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# Is 0.999... repeating = 1.0 ?

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after 1 vote the winner is...
dmussi12
 Voting Style: Open Point System: 7 Point Started: 2/8/2014 Category: Education Updated: 6 years ago Status: Post Voting Period Viewed: 919 times Debate No: 45478
Debate Rounds (3)
 http://www.khanacademy.org... This video states that 0.99999... repeating is equal to 1. Please only respond with your arguments, not the ones from the video because I have already watched the video. I stand opposing the equality of the two numbers.Report this Argument The number .999... can be expressed as this sum:Sn=9/(10^n)=9/10+9/100+9/1000...This is a convergent series so, according to the convergence theorem,Sn=(9/10)/(1-(1/10))=.9/.9=1Therefore, .999...=1http://en.wikipedia.org...This actually uses more 'correct' math than the video, which lies on several assumptions as it is intended for the general populace. However, I extend all arguments presented in the video to my argument as supplementary evidence.Report this Argument First of all, you know that the number 1 is rational. You also know that 0.9999... is irrational. If you were to say that 0.999... is equal to 1 than 1 is an irrational number (and vise versa.) Also, you cannot "prove" 0.999 repeating =1 using a mathematical formula. To do that you would need the fully complete number, and that is simply impossible. The fact is that 0.999 repeating will never reach 1, with every digit, it will get 90% closer, that gap just keep shrinking and shrinking and shrinking infinitely (which is why it is repeated infinitely.) If you could solve it mathematically, using simple subtraction, when you took 1 and you subtract 0.999... from it you would get 0.000...(1). Most people would say that with all those forever repeating zeros you would never reach the one. However, You don't have to write an infinite amount of zeroes, you only have to put the repeat symbol above the 0 to represent the fact that the 0s are infinitely repeating. The 1 is there the whole timeReport this Argument Rational number: a number that can be expressed as a fraction; the decimal form ends/repeats infinitely. 1 is rational, so is 0.999. It's 1's dec. expansion. Is .333 irrational? No, it equals 1/3. The relationship is the same.You can't prove 0.999 =1 using a mathematical formula.That's exactly what convergence theorem does, so you CAN apply the formula I gave. These theorems exist so we can rationalize complex problems and establish intrinsic problems of math It is the same thing with limits. These laws are accepted universally.You don't have to write an infinite amount of zeroes, the 1 is there the whole timeYou said that .999 would never reach 1, but now you're saying that 0.000 will eventually reach 0.001. The 1 is not just there, it doesn't exist. To put it in other words, picture the 1 at the end. Put a 0 in front of it. Repeat this infinitely, and you will never find the 1. Before you try to apply this argument to .999, try applying the convergence theorem formula to .000.Report this Argument F22Raptor forfeited this round. As I have provided mathematical proof that remains unrefuted, and as my opponent has forfeited Round 3, I extend my arguments and implore at least one person to see this and vote PRO (I don't really want this to end in a tie when it certainly was not).Report this Argument  