AlexL Posted January 1, 2011 Author Report Share Posted January 1, 2011 Another way of justifying that 1=0.99999999... : note that 1=11/11=(1+10)/11=1/11 + 10/11 Now, in decimal representation (obtained by long division) 1/11 =0.0909090909090909 and so on, and 10/11=0.9090909090909090 and so on Adding the right hand sides (from right to left or from left to right), one gets 1=0.9999999999999999 and so on Alex Quote Link to comment Share on other sites More sharing options...
Darylium Posted January 1, 2011 Report Share Posted January 1, 2011 All 'proof' that 1 equals 0.999... is based on discarding precision when switching from natural numbers to rational numbers. As such it is nothing but a mathematical illusion. Nice trick though. Quote Link to comment Share on other sites More sharing options...
AlexL Posted January 1, 2011 Author Report Share Posted January 1, 2011 (edited) All 'proof' that 1 equals 0.999... is based on ... If this is about my post, please note that I used the word "justification". As to the proof that 1 equals 0.999... : I see you are new to this thread and I suppose you didn't read it in its entirety, but I don't blame you :-) To even begin to think of a proof, one has first to define what one means by "0.999...", which is the very title of this thread; if interested, just read the opening post and some of my subsequent ones. The result is: if "0.999..." means something called in analysis a "limit", of a specific sequence, then it can be proved that it is equal to exactly 1. If you need the details of the proof, you can find them in another thread about 0.99999..., or I can PM them to you. Nice trick though. It's not mine, I just found it in a book and thought it was nice too. Alex Edited January 1, 2011 by AlexL Quote Link to comment Share on other sites More sharing options...
Darylium Posted January 1, 2011 Report Share Posted January 1, 2011 If this is about my post, please note that I used the word "justification". As to the proof that 1 equals 0.999... : I see you are new to this thread and I suppose you didn't read it in its entirety, but I don't blame you :-) To even begin to think of a proof, one has first to define what one means by "0.999...", which is the very title of this thread; if interested, just read the opening post and some of my subsequent ones. The result is: if "0.999..." means something called in analysis a "limit", of a specific sequence, then it can be proved that it is equal to exactly 1. If you need the details of the proof, you can find them in another thread about 0.99999..., or I can PM them to you. It's not mine, I just found it in a book and thought it was nice too. Alex I am sorry for jumping the gun on the proof thing, I honestly thought it was just a trick. I did a bit more research this time, and the most intuitive strategy I could find was multiplying 0.333... = 1/3 by three. Though I must admit that 0.999... = 1 still seems a bit odd. Quote Link to comment Share on other sites More sharing options...
AlexL Posted January 2, 2011 Author Report Share Posted January 2, 2011 (edited) ... the most intuitive strategy I could find was multiplying 0.333... = 1/3 by three. Beside yours, or 1=1/11 + 10/11 etc., another intuitive strategy is multiplying 1/9=0.111111 and so on by 9. In my view, none of these are really proofs, only plausibility checks. Alex Edited January 2, 2011 by AlexL Quote Link to comment Share on other sites More sharing options...
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