In reality, is .999 repeating actually equal to 1?
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Alexby1
Voting Style:  Open  Point System:  7 Point  
Started:  11/20/2009  Category:  Miscellaneous  
Updated:  7 years ago  Status:  Post Voting Period  
Viewed:  2,531 times  Debate No:  10190 
Debate Rounds (4)
Comments (3)
Votes (1)
Hello everyone.
I have been searching through the website's old debates and I had come upon multiple interesting mathematical debates covering this topic. However, they are long gone and the one I looked through had bad grammar, spelling, and behavior. Thus, I would like to redo this debate with a hopefully better behaved person. Thank you in advance for whoever accepts this debate, and I hope that you enjoy it. Please be aware that when I state a decimal with three numbers following the actual decimal point, I do mean a repeating decimal. (EX: .999 = .9 repeating)  = Round 1 =  The essential idea is that .999 is the same as 1.0. Now, when you think about this it is impossible for two different numbers to be equal to each other. Note that I say two DIFFERENT numbers. If 0.9 repeating was the same as 1.0 then all of these following equations would be true: .333 = 1/3 .666 = 2/3 .999 = 1 Now although it is common usage, 1/3, 2/3, and 1, are merely place holders, not ACTUAL EQUALITIES, for .333, .666, and .999 respectively.  = Round 1 [Con] End=  Yes, this may be a short start, but I do have much to back me up. I hope to have an interesting debate with someone. Thank you all. Atonement
I affirm, that "In reality, .999 repeating [is] actually equal to 1." Thanks to my opponent for the interesting debate. DEFINITIONS Like my opponent has said, for the purposes of this debate, .999 = .9 repeating, .333 = .3 repeating, and so on. ARGUMENT I plan to prove that .999 equals one through two algebraic proofs: 1) .333 = (1/3) 3 * .333 = 3 * (1/3) = (3*1 / 3) .999 = 1 (http://en.wikipedia.org......)  2) x = .999 10x = 9.999 10x  x = 9.999  .999 9x = 9 x = 1 .999 = 1 (http://en.wikipedia.org......)  Both of these proofs show conclusively that .999 is equal to 1. REFUTATIONS "Now, when you think about this it is impossible for two different numbers to be equal to each other. Note that I say two DIFFERENT numbers." .999 and 1 are not two different numbers: the above proofs show this idea to be false.  "If 0.9 repeating was the same as 1.0 then all of these following equations would be true: .333 = 1/3 .666 = 2/3 .999 = 1" Well, those equations are all true. (http://en.wikipedia.org...)  "Now although it is common usage, 1/3, 2/3, and 1, are merely place holders, not ACTUAL EQUALITIES, for .333, .666, and .999 respectively." (1/3) is equal to .333 . It is not simply common usage. CONCLUSION The two algebraic proofs I demonstrated both prove that .999 is, in reality, equal to 1. Until my opponent can find fault with both of these proofs and their logic, I urge you to vote PRO. 

Thank you, Alex, for the intriguing response. After reviewing your profile I see that you will be a good competitor, and I look forward to the next three rounds.
 = Round 2 =  =Preliminary= Though I am happy to see a competitor use sources, I would like to point how to the public that Wikipedia was used, and is not always the most reliable choice. However, after reviewing the specific topic, it seems reasonable. =Argument= "1) .333 = (1/3) 3 * .333 = 3 * (1/3) = (3*1 / 3) .999 = 1" I would honestly have to agree with this statement, with one mere exception. This assumes that .333 eventually ends, because .3 does, in fact equal 1/3. However, the final equation is the main defect. I will admit that once .999 terminates, it would equal one, when rounded up. However, the human mind does not easily understand the concept of infinite. Everything we have in life is limited by a finite number. We have everything measured or the ability to be measured, from the distance of one end of the universe to the other to anything as simple as the distance from your house to your place of work and/or school. == "2) x = .999 10x = 9.999 10x  x = 9.999  .999 9x = 9 x = 1 .999 = 1" This however, I cannot agree with. From my knowledge as an algebra student, I would be forced to assume that once you have a set number for a variable, you should use the aforementioned number. Thus: x = .999 10x = 9.999 10x  .999 = 9.999  After which you would continue by adding .999 to each side. 10x = 10.9998  Please note that I mean the .999 repeats, however when you decide to terminate the decimal, it ends in an eight. And upon dividing both sides by ten, you end with: x = 1.09998  Again, this is in coordinance with the previous statement that this decimal repeats until finally terminating with an eight. Please, if I am wrong, do correct me, but this is a simple counteraction against your statement. == "(1/3) is equal to .333 . It is not simply common usage." After simple calculations, you can find that .3 is equal to one third. However, the topic of argument is .999 equating to 1. Though, I would like to withdraw my statement with which I said that .333 ≠ 1/3. I do wish for my other two beliefs to stand. .666 is not equal to 2/3 .999 is not equal to 3/3 or 1. ==Rebuttal== As I do not entirely enjoy being on the defensive, I do have two quick statements for you to take your best shot at. If .999 = 1 Then .999+.999 = 1+1 However, .999+.999 = 1.9998 And 1 + 1 = 2 Finally, 1.9998 ≠ 2 Also, think about the following: If .999 = 1 Then .999 * .999 = 1*1 However, .999 * .999 = .9998 And 1*1 = 2 Finally, .9998 ≠ 1 Although both equations become .0002 (with the zero repeating, of course) away from becoming a true equation, they do not quite make it without rounding up.  [Sources] I have not used any site in particular for extracting information, however I have gotten ideas from various other debates here at debate.org, as well as http://www.abovetopsecret.com...  =Conclusions= I hope that I have swayed your opinion away from voting pro. I believe that I have provided enough information to disprove my opponent's equations. I trust that you, the voters, will choose wisely. If you believe that I had the more convincing argument, then please, vote Con. However, if you deny that I had the more convincing argument, then vote for my opponent.
I am glad to see that we agree on the value of this debate. I must, however, continue to disagree with my opponent. NOTE ON SOURCES Wikipedia is generally fine to use for general information. I, too, think it is reasonable to use it here. The source does not negate the mathematical reliability of my proofs. REFUTATION "This [my first proof] assumes that .333 eventually ends, because .3 does, in fact equal 1/3." .3 does not equal 1/3. .333 (repeating) equals 1/3.  "However, the final equation is the main defect. I will admit that once .999 terminates, it would equal one, when rounded up. However, the human mind does not easily understand the concept of infinite. Everything we have in life is limited by a finite number. We have everything measured or the ability to be measured, from the distance of one end of the universe to the other to anything as simple as the distance from your house to your place of work and/or school." .999 (repeating) does not terminate. It repeats forever. No rounding is necessary to prove my side. The fact that a concept is not easily understood by the human mind does not make the idea false. If everything in life is limited, how can numbers like .333 or .999 (repeating, of course) exist?  "From my knowledge as an algebra student, I would be forced to assume that once you have a set number for a variable, you should use the aforementioned number." I did use the same number for the same variable constantly. Please show me where I did not.  "Thus: x = .999 10x = 9.999 10x  .999 = 9.999  After which you would continue by adding .999 to each side. 10x = 10.9998  Please note that I mean the .999 repeats, however when you decide to terminate the decimal, it ends in an eight." There is a very glaring error in this proof. My opponent states that "10x  .999 = 9.999," which is clearly false. It should read: "10x  .999 = 9," because 10x is equal to 9.999. This mistake shows my opponent's counterproof to be false.  "After simple calculations, you can find that .3 is equal to one third." Again, this is entirely incorrect. .3 = 3/10. .333 (repeating) = 1/3.  ".666 is not equal to 2/3 .999 is not equal to 3/3 or 1." .666 (repeating) is equal to 2/3. .999 (again, repeating) is equal to 1, as shown through my two initial proofs  which as of yet have not been adequately refuted. "If .999 = 1 Then .999+.999 = 1+1 However, .999+.999 = 1.9998 And 1 + 1 = 2 Finally, 1.9998 [does not equal] 2" .999 + .999 does not equal 1.9998. My opponent has failed to recognize that .999 (in this debate) means .9 repeating  a guideline that he himself established. .999 (repeating) + .999 (repeating) = 2.  "If .999 = 1 Then .999 * .999 = 1*1 However, .999 * .999 = .9998 And 1*1 = 2 Finally, .9998 [does not equal] 1" .999 (repeating) * .999 (repeating) does not equal .9998. Again, my opponent is using .999 as a nonrepeating decimal. CONCLUSION My two proofs from Round 1 still stand. I have shown that the proofs presented by my opponent both contain fundamental errors or misunderstanding. For these reasons, I continue to urge the voters to support the PRO. I await my opponent's next argument with excitement. 

Atonement forfeited this round.
As my opponent forfeited, please extend my arguments. Vote PRO. 

I am sorry for forfeiting the previous round, I was overwhelmed with real life and was not able to make it onto the website. However, I am challenged for time today and tomorrow as well, so I must forfeit again.
However I did mean to say 2 rounds ago that 10x  .999 = 9 which would return to 10x = 9.999 which is then not equal. Thanks for the first two rounds and hopefully we can debate on perhaps a different topic when I am not pushed for time.
Extend all of my arguments.  The one thing I would like to clarify: "10x  .999 = 9 which would return to 10x = 9.999 which is then not equal." If x = .999, then 10 * x = 9.999. It's simple multiplication.  .999 is equal to 1. Please join me in voting PRO. Sorry to my opponent about his busy schedule  maybe we can do this another time. 
1 votes has been placed for this debate.
Vote Placed by Alexby1 7 years ago
Atonement  Alexby1  Tied  

Agreed with before the debate:      0 points  
Agreed with after the debate:      0 points  
Who had better conduct:      1 point  
Had better spelling and grammar:      1 point  
Made more convincing arguments:      3 points  
Used the most reliable sources:      2 points  
Total points awarded:  0  6 
Sorry about the whole forfeit thing. I know it wasn't your fault.
S&G: Tied. I didn't notice anything.
Arguments: Pro. I still agreed with myself by the end, and my rebuttals were never countered because of the forfeit. Con's math had some errors.
Sources: Pro. Even Wikipedia beats a random forum.