JackC 0 #26 December 27, 2007 I know the proof is correct, I just think they way you explained it was misleading. For any finite number of decimal places the proof fails and most people haven't got the hang of working with infinities. Quote Share this post Link to post Share on other sites
KelliJ 0 #27 December 27, 2007 If I write a short computer program that starts at 0 and adds, in succession, 9x10^-1, 9x10^-2, 9x10^-3, etc., and would continue until the sum equaled 1, when would the program stop? Quote Share this post Link to post Share on other sites
shropshire 0 #28 December 27, 2007 depends upon the resolution of the computer (.)Y(.) Chivalry is not dead; it only sleeps for want of work to do. - Jerome K Jerome Quote Share this post Link to post Share on other sites
kallend 2,147 #29 December 27, 2007 QuoteI know the proof is correct, I just think they way you explained it was misleading. For any finite number of decimal places the proof fails and most people haven't got the hang of working with infinities. More infinite wierdness: the number of even integers = the number of all integers (even and odd) = aleph0 aleph0*aleph0 = aleph0 aleph0^aleph0 = C (C>aleph0) C*aleph0 = C C*C = C but C^C >C there are no known transfinite numbers between aleph0 and C, but whether any exist is undecidable. etc.... The only sure way to survive a canopy collision is not to have one. Quote Share this post Link to post Share on other sites
kallend 2,147 #30 December 27, 2007 QuoteIf I write a short computer program that starts at 0 and adds, in succession, 9x10^-1, 9x10^-2, 9x10^-3, etc., and would continue until the sum equaled 1, when would the program stop? If you used all the atoms in the universe you still wouldn't have a computer big enough.... The only sure way to survive a canopy collision is not to have one. Quote Share this post Link to post Share on other sites
JackC 0 #31 December 27, 2007 That depends on what program you wrote. A simple loop that adds 9x10^-n for each n-th trip round the loop would never reach 1. Computers don't get infinities either. You have to catch them or the program either crashes, gets stuck in a loop or outputs gibberish. Quote Share this post Link to post Share on other sites
KelliJ 0 #32 December 27, 2007 I guess infinities are an abstract I'll never grasp. I remember from calculus class a subject called, if I remember right, Gabriel's Horn. It was integrating to find the surface area of a horn of infinite length. The surface area was infinite, yet the volume was finite. Though, at the time I understood the math behind it, I still got headaches thinking of the paradox. Quote Share this post Link to post Share on other sites
pop 0 #33 December 27, 2007 QuoteQuoteDoes .999... (repeating) equal 1.0? Let s = 0.9999... then 10s = 9.99999... then (10s - s) = 9 9s = 9 therefore s = 1 If s=1 then 10s=10. 10 does not equal 9.999999999999999999999.7 ounce wonders, music and dogs that are not into beer Quote Share this post Link to post Share on other sites
jakee 1,594 #34 December 27, 2007 QuoteIf s=1 then 10s=10. 10 does not equal 9.999999999999999999999. 10 does equal 9.999... - that's the whole point. 0.999...=1 9.999...=10Do you want to have an ideagasm? Quote Share this post Link to post Share on other sites
JackC 0 #35 December 27, 2007 I remember a similar problem. A Koch Snowflake has an infinite perimeter but a finite area. So if you extruded the shape into a Koch Vase you could fill it with paint but you wouldn't have enough to paint it. But if you made the walls infinitely thin the inside surface area would be the same as the outside surface area and because it's full of paint, you've already painted the inside and that's impossible. Quote Share this post Link to post Share on other sites
pop 0 #36 December 27, 2007 QuoteQuoteIf s=1 then 10s=10. 10 does not equal 9.999999999999999999999. 10 does equal 9.999... - that's the whole point. 0.999...=1 9.999...=10 A 10 is a 10, and 9.9999 to infinity is always slughtly less then 10. It will never reach 10.7 ounce wonders, music and dogs that are not into beer Quote Share this post Link to post Share on other sites
JackC 0 #37 December 27, 2007 QuoteA 10 is a 10, and 9.9999 to infinity is always slughtly less then 10. It will never reach 10. Ermm.. no. But I think the real moral of this story is don't fuck with infinities. There be dragons... Quote Share this post Link to post Share on other sites
jakee 1,594 #38 December 27, 2007 QuoteQuoteQuoteIf s=1 then 10s=10. 10 does not equal 9.999999999999999999999. 10 does equal 9.999... - that's the whole point. 0.999...=1 9.999...=10 A 10 is a 10, and 9.9999 to infinity is always slughtly less then 10. It will never reach 10. You don't get to decree the proof invalid because you don't like it. Do you agree that 0.333... is 1/3?Do you want to have an ideagasm? Quote Share this post Link to post Share on other sites
KelliJ 0 #39 December 27, 2007 Jeez, my headache keeps getting worse. 1/9=.111... .111...X9=.999... 1/9x9=9/9=1 Are both right? Quote Share this post Link to post Share on other sites
kallend 2,147 #40 December 27, 2007 QuoteQuoteQuoteDoes .999... (repeating) equal 1.0? Let s = 0.9999... then 10s = 9.99999... then (10s - s) = 9 9s = 9 therefore s = 1 If s=1 then 10s=10. 10 does not equal 9.999999999999999999999. I guess you simply didn't bother to read the proof.... The only sure way to survive a canopy collision is not to have one. Quote Share this post Link to post Share on other sites
kallend 2,147 #41 December 27, 2007 QuoteJeez, my headache keeps getting worse. 1/9=.111... .111...X9=.999... 1/9x9=9/9=1 Are both right? Yes.... The only sure way to survive a canopy collision is not to have one. Quote Share this post Link to post Share on other sites
champu 1 #42 December 27, 2007 Pop: lim(sum(9*10^-m,m,1,n),n,inf) = 1 Kallend: Please stop talking about the cardinality of infinite sets, that's how brains melt. (plus I'm still bitter that they they released a new edition of Kenneth Rosen's text at the end of that semester so the bookstores wouldn't buy mine back... grumbles...) Quote Share this post Link to post Share on other sites
kallend 2,147 #43 December 27, 2007 Quote Pop: lim(sum(9*10^-m,m,1,n),n,inf) = 1 Kallend: Please stop talking about the cardinality of infinite sets, that's how brains melt. Maybe yours already melted.... The only sure way to survive a canopy collision is not to have one. Quote Share this post Link to post Share on other sites
champu 1 #44 December 27, 2007 Quote That depends on what program you wrote. A simple loop that adds 9x10^-n for each n-th trip round the loop would never reach 1. Computers don't get infinities either. You have to catch them or the program either crashes, gets stuck in a loop or outputs gibberish. All of the floating point architectures that I'm aware of would call it 1 and stop as soon as the difference became smaller than the resolution of the FP register you were using. At which point I'm sure the VLSI designers said what any electrical/computer engineer says when asked to accommodate a pathological situation, "Let 'em fix it in software if they want that to work." /edited to add: Quote Maybe yours already melted. It's quite possible. There are many topics in mathematics I tried to stop thinking about years ago. But even after I've taken the keys out my brain is still dieseling. Quote Share this post Link to post Share on other sites
JackC 0 #45 December 27, 2007 Quote All of the floating point architectures that I'm aware of would call it 1 and stop as soon as the difference became smaller than the resolution of the FP register you were using. I'm sure you're right but at what point that would happen depends in the implementation you are using. If in doubt, test it. The point is: if you're going to get a computer to do your maths, make sure you know how and when it will fall over and give you gibberish. Quote At which point I'm sure the VLSI designers said what any electrical/computer engineer says when asked to accommodate a pathological situation, "Let 'em fix it in software if they want that to work." That's fine provided software know what the problem is and how to fix it. There's not too much evidence to suggest they do Quote Share this post Link to post Share on other sites
KelliJ 0 #46 December 27, 2007 Quote Quote Jeez, my headache keeps getting worse. 1/9=.111... .111...X9=.999... 1/9x9=9/9=1 Are both right? Yes. No wonder the stock price of aspirin companies jumps at the start of each semester. Quote Share this post Link to post Share on other sites
kallend 2,147 #47 December 27, 2007 Quote Quote Quote Jeez, my headache keeps getting worse. 1/9=.111... .111...X9=.999... 1/9x9=9/9=1 Are both right? Yes. No wonder the stock price of aspirin companies jumps at the start of each semester. Why? You just proved it for yourself!... The only sure way to survive a canopy collision is not to have one. Quote Share this post Link to post Share on other sites
grannyinthesky 0 #48 December 28, 2007 This kind of discussion is what makes me like math, as well as teaching math, so much. "safety first... and What the hell..... safety second, Too!!! " ~~jmy POPS #10490 Quote Share this post Link to post Share on other sites
asmund 0 #49 December 28, 2007 QuoteMath nerds should be tarred and feathered. (Now where's some grammar I can correct?) The question he asks is hardly math nerd material. Trying to comprehensively answer the opening question of this post, will lead to an insight to the meaning and existence of life, the construct of the universe and what may happen after death. Don't worry about it.I like subway. Quote Share this post Link to post Share on other sites
KelliJ 0 #50 December 28, 2007 How big IS the universe? What is outside the universe? If there is something-or nothing-how far does it go? Does .999... equal 1? How many licks DOES it take to get to the center of a Tootsie-Roll Tootsie-Pop? Quote Share this post Link to post Share on other sites
