5 years ago