----------------------------------- Ktomislav Wed Sep 24, 2008 7:54 am 1 = -1 ----------------------------------- Maybe some of you already now this. But I will prove to the rest of you that 1 = - 1. There's something wrong in that, and you have to figure out what is it. i is an imaginary unit and i? = -1 i? = i * i -1 = sqrt (-1) * sqrt (-1) -1 = sqrt (-1 * (-1)) -1 = sqrt (1) -1 = 1 :D ----------------------------------- Zeroth Wed Sep 24, 2008 8:26 am Re: 1 = -1 ----------------------------------- Thats easy. You can't pull the two sqrt(-1)'s together into one sqrt. ----------------------------------- Ktomislav Wed Sep 24, 2008 8:32 am Re: 1 = -1 ----------------------------------- Thats easy. You can't pull the two sqrt(-1)'s together into one sqrt. Yes, that's it. :cry: And how about this one. This one is easier. 2a-a+2b-b=2c-c 2a+2b+2c=a+b+c 2(a+b+c)=a+b+c 2=1 :D ----------------------------------- jbking Wed Sep 24, 2008 10:47 am Re:1 = -1 ----------------------------------- In the second one, there is the point that a+b+c=0 which is why you couldn't divide both sides by that and still have a valid equation. ----------------------------------- Ktomislav Wed Sep 24, 2008 10:52 am Re: 1 = -1 ----------------------------------- Yes, that's it. :lol: You are too smart. :mrgreen: :mrgreen: And you probably new that before. :) ----------------------------------- Zeroth Wed Sep 24, 2008 11:33 am Re: 1 = -1 ----------------------------------- Theres also the fact that you mixed up some of the signs in the moving of variables. :P ----------------------------------- gitoxa Wed Sep 24, 2008 9:19 pm Re: 1 = -1 ----------------------------------- My favourite mathematical proof. x = 0.999... 10x = 9.999... 10x - x = 9.999... - 0.999... 9x = 9 x = 1 1 = 0.999... ----------------------------------- CodeMonkey2000 Wed Sep 24, 2008 9:35 pm Re: 1 = -1 ----------------------------------- Thats easy. You can't pull the two sqrt(-1)'s together into one sqrt. Yes, that's it. :cry: And how about this one. This one is easier. 2a-a+2b-b=2c-c 2a+2b+2c=a+b+c 2(a+b+c)=a+b+c 2=1 :D That's assuming that a+b+c is not equal to zero. So it's not a real proof :D @Zeroth I believe you can. But the fallacy in that one was that sqrt has 2 values, +/- and he didn't evaluate both, and assumed positive. ----------------------------------- Tony Wed Sep 24, 2008 10:17 pm Re: 1 = -1 ----------------------------------- My favourite mathematical proof. 1 = 0.999... @Ktomislav -- I was contemplating putting that one up for you, and I guess gitoxa went ahead. Try to find a flaw with that. :wink: ----------------------------------- CodeMonkey2000 Wed Sep 24, 2008 10:32 pm Re:1 = -1 ----------------------------------- I not sure about that one. Real numbers vs. integers? ----------------------------------- Tony Wed Sep 24, 2008 11:57 pm Re:1 = -1 ----------------------------------- Integers are real numbers (are a subset of). ----------------------------------- r691175002 Thu Sep 25, 2008 12:27 am Re: 1 = -1 ----------------------------------- 1 = 0.999... There is nothing wrong here. ----------------------------------- Dan Thu Sep 25, 2008 1:03 am Re:1 = -1 ----------------------------------- For thous wondering: http://en.wikipedia.org/wiki/0.999... ----------------------------------- Tony Thu Sep 25, 2008 9:59 am Re:1 = -1 ----------------------------------- Thus the emphasis and the winking in the original question. Yes, it was a trick question -- there is no flaw. Although it appears that some might think that there might be. :P ----------------------------------- Ktomislav Thu Sep 25, 2008 11:05 am Re:1 = -1 ----------------------------------- Thus the emphasis and the winking in the original question. Yes, it was a trick question -- there is no flaw. Although it appears that some might think that there might be. :P But 1 is not equal to 0.999... If you use that formula it may be is, but in fact it is not. :) It is an exception. (it must be) :) ----------------------------------- Tony Thu Sep 25, 2008 12:00 pm Re:1 = -1 ----------------------------------- but in fact it is not Oh, but now you have to prove such a statement :wink: ----------------------------------- Zeroth Thu Sep 25, 2008 12:04 pm Re: 1 = -1 ----------------------------------- Moderators, can we please lock this thread, and ban this topic? (Specifically 1=0.9999....). In a lot of places, like the XKCD forums, this topic is banned, since there is no real right way we can prove it, by Godel's Incompleteness Theorem(which states that there will exist for any mathematical system/algebra statements that cannot be proven within that algebra, and can only be proven true by going outside of that realm.) In this case, we cannot prove 1=0.999 or not, not with the mathematics we have at our disposal. Its ambiguous, and so.... it can get into pretty heated discussions, arguments, etc. ----------------------------------- Tony Thu Sep 25, 2008 12:36 pm Re:1 = -1 ----------------------------------- I suppose you are right. 1 is an infinitely precise point on a real number line. 0.999... is a point infinitely close to the one above. It's an indeterminate situation, as we cannot be certain if the two mentioned infinities are the same or not. Although in calculus we take 1 = 0.999... as being true. per limit of infinite geometric series. Still... there's some room for ambiguity at a higher level. ----------------------------------- gitoxa Thu Sep 25, 2008 4:07 pm Re:1 = -1 ----------------------------------- Zeroth, I don't see the point of locking a topic like this. Sure, it could get out of hand, but it can be locked then. And Ktomislav, that equation isn't an exception to any math rules. There are many "proofs" (lolquotes) to prove that .999... = 1 ----------------------------------- Ktomislav Thu Sep 25, 2008 4:37 pm Re:1 = -1 ----------------------------------- Zeroth, I don't see the point of locking a topic like this. Sure, it could get out of hand, but it can be locked then. And Ktomislav, that equation isn't an exception to any math rules. There are many "proofs" (lolquotes) to prove that .999... = 1 But that's stupid. How can two different numbers be equal to each other? And btw people invented that formula and people are not perfect. Only God knows what is the truth. :D ----------------------------------- [Gandalf] Thu Sep 25, 2008 4:49 pm Re:1 = -1 ----------------------------------- How can two different numbers be equal to each other? You're the one saying they're two different numbers, who says they can't be two different representations of a number? After all, I'm pretty sure you'd be fine with accepting 1/3 as being 0.333... No? ----------------------------------- gitoxa Thu Sep 25, 2008 4:52 pm Re:1 = -1 ----------------------------------- Please (PLEASE) don't get started with that, that's a whole other sack of potatoes. Like I said, many other '"proofs" exist, go look them up. Gandalf reminded me of another proof. 1/3 = .333... 3 * (1/3) = 3 * .333... 3/3 = .999... 1 = .999... ----------------------------------- Insectoid Thu Sep 25, 2008 4:55 pm Re:1 = -1 ----------------------------------- I think threads like this are great, as long as everybody is open to other people's ideas. It's only when people get defensive or say that 'I'm just right, and you're wrong because I'm a bull-headed stubborn buffoon" that it gets out of hand. As a community of programmers, most of us will enjoy a question like this. I have always wondered about this. My opinion is that the human brain is incapable if computing infinity. So, 0.99... has been given the value of 1 to adapt to our mental restrictions. ----------------------------------- gitoxa Thu Sep 25, 2008 4:57 pm Re:1 = -1 ----------------------------------- I have always wondered about this. My opinion is that the human brain is incapable if computing infinity. So, 0.99... has been given the value of 1 to adapt to our mental restrictions. If that was true, there wouldn't be people that disagreed with it. :P ----------------------------------- Insectoid Thu Sep 25, 2008 4:58 pm Re:1 = -1 ----------------------------------- My opinion is that those are the people who haven't come to terms with the fact that the human brain does indeed have finite limits. ----------------------------------- Dan Thu Sep 25, 2008 5:19 pm Re:1 = -1 ----------------------------------- My opinion is that those are the people who haven't come to terms with the fact that the human brain does indeed have finite limits. I think it is more the universe being finite in terms of how much energery and matter it has then the human brain being unable to visualize finite. In real life there is a minimum size partical so you can't have endless fractions if you are in the smallest possible unit of matter. It's like [url=http://en.wikipedia.org/wiki/Zeno%27s_paradoxes]Zeno's paradoxes. Calculus prity much proves that humans can deal with the idea of infinity and try to make a math system that works with it even if it can't ocure in real life. ----------------------------------- Tony Thu Sep 25, 2008 5:26 pm Re:1 = -1 ----------------------------------- infinity is one of those concepts that is difficult to grasp, until it finally clicks at some point. Though that's the point -- infinity is a concept, not a number. As I've mentioned earlier, integers are a subset of real numbers. They are specific points on a number line. Go ahead, pick one. Come up with a definition for it -- "all points within a certain range shall be deemed to be an integer of 1". This is essentially your basic delta-epsilon proof. Pick a range, any range of how close you want to be to 1.0, and I'll tell you how many 9s need to be in 0.99.. to be in that range. The point of integers is that they are infinitely precise, but then I also have an infinite number of 9s to get me to the same point. Though I think it's the visual difference that throws people off. If we were to approach the same value from another end. That is 1 = 1.000... I suspect that less people will have trouble accepting that. ----------------------------------- [Gandalf] Thu Sep 25, 2008 5:30 pm Re:1 = -1 ----------------------------------- In real life there is a minimum size partical Oh? ----------------------------------- Insectoid Thu Sep 25, 2008 5:31 pm Re:1 = -1 ----------------------------------- Subatomic particals. Very small, those things. ----------------------------------- [Gandalf] Thu Sep 25, 2008 5:45 pm Re:1 = -1 ----------------------------------- Now I'm not a physicist, however people used to think atoms were so-called elementary particles, however now we have quarks and leptons and such, and new particles are being constantly being discovered. Remember the LHC particle accelerator deal? Ya, because of those things. I can't say Dan is wrong about there being finitely small particles, however I don't believe it would be prudent to simply assume there were. After all, that's the whole (or part of) the point of this discussion, no? ----------------------------------- Dan Thu Sep 25, 2008 6:05 pm Re:1 = -1 ----------------------------------- Now I'm not a physicist, however people used to think atoms were so-called elementary particles, however now we have quarks and leptons and such, and new particles are being constantly being discovered. Remember the LHC particle accelerator deal? Ya, because of those things. I can't say Dan is wrong about there being finitely small particles, however I don't believe it would be prudent to simply assume there were. After all, that's the whole (or part of) the point of this discussion, no? Acording to our current understanding of physicist (or at least my interation of it) quarks and leptons are the smallest and most basic compoents of matter so the asumpotion that there is somthing smaller is more out side current physicics then the one that there is not. As for the LHC i blive it is coilding hadrons witch are made out of quarks so i don't it is going to find somthing smaller then quarks and i don't think it is theroirized that it will either. Also at some point you are going to be coverting the particals to energery rather then making them smaller. Edit: also even if there where an infitent number of small particals it would not matter to this debate becues mesuring things in the number of elments/mouluces would still give you a finite interger number as once elementes are broken up or colidenide they are unstable and do not last for long enought to matter. Edit 2: i also realsies string theroy says there are even smaller things, strings, but that is not nessarly a partical and just gets more complicated and off topic from there. My point simply was the hummans can theroirze and try to visualize inffinity in some ways and it is not behond our ablity. ----------------------------------- [Gandalf] Thu Sep 25, 2008 8:07 pm Re: 1 = -1 ----------------------------------- Yes, but that's the nature of this discussion. Modern physics isn't necessarily "correct", neither is math for that matter, however math is far less likely to change. Anyway, I was going to say more about my crackpot theories of infinitely small particles and such, but I'm feeling lazy so I'll just link Wikipedia: [url=http://en.wikipedia.org/wiki/Elementary_particles#Beyond_the_Standard_Model]Beyond the Standard Model *edit* As for your 2nd edit, the point I was trying to argue against was that we have a hard time with infinity because the universe is finite in various ways. ----------------------------------- Tony Thu Sep 25, 2008 8:18 pm Re: 1 = -1 ----------------------------------- we have a hard time with infinity because the universe is finite in various ways. But we don't necessary need a physical example to understand the process by which we approach infinity. Something we all understand here are infinite loops, non-terminating recursion, etc. We don't require such programs to reach the end, to understand their behaviour. ----------------------------------- [Gandalf] Thu Sep 25, 2008 8:27 pm Re:1 = -1 ----------------------------------- I agree. However, in addition, I was saying that it's not some physical impossibility of infinity (as I interpret what Dan was saying) that limits our ability to understand infinity. ----------------------------------- Dan Thu Sep 25, 2008 9:08 pm Re:1 = -1 ----------------------------------- I agree. However, in addition, I was saying that it's not some physical impossibility of infinity (as I interpret what Dan was saying) that limits our ability to understand infinity. I was not trying to say infinity is imposable thats why we don't understand it, in fact i do think some things are infinit such as space (tho is an infinit amount of nothing realy infinity?). What i was trying to say is that we are visulasing something that goses beyond infinity in reality (or is not realy infinity in reality and is finite, at least in my option) so this is why the confusion comes up. I think things like calculus proves that not only can we come close to understating infinity mathematical but that we made a math system that can solve problems with infinity. ----------------------------------- Brightguy Fri Sep 26, 2008 4:10 pm Re: 1 = -1 ----------------------------------- But the fallacy in that one was that sqrt has 2 values, +/- and he didn't evaluate both, and assumed positive. Usually the radical symbol means a function which returns the principal square root, though it wasn't always this way. Last year there was an article in the American Mathematical Monthly about how some people thought Euler mistakenly claimed Try to find a flaw with that. :wink: It does assume high-level properties of the real numbers. Formally, you must show 0.999... converges. For those with the opinion that 0.999... != 1, you have a luxury that isn't possible with a lot of other opinions: you can find out why your opinion is wrong! It may just be misunderstood notation or definitions, or perhaps a fundamental misconception about the reals. (Don't be ashamed if this is the case -- the real numbers may seem obvious but they aren't really.) The good thing about math is that you have the ability to study it and check for yourself how things work out. (Maybe not immediately, but in time...) It's an indeterminate situation, as we cannot be certain if the two mentioned infinities are the same or not. :naughty: since there is no real right way we can prove it, by Godel's Incompleteness Theorem(which states that there will exist for any mathematical system/algebra statements that cannot be proven within that algebra, and can only be proven true by going outside of that realm.) In this case, we cannot prove 1=0.999 or not, not with the mathematics we have at our disposal. :!: This is plain BS. ----------------------------------- r691175002 Fri Sep 26, 2008 11:16 pm Re: 1 = -1 ----------------------------------- This isn't really an open topic. As Wikipedia states "The equality has long been accepted by professional mathematicians and taught in textbooks." There is an abundance of both proof and explanations of why this is true. On the topic of minimum size particles one thing that has always intrigued me is the set of plank units. A plank length, for example, is considered the smallest length that is measurable essentially splitting up the universe into discrete steps (It really doesn't matter if it is discrete or not if it is not measurable beyond that point). ----------------------------------- Ktomislav Sat Sep 27, 2008 6:09 am Re: 1 = -1 ----------------------------------- How can 2 numbers be equal if one is an integer and other one is not? :D ----------------------------------- Insectoid Sat Sep 27, 2008 7:47 am Re:1 = -1 ----------------------------------- 1=1.000=1 1 is an integer. 1.00 is real. 1 is natural. all different types, all equal. ----------------------------------- gitoxa Sat Sep 27, 2008 2:00 pm Re: 1 = -1 ----------------------------------- How can 2 numbers be equal if one is an integer and other one is not? :D Because they are all real numbers. ----------------------------------- Brightguy Sat Sep 27, 2008 7:12 pm Re: 1 = -1 ----------------------------------- Think of 1 as a real number and 0.999...=0.9+0.09+0.009+... as a expression involving real numbers. A priori you might not know if 0.999... is real or not; to show it is in fact real you must show a certain limit exists (limits can be used to explicitly construct the real numbers). ----------------------------------- Ktomislav Sun Sep 28, 2008 3:07 am Re: 1 = -1 ----------------------------------- How can 2 numbers be equal if one is an integer and other one is not? :D Because they are all real numbers. Thats not the point. I know they are both real. But I say that one is integer and other one is not. ----------------------------------- gitoxa Sun Sep 28, 2008 12:00 pm Re: 1 = -1 ----------------------------------- Thats not the point. I know they are both real. But I say that one is integer and other one is not. That's like saying we can't compare -1 and 1, because one is a natural number, and the other is not. ----------------------------------- Ktomislav Sun Sep 28, 2008 2:43 pm Re: 1 = -1 ----------------------------------- Thats not the point. I know they are both real. But I say that one is integer and other one is not. That's like saying we can't compare -1 and 1, because one is a natural number, and the other is not. You can compare them, but they are not equal. ----------------------------------- Tony Sun Sep 28, 2008 3:13 pm Re:1 = -1 ----------------------------------- equality is a type of comparison. We can compare the values of 0.999... and 1 and see which one (if any) is greater or smaller than another. If the value is neither greater nor smaller (but the two values are comparable to each other), then they must be equal. If you are thinking of claiming that one is smaller or greater than another, than be prepared to quantify that (that is -- by how much?) ----------------------------------- Ktomislav Mon Sep 29, 2008 4:29 pm Re:1 = -1 ----------------------------------- If you are thinking of claiming that one is smaller or greater than another, than be prepared to quantify that (that is -- by how much?) It doesn't matter how much is it smaller by. Then you tell me is pi smaller than 4 (by how much?). You don't know, do you? What does it mean? ----------------------------------- Tony Mon Sep 29, 2008 7:06 pm Re:1 = -1 ----------------------------------- I do know. The difference between pi and 4 is at least 0.8 Which is enough to make a pretty clear distinction between the numbers, without the need for greater precision. ----------------------------------- S_Grimm Tue Sep 30, 2008 10:36 am Re:1 = -1 ----------------------------------- there are a million different mathimatical "Truths", some of which would lead to a thread of several hundred pages. i believe that the true question here is, "can we really believe that our kindergarden teachers lied to us and said 1 = 1 not 2?" ----------------------------------- Vermette Fri Oct 03, 2008 1:08 pm Re:1 = -1 ----------------------------------- 1 is equal to 2 for extremely large values of 1 :D ----------------------------------- Ktomislav Fri Oct 03, 2008 6:46 pm Re: 1 = -1 ----------------------------------- This is pointless. And BTW I know, 1 is larger than 0.999... for 1 - 0.999... :D OK. End of discussion. ----------------------------------- Dan Fri Oct 03, 2008 6:49 pm Re:1 = -1 ----------------------------------- 1 is larger thean 0.999... for 0?? (I don't get it). ----------------------------------- Insectoid Fri Oct 03, 2008 8:14 pm Re:1 = -1 ----------------------------------- It isn't pointless, this is the kind of question that provokes a discussive argument. Especially on a website full of math-y people and programmer-y people. ----------------------------------- gitoxa Fri Oct 03, 2008 9:52 pm Re:1 = -1 ----------------------------------- OK. End of discussion. You can't say that and think that we'll all give up and see you're way. :P I think the only thing pointless is trying to make you understand. ----------------------------------- Ktomislav Tue Oct 07, 2008 7:19 pm Re:1 = -1 ----------------------------------- 1 is larger thean 0.999... for 0?? (I don't get it). 1 - 0.9999999... is not 0. ----------------------------------- Tony Tue Oct 07, 2008 8:37 pm Re:1 = -1 ----------------------------------- 1 - 0.9999999... is not 0. You forgot to explain as to why not, let alone what is it instead of 0. Can you reasonably expect anyone to take you seriously at this point? :lol: