tryto Posted April 17, 2009 Share Posted April 17, 2009 Recently, I've had a couple arguments with people disputing the fact that 0.999...=1, a couple on the internet and a couple in real life. It seems to me that this is one of the most commonly disputed mathematical facts, along with the Monty Hall problem, and it is also by far one of the simplest. Even after repeated proofs I've sent to people, they still appear to intuitively distrust this. I've come to the conclusion that most of these people are ignorant that 0.999.... indicates that the 9's go on forever, infinitely, and thus never reach an end point, and thus, in fact, do equal one. To dissuade people from changing the topic and trying to argue that 0.999 !=1, I've included a couple of proofs. [hide=]1. 1=9/9(fractional laws) =9*(1/9)(because 9*1=9, thus 9/9) =9*0.1111...(because 1/9=0.111....) 1=0.999.....(because 9*0.111...=0.999....) 2. x=0.999... 10x=9.999... 10x-x=9.999...-0.9999... so 9x=9 x=1 therefore 0.999....=1[/hide] Can anyone think of an alternate explanation for this phenomen? Why would otherwise rational people, who can accept easily that 0.333....=(1/3) and that 0.333...*3=1 (by fractional definitions), be unable to accept this particular mathematical fact? When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. ~Jonathan SwiftWebsite Updates/Corrections here. WE APPRECIATE YOUR INPUT! Crewbie's Missions!Contributor of the Day!Thanks to artists: Destro3979, Guthix121, Shivers21, and Unoalexi. Link to comment Share on other sites More sharing options...
Riku3220 Posted April 17, 2009 Share Posted April 17, 2009 I think that 0.9999... = 1 but anything I would say math wise would get demolished here. Link to comment Share on other sites More sharing options...
Mr_Adam Posted April 17, 2009 Share Posted April 17, 2009 I believe graphing y = 1/x+4 will give you an example of such. x =/= -4, so it is able to get to any number as close as possible, without ever actually touching the -4 line. Putting in -4 will divide by zero - impossible. But doing -3.99999... will not destroy the equation. That was probably explained terribly. But I hope you got my message overall. ALSO: .9999... / 3 = .3333.... 1 / 3 = .3333... + .0000....1 The latter is impossible, so I assume for the sake of easyness we just round to .3333.... Or just use fractions. Link to comment Share on other sites More sharing options...
Madmanpur3 Posted April 17, 2009 Share Posted April 17, 2009 0.999... != 1, however, the LIMIT of 0.999... = 1. Anything involving infinity is a limit, not an actual number. May the presents of our lord and savior, Santa, be with you this holiday season!First annual Clausmas - 2009 December 25 Link to comment Share on other sites More sharing options...
ElkNight Posted April 17, 2009 Share Posted April 17, 2009 Most people I know dont think .999.... is equal to one because they dont get it. They just go "No .999 is .999 not 1" 8,180WONGTONG IS THE BEST AND IS MORE SUPERIOR THAN ME#1 Wongtong stalker.Im looking for some No Limit soldiers! Link to comment Share on other sites More sharing options...
tryto Posted April 17, 2009 Author Share Posted April 17, 2009 0.999... != 1, however, the LIMIT of 0.999... = 1. Anything involving infinity is a limit, not an actual number. Yes, but the limit of a sequence is a fixed value, not an infinite process, so saying this is misleading. I believe graphing y = 1/x+4 will give you an example of such. x =/= -4, so it is able to get to any number as close as possible, without ever actually touching the -4 line. Putting in -4 will divide by zero - impossible. But doing -3.99999... will not destroy the equation. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between 0.999...9 and 1. That is, if you do the subtraction, 1 0.999...9 will not equal zero. But the point of the "..." is that there is no end; 0.9999... is infinite. 1 / 3 = .3333... + .0000....1 This is just untrue; 1/3 is exactly equal to 0.333.....; if you include the 0.0000...1 you won't equal 1/3. This can be shown using infinite series. When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. ~Jonathan SwiftWebsite Updates/Corrections here. WE APPRECIATE YOUR INPUT! Crewbie's Missions!Contributor of the Day!Thanks to artists: Destro3979, Guthix121, Shivers21, and Unoalexi. Link to comment Share on other sites More sharing options...
Madmanpur3 Posted April 17, 2009 Share Posted April 17, 2009 0.999... != 1, however, the LIMIT of 0.999... = 1. Anything involving infinity is a limit, not an actual number.Yes, but the limit of a sequence is a fixed value, not an infinite process, so saying this is misleading.The limit of a sequence represents a fixed value, and in this case that fixed value is equal to 1 - but the limit itself is not equal to 1. May the presents of our lord and savior, Santa, be with you this holiday season!First annual Clausmas - 2009 December 25 Link to comment Share on other sites More sharing options...
Robert_de_Sable Posted April 17, 2009 Share Posted April 17, 2009 Since I'm terrible at math, my favorite way to approach this one is actually pretty simple, and thankfully in layman's terms. Find a number that is larger than 0.999... but smaller than 1.0. You can't do it. They're the same number. [English translation needed] Link to comment Share on other sites More sharing options...
mmmcannibalism Posted April 17, 2009 Share Posted April 17, 2009 I like the second shown proof, and robert's method this was the second thing I learned in pre calc, right after there are more numbers between 0 and 1 then there are numbers in the series 1, 2, 3, 4, 5, 6, 7, .... edit-- another good way of looking at it is the fact that with an infinite amount of 9s the gap becomes infinitely small and is therefore 0 Orthodoxy is unconciousnessthe only ones who should kill are those who are prepared to be killed. Link to comment Share on other sites More sharing options...
deathdrow Posted April 17, 2009 Share Posted April 17, 2009 wait, so does that mean 1.999... is equal to two? I don't quite get this. Link to comment Share on other sites More sharing options...
Quoi_Tu Posted April 17, 2009 Share Posted April 17, 2009 The first proof is terrible. I'd reword it to connect the bits about 1/9 and 1/3 because right now they're meaningless and the last statement "so 0.9999...=1" seemingly comes from nowhere. In otherwords, it isn't proven with that proof. You just proved "3*0.333....=1" and "9*0.111...=9*(1/9)=(9*1)/9" not "0.9999...=1". Beer Link to comment Share on other sites More sharing options...
mmmcannibalism Posted April 17, 2009 Share Posted April 17, 2009 wait, so does that mean 1.999... is equal to two? I don't quite get this. yes to try to give the simplest explanation possible(look at proof 2 in the first post its neat) .9 is .1 away from equaling 1 .99 is .01 away from eqauling 1 .999 is .001 away from equaling 1 so an infinite amount of 9s would have an infinite amount of zeroes before a 1; subtracting .999... from 1 will give a difference of an infinite amount of zeroes. 0.000...=0 so 1=.999... Orthodoxy is unconciousnessthe only ones who should kill are those who are prepared to be killed. Link to comment Share on other sites More sharing options...
Guest Posted April 17, 2009 Share Posted April 17, 2009 Well, 0.9999... Exists only in theory, correct? Therefore, it does equal one (using your logic, and evidence you have given) But in one sense it will always be less then 1. So yes and no. Link to comment Share on other sites More sharing options...
mmmcannibalism Posted April 17, 2009 Share Posted April 17, 2009 Well, 0.9999... Exists only in theory, correct? Therefore, it does equal one (using your logic, and evidence you have given) But in one sense it will always be less then 1. So yes and no. sort of; since you cant actually write out an infinite amount of 9's any form of this you write out excluding the .999... as infinity notation will be less then one. With an infinite amount of 9's it does however=1 Orthodoxy is unconciousnessthe only ones who should kill are those who are prepared to be killed. Link to comment Share on other sites More sharing options...
tryto Posted April 17, 2009 Author Share Posted April 17, 2009 The first proof is terrible. I'd reword it to connect the bits about 1/9 and 1/3 because right now they're meaningless and the last statement "so 0.9999...=1" seemingly comes from nowhere. In otherwords, it isn't proven with that proof. You just proved "3*0.333....=1" and "9*0.111...=9*(1/9)=(9*1)/9" not "0.9999...=1". It's basically 1=9/9(fractional laws) =9*(1/9)(because 9*1=9, thus 9/9) =9*0.1111...(because 1/9=0.111....) 1=0.999.....(because 9*0.111...=0.999....) and you're correct; I left out a couple of lines; I'll just replace the first one with this one #-o When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. ~Jonathan SwiftWebsite Updates/Corrections here. WE APPRECIATE YOUR INPUT! Crewbie's Missions!Contributor of the Day!Thanks to artists: Destro3979, Guthix121, Shivers21, and Unoalexi. Link to comment Share on other sites More sharing options...
mmmcannibalism Posted April 17, 2009 Share Posted April 17, 2009 The first proof is terrible. I'd reword it to connect the bits about 1/9 and 1/3 because right now they're meaningless and the last statement "so 0.9999...=1" seemingly comes from nowhere. In otherwords, it isn't proven with that proof. You just proved "3*0.333....=1" and "9*0.111...=9*(1/9)=(9*1)/9" not "0.9999...=1". It's basically 1=9/9(fractional laws) =9*(1/9)(because 9*1=9, thus 9/9) =9*0.1111...(because 1/9=0.111....) 1=0.999.....(because 9*0.111...=0.999....) and you're correct; I left out a couple of lines; I'll just replace the first one with this one #-o ah, that proof makes a lot more sense now, confused me the first time I read it. Orthodoxy is unconciousnessthe only ones who should kill are those who are prepared to be killed. Link to comment Share on other sites More sharing options...
compfreak847 Posted April 17, 2009 Share Posted April 17, 2009 The proof is interesting, but I think it's just another way of looking at it. Infinity, essentially a nonreal concept cannot be compared to real numbers on our decimal system; it can easily be said that .999... is infinity increasing but never actually reaching 1. It's the same concept as approaching the event horizon of a black hole; to observers, it goes infinity slowly and will never reach the actual horizon. Basically, yes, it is increasing - but by an infinitely small amount. Your proof represents another way of looking at it, but it isn't accurate to treat infinity as a number when doing comparisons. Essentially, 1/9 and .999... are numbers that cannot be compared in the standard decimal system, as they represent a concept instead of an actual value. Drops: 1x Draconic Visage, 56x Abyssal Whip, 5x Demon Head, D Drops: 37, Barrows Drops: 43, DK Drops: 29GWD drops: 14,000x Bars, 1x Armadyl Hilt, 2x Armadyl Skirt, 4x Sara Sword, 1x Saradomin Hilt, 8x Bandos Hilt, 8x Bandos Platebody, 9x Bandos Tassets, 4x Bandos Boots, 43x Godsword Shard, 82x Dragon BootsDry streak records: Saradomin 412 kills Bandos 988 kills Spirit Mages 633 kills - Slayer Sucks Link to comment Share on other sites More sharing options...
aquariusman Posted April 17, 2009 Share Posted April 17, 2009 I believe that 0.999... = 1. My proof is that if you were to infinitely divide a number, you would get 0. However, if you look at in a different way, 1/(2^n), you would see that there must be a numerical value greater than 0, since everything (Excluding 0), divided by anything (Again, excluding 0), must have a value greater than 0. So, would you not agree that 0.999...9 + 0.000...1 = 1? Since 0.000...1 equals zero... 0.999... + 0 = 1 0.999... = 1 I doubt this isn't flawed though, but it makes sense to me. There's no such thing as regret. A regret means you are unhappy with the person you are now,and if you're unhappy with the person you are, you change yourself. Thatregret will no longer be a regret, because it will help to form the new,better you. So really, a regret isn't a regret. It's experience. Link to comment Share on other sites More sharing options...
Zierro Posted April 17, 2009 Share Posted April 17, 2009 Whoa... Madman and Compfreak agree with me on this topic? I wasn't that expecting at all. Link to comment Share on other sites More sharing options...
Captainkidd Posted April 17, 2009 Share Posted April 17, 2009 It's a simple matter of logic. 0.999... Will ALWAYS be 0.00...1 away from 1, no matter how many zeros you put in. ~ Captainkidd Link to comment Share on other sites More sharing options...
Riku3220 Posted April 17, 2009 Share Posted April 17, 2009 It's a simple matter of logic. 0.999... Will ALWAYS be 0.00...1 away from 1, no matter how many zeros you put in. ~ Captainkidd But 1/3 = .3333... And 1/3*3 equals 3/3 which equals 1 So therefore .333...*3 = 1. Link to comment Share on other sites More sharing options...
mmmcannibalism Posted April 17, 2009 Share Posted April 17, 2009 It's a simple matter of logic. 0.999... Will ALWAYS be 0.00...1 away from 1, no matter how many zeros you put in. ~ Captainkidd yes, but with an infinite amount of nines the difference between 1 and .999... is a 1 an infinite amount of places to the right of the decimal. Since you cant go an infinite amount of places to the right you will only be able to read 0.00000000000.... forever; therefore, .999... is 1 edit--and what riku said. Orthodoxy is unconciousnessthe only ones who should kill are those who are prepared to be killed. Link to comment Share on other sites More sharing options...
Zierro Posted April 17, 2009 Share Posted April 17, 2009 Since you cant go an infinite amount of places to the right you will only be able to read 0.00000000000.... forever; therefore, .999... is 1 1.00000000-> -0.99999999-> -------------- 0.<-00000001 The infinity goes in the direction of the decimal point forever, but there is still always going to be that 1 on the far right. Link to comment Share on other sites More sharing options...
Laura Posted April 17, 2009 Share Posted April 17, 2009 I never understood this concept, even as my last course being AP Calc. If I walk precisely 99.999...% am I touching the wall? I assumed that I would begin walking infinitely closer to the wall, without ever touching it. Assuming that based on your proof that 99.999... also equals a value equal to that of 100. I understand the proof, don't get me wrong, I don't understand the application. Link to comment Share on other sites More sharing options...
Nom Posted April 17, 2009 Share Posted April 17, 2009 I'm not the best math student, but I make it easy for myself. "Find me a number between .999... and 1!" "Okay! .0000000000000000 . . . 0 . . . 0 . . . oh . . ." Link to comment Share on other sites More sharing options...
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