said_alyami Posted April 26, 2006 Posted April 26, 2006 There is a new theory that may end the endless debate about 1=0.999... the whole theory can be found in this link http://www.geocities.com/humood_theory here is a Quote about cracking the famouse proof: Proof (A) x=0.999... 10x= 9.999... 10x-x=9.999... - 0.999...=9 9x=9 x=1 --------------------------------------------------------------I will start by cracking this proof and show it is in fact a Paradox not a proof. Let x = 0.999... 10x = 9x + x = [ 9* (0.999...) ] + 0.999… I expressed 10x as (9x +x) 10x – x = ( 9x + x ) – x = [ 9* (0.999…) ] + 0.999… - 0.999… Positive 0.999... goes with Negative 0.999... 9x = 9* (0.999…) x = 9* ( 0.999…) / 9 = 1 * 0.999… x =0.999 We did not end with x=1 In fact what made it looks like a proof was the mistake we did in this step: 10 x – x = 9.999… - 0.999... = 9 This step assumes the value of infinite 9s after the decimal point in ( 10x ) equals the value of infinite 9s in (x) The infinite decimals in x = 0.999... have different behavior than the infinite 9s in 10x =9.999... Because 9s in x= 0.999... Go to infinity faster than 9s in 10x=9.999... (10 x ) is always one decimal behind (x) When x = 0.99 then 10x=9.9 When x = 0.9999 then 10x=9.999 When x = 0.999999999 then 10x=9.99999999 Value of Infinite decimals in (x) – Value of infinite decimals in (10x) = Lim x-> ∞ 9 / (10^x) So the difference between the two values equals an infinitely small fraction which is Lim x-> ∞ 9 / (10^x) This means in Proof (A) when we say: 10 x – x = 9.999… - 0.999... = 9 We are already Assuming that a number minus an infinitely small fraction is the number itself. Which means we are already assuming that Lim x-> ∞ 1 - ( 1/x ) = 0.999... = 1 So, in Proof (A) we already assumed 1= 0.999... Then no wonder we finally ended with our assumption, which make it not a Proof at all. You cannot prove any statement by assuming it to be true from the beginning. But when we explain (10 x ) in a very fundamental method no body can argue with, we will say: 10x =0.999… + 0.999… + 0.999… + 0.999… +0.999… +0.999… +0.999… +0.999… +0.999… +0.999… And when we subtract x = 0.999… from it : 10x - x =0.999… + 0.999… + 0.999… + 0.999… +0.999… +0.999… +0.999… +0.999… +0.999… + 0.999… - 0.999…= 0.999... * 9 The infinite 9s in x after the decimal is gone with only one of those 0.999... Not all of them as we did in Proof (A). ------------------------------------------------------------------ All the theory and results can be seen in the link: http://www.geocities.com/humood_theory For comments please write it here in this forum or send it to the e-mail said_yam@yahoo.com
matt grime Posted April 27, 2006 Posted April 27, 2006 There is a new theory that may end the endless debate about 1=0.999... There is no debate, at least by people who know what they are talking about. Locked thread, please.
Dave Posted April 27, 2006 Posted April 27, 2006 I'm fed up with debating this over and over and over again. There are loads of threads on the topic. Thread closed.
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