Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20080720 09:01:50
Zero !!!Invented (or Discovered) by an Indian, 'ZERO' is believed to be one of the 'Greatest Inventions' of all times! Recall that Roman numerals don't have any 0! If two or more thoughts intersect with each other, then there has to be a point. #2 20080721 07:36:05#3 20080721 12:58:12
Re: Zero !!!How?? Like.. Fixed LaTeX  Ricky Last edited by ZHero (20080721 12:59:57) If two or more thoughts intersect with each other, then there has to be a point. #4 20080721 13:15:37
Re: Zero !!!because division by exactly zero is undefined. The operation has no meaning. And I think I can hear Ricky groaning when you say 1/∞ is equal to 0. Last edited by mikau (20080721 13:16:00) A logarithm is just a misspelled algorithm. #5 20080721 13:33:05
Re: Zero !!!The question is what do we really mean by something 1/a? 1/a is related to a by the following property: That is, 1/a is simply some number such that when multiplied by a, it gives the value 1. Note that all the little details about arithmetic, different properties of numbers, and their meanings (when we interrupt numbers as quantities) are not used here. 1/a is simply some number (any number) such that when multiplying it by a, we get 1. It turns out with a few very simple algebraic properties, 1/a must be unique. So my "any number" remark doesn't really come up in the vast majority of mathematics. But I suppose under a really ugly and chaotic system, it could. That being said, what is 1/0? Well, going by the above, it is some number such that 1/0 * 0 = 1. We now prove that this is not possible. Let a by any number. First, note that certainly 0 + 0 = 0. Then multiply both sides by a. We get that a * (0 + 0) = a*0. Using the distributed property, a*0 + a*0 = a*0. Subtracting a*0 from both sides, we are left with a*0 = a*0  a*0. However, it must be that a*0 = a*0 (multiplication is well defined). Therefore, a*0 = a*0  a*0 = 0. Therefore, for any number a, we have a*0 = 0. So there is no such number where a * 0 = 1. Now you may say, "What about infinity"? When we introduce infinity, so many things get destroyed that make you want to go back and say, "Ok, let's not add in infinity". We can try to define infinity in a way that intuitively makes sense: For any positive real number a, a * infinity = infinity, a * infinity = infinity, infinity * infinity = infinity, infinity * infinity = infinity, infinity * infinity = infinity. But what about 0? If we have 1/0 * 0 = 1 and 1/0 = infinity, then infinity * 0 = 1. Now doesn't that just strike you as a bit odd? 0*infinity = 1, but 0*infinity = (aa)*infinity as well. If we were to want the distributive properly to work on infinity, then a*infinity  a*infinity = 1? No, no, that can't be. For the love of God! So the distributive properly doesn't work on infinity. Everything else is still ok... There is a well know theorem in mathematics that if a*b = c*b, then a = c. It should be obvious that this doesn't hold since 1*infinity = 2*infinity. Imagine trying to solve equations that involve infinity! Distribution doesn't work and neither do normal cancellation laws. Things only get worse as you consider the more "pure" aspects of algebra. And I've been trying to say away from the "obvious" problems with introducing infinity (0/0, 0^0, etc). It's best just to leave infinity where it belongs: A thing that numbers can approach, but not be. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #6 20080721 15:48:29
Re: Zero !!!What a nice fun explanation!
Love it "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #7 20080721 15:52:51
Re: Zero !!!
Indeed! #8 20080721 18:40:49#9 20080721 21:29:29
Re: Zero !!!Works if you take limits: Otherwise it's not true Last edited by Identity (20080721 21:29:47) #10 20080721 21:45:25#11 20080721 22:11:18
Re: Zero !!!Jane! Not so harsh, please. You are right, though. See Limits (An Introduction): "When it is different from different sides". "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #12 20080722 02:22:44
Re: Zero !!!Above ideas bout Zero are nonetheless but an obscure thought! If two or more thoughts intersect with each other, then there has to be a point. #13 20080722 03:43:36
Re: Zero !!!hahaha, I even caught my calc teacher at school making that exact mistake, marked it wrong when I said wrote infinity in red ink. A lot of people thanked me when I pointed out that 'does not exist' was indeed the correct answer. The two sided rule is just a technicality thats easy to forget. Just an honest mistake, not bs. Did you get enough sleep, Jane? Last edited by mikau (20080722 03:44:17) A logarithm is just a misspelled algorithm. #14 20080722 04:16:45
Re: Zero !!!
Looks like we have another Anthony R. Brown in our midst. Time for another ban? #15 20080722 04:41:33
Re: Zero !!!ZHero, don't listen to Jane. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #16 20080722 04:44:32
Re: Zero !!!
Mathematically acceptable? No, not really. Perhaps you like to think of it that way, and as long as you know the idea is merely a way to think of it rather than actual mathematics, that's OK. But division by 0 is not used in any calculations. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #17 20080722 04:59:14
Re: Zero !!!I can see the difference between saying 'to divide by 0 (which is NOT DEFINED)' and to just say 1/0 equals infinity! Last edited by ZHero (20080722 05:25:57) If two or more thoughts intersect with each other, then there has to be a point. #18 20080722 05:19:05
Re: Zero !!!From the above discussions, all that i can make out is that 'limit x tends to infinity of 1\x' is different from 1\infinity ! Last edited by ZHero (20080722 05:29:09) If two or more thoughts intersect with each other, then there has to be a point. #19 20080722 05:38:50
Re: Zero !!!
Really? Because I can't. Those two things mean exactly the same, yet you are saying one is not defined and one is infinity.
They are ommitted because mathematicians, as a whole, are lazy. It does not mean they aren't their in the heads of mathematicians who read them. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #20 20080722 06:53:06
Re: Zero !!!
oh! What i really wanted to say was.. 'difference between . . . and limit x tends to 0 of 1\x' ! If two or more thoughts intersect with each other, then there has to be a point. #21 20080722 07:48:03
Re: Zero !!!
Cool. But it should never have been "nasty". "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #22 20080722 12:36:00
Re: Zero !!!That was a really Cool (and obviously very Patient) answer! If two or more thoughts intersect with each other, then there has to be a point. #23 20080722 15:32:56
Re: Zero !!!You are forgetting the kind of people who think they know a lot... but don't really. There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction. #24 20080722 15:51:41
Re: Zero !!!Would love to hear about it ... maybe start a new topic? "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #25 20080722 16:00:06
Re: Zero !!!
The integers modulo n always form a ring, and in such an algebraic structure, my proof that a*0 = 0 holds. Thus, division by zero again does not make sense. Now you can change definitions around to make it work (for example, having 1/a no longer meaning "the multiplicative inverse of a" would do it), but that isn't exactly modular arithmetic anymore. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." 