.999... is equal to 1
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Voting Style:  Open  Point System:  7 Point  
Started:  9/18/2008  Category:  Science  
Updated:  8 years ago  Status:  Post Voting Period  
Viewed:  2,964 times  Debate No:  5431 
Debate Rounds (3)
Comments (17)
Votes (2)
.999... = 1 is a false statement.
Definitions .999... refers to .9 with recurring nines 1 refers to the real number, 1 = means is exactly equal to .999... is actually equal to .999... not one. Many people simply round, however some say it actually equals 1, which is false Most people that contend 1 is = to .999... usually provide the below mathematical proof. Step 1) Let x = .999... Step 2) 10x = 9.999... (multiplying RHS and LHS by 10) Step 3) 10x  x = 9.999...  x (subtracting x from both sides) Step 4) 9x = 9 Step 5) x = 1 Conclusion .999... = 1 HOWEVER if you will notice step 4 is incorrect. It refers to 9x=9. This is wrong it actually equals 9x=8.999...1 or 8.99forever with a one at the end. Lets deal with finite to illustrate. Step 1) Let x = .999 (just 3 of them, not infinite) Step 2) 10x = 9.99 (multiplying RHS and LHS by 10) Step 3) 10x  x = 9.99  x (subtracting x from both sides) Step 4) 9x = 8.991 (there is always one 9 less than x and a 1 at the end) Step 5) x = .999 Conclusion .999 doesn't equal 1
My opponent's arguments can be dismantled rather easily. First off, he makes no actual attempt to question the process of the proof, that is, he doesn't deny that 10x  x = 9.999...  x and that the next logical process is 9x = 9. His main point is that 9x actually equals "8.99 forever with a one at the end". I don't even feel that the logical fallacy needs to be pointed out here, but I'll do it anyway. If it is 8.99 forever then there is no 'one at the end'. Infinity by definition is not a finite number, it is not a really large number, it is not even a number. There is no 'one at the end' at all. Now let me pose a different proof that I'm sure my opponent will find hard to poke through: 1/3 = 0.333... [basic math, check your calculator] 3*(1/3) = 3*(0.333...) [multiply each side by 3] 3/3 = 0.999... 1 = 0.999... The same proof works for 1/9 1/9 = 0.111... 9*(1/9) = 9*(0.111...) 9/9 = 0.999... 1 = 0.999... The fact that my opponent points out that the proof doesn't work for finite numbers is exactly my point. It will only work for 0.999... I was going to make that point, but my opponent has done it for me. Instead, I'll show that only 0.999... = 1, not any other recurring decimal. 0.888... = x 8.888... = 10x 8.888...  x = 10x  x 8 = 9x x = 0.888... 0.888... = 0.888... Note how it only works for 0.999... = 1. My opponent's key flaw is that he does not instand the concept of a recurring number, there is no 'one at the end' for such a number because there is no end. There are no mathematical or algebraic errors in the proof presented, and my opponent has not pointed out any. 0.999... = 1 This is mathematically true and it is not even open for debate. Vote PRO. 

Now I know full good and well that most of you reading this debate actually believe that .999... is equal to 1. I will now help you understand why you and my opponent are wrong entirely. There is no question in my mind that 1 and .999... recurring forever is not equal but entirely two different numbers.
The first logic most people refer to, and correctly so, is the concept that .999... is a theoretical number in which it is the number closest to one but still not 1. This is a wise individuals first thoughts when someone attempts to suggest that 1 is equal to .999... Now my opponent attempts to prove this silly notion with mathematics, and this is the point where most people sacrifice their intellect and simply accept that the Theoretical number .999... is equal to one. So before I move on logically, at first glance, 1 does not equal .999... This is going to get complex, but if you think about it you will understand. Now here is the flaw in the algebraic examples my opponent is providing. They are numbering systems that are based on 10. Most of you only have conceptualized numbering systems based on 10 but it doesn't have to be based on 10. It could just as easily based on lets say 8, this is where you must really use your intellect to understand. If the numbering system was 8 lots of things remain the same, however a few important things change, namely decimals. Now lets say we have the number system of 8. 1/8 now would equal .1 and 1/4 would equal .2. 1/3 would now equal .375 as opposed to .333... This may sound silly but you have to understand that decimals simply represent the numbering system chosen and my opponent is using 10. Now regardless of which numbering system you use whether it is 8 or 10 one third (1/3) will be equal to (1/3) however the way the number is represented in decimal form changes. So if my opponent was using the numbering system based on 8 he would be arguing that .777... recurring forever is equal to 1, which of course is also incorrect. To help you better understand numbering systems our calendar months per year is based on 12. If you wanted to represent half a year it would be 6 months. Represented using a numbering/numeral system based on 10 the decimal representation of half would be .5. A 12 numbering system would represent the decimal of .6. A 8 numbering system would have 6 months represent .4 in decimal format. All systems would still represent the 6 months as 6/12 in fraction representation. So if you can see it is simply the decimal representation of the fraction that varies according to the numbering/numerical system. http://en.wikipedia.org... http://en.wikipedia.org... The error with in my opponents algebraic expressions is the decimal system based on 10. The 10 system is the most widely used system in the world today, and many find it difficult to understand other systems because it is like learning math all over. So to close the Logical answer is: The theoretical number that is closest to One is not the equal to one.
Interesting strategy by my opponent in essentially changing the entire debate just by his choosing. But whatever, hopefully the voting audience notices that my opponent has: A) Seemingly conceded that BOTH my mathematical proofs hold, as there has been no challenge to their legitimacy B) Has accepted my rebuttal to his claim that 0.999... is a recurring decimal with an end (wat). But whatever, totally change the debate if you want. The thing with the 8 number system is pretty easy to respond to, like you said the phenomenon occurs with 0.777... instead of with 0.999... This just shows that in that numbering system, there is no excluding factor that says that recurring decimals cannot equal one. It really makes no difference on the matter that 1 = 0.999... (A point that you're seemingly and confusingly conceding). As someone in the comments section points out, this debate isn't entitled "For all numbering systems, especially eight there is only one value for one, although it can change sometimes" This just isn't what we are debating. The resolution is the RECURRING DECIMAL 0.999... is equal to ONE. As it happens, 0.999... can be represented in many forms in different numbering systems, but it still holds that 0.777..., 0.111... or whatever you want to call it, is equal to one. If 0.777... is the same thing as 0.999... in the octal numbering system and that still equals one (which it does) then this is nothing but a point FOR my contention that 0.999... = 1. There isn't a huge difference in the different number systems in regards to the various truisms. 0.999... equalling one is the same. To take the octal numbering system as an example, 0.777... is equivalent to 0.999... of the widely used system, it is also equal to one. It doesn't matter what system you use, it still holds. All this, of course is totally mute because the resolution makes no mention of different numbering systems or anything of the sort. It simply reads that 0.999... is equal to one. Given that 0.999... can be represented in different number systems and those can also be proven to equal one, then I really don't see how my opponent has a point. Once again I remind the voting audience to note that my opponent has ignored my rebuttals. Thank you, and sorry if this seems rushed, it is. 

jmlandf forfeited this round.
Well my opponent has forfitted the last round, essentially sealing my victory as if it weren't abundantly clear already. I'm not quite sure why I accepted this debate, because it isn't really up for debate. To debate whether 0.999... = 1 is comparable to debating whether 2+2 = 4. My opponent had tried to change the definition of '2' so to speak in order to twist his way into winning. But there are serveral flaws to this. His 'number system' idea doesn't work, as varying representations of 0.999... in those systems equal the representation of one in the base 10 system. He did not allude at all that we were to debate with varying number systems, to assume this is ludicrous. Base 10 is the universal system. Thus, I will finish by posting my still unconstested proof for the fact that 0.999... = 1 x = 0.999... 10x = 9.999... 10x  x = 9.999...  x 9x = 9 x = 1 Also 1/3 = 0.333... 3*[1/3] = 3*[0.333...] 1 = 0.999... This will be my last 'debate on this topic' Thank you ladies and gentlemen. 
2 votes have been placed for this debate. Showing 1 through 2 records.
Vote Placed by InquireTruth 8 years ago
jmlandf  Biowza  Tied  

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Vote Placed by TheSkeptic 8 years ago
jmlandf  Biowza  Tied  

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If you want to argue that your original proposition that 0.999... is not equal to 1 was not based on a DECIMAL number 0.999.... then go for it. I suggest you will have difficulties and it will look like a cheap copout. It may look like you deliberately ommitted to specify what system you are using in your original proposition. I dare say that, in absence of any specific stipulation to the contrary, any number expressed (orally or in writing) in today's society (anywhere in the world) will be assumed to be an expression in the decimal system.
I mentioned octal because you mentioned octal in your debate when dealing with different number systems. What I have posted below shows proof that 0.99.... (decimal), if expressed in octal (0.77...) still equals ONE.
To an extent you are right... 0.999... equalling ONE is a result of the decimal system. But so what? The proposition you are arguing against is that the DECIMAL expression 0.99...recurring is equal to the DECIMAL expression "1".
You see, it so happens that 0.777... in octal is a similar represenation to 0.999.. in decimail. But even if the representation did not exist in octal, that would not negate the argument that 0.999... rec in Decimal is equal to 1 in decimal.
8x=7.7777...
8xx=7
7x=7
x=1
Isn't this how it looks in Octal?