.9999 repeating = 1

DEBUNKING OPPOSING ARGUMENT: First, I must insist on legitimate sources or mathematical proofs verifying that infinity is a finite number. My opponent example of defining infinity to be finite never addressed my proof of flawed conception (a != a), and uses an interesting thought experiment to try to support his definition. Yes it is true that there is an infinite amount of numbers between 4 and 5 but any one of those numbers in not infinitely large. By definition, there is an infinite of numbers between every pair nonequivalent numbers. None of these numbers are actually infinite, they are only mirroring the fact that they are on a continuum of infinite numbers. On this note, my opponent states that .99999 repeating and 1 are not equivalent, therefore and an infinite number of numbers must be in between. Simply, name one of them. The youtube video my opponent posted started off by stating it was a semantic debate rejected by mathematicians alike. Trying to redefine infinity as a number is flawed in that it is boundless, concept that can't be quantified in any situation unless it can be taken out of the equation. For instance, infinities can cancel one another under certain circumstances. To understand infinity while applying it to mathematics, we apply it as a rate of growth. A rapidly growing number that never ceases to end, hence IN-finite. When my opponent stated that infinity was finite, he responded by saying it was meant as humorous and not meant to be a real point, despite the fact all of his attempts were based on the tangibility of infinity. His defense was this statement: "the reason that it is called infinity is because we think of it a something that can't be reached even though it is a real number." Incorrect. Finite is an attribute assigned to anything that ends, or is bounded. Infinite is derived as NOT finite, or continuing without bounds. These are all the articles given to finite based on these all encompassing definitions: http://i237.photobucket.com... If I may direct your attention to the second bullet point which differentiates the obvious difference between infinite and finite. When trying to support his claim, my opponent made a link to a reference which never stated infinity was a number nor that it was 'finite'. In response he posted a quote from Wikipedia: "In mathematics, "infinity" is often used in contexts where it is treated as if it were a number". The flaw is apparent: "Treated as a number" is true because notion for continuing without bounds do appear in high level math. "Treated as" a number does NOT mean "is" a number. My opponent cites two more links which supposedly support his claims, and AGAIN fails to direct any quotes from them. Giving my opponent the benefit of the doubt, I will assume he is not trying to just make it look like he has evidence and he believes that there is relevant information to our discussion. The first link seems to center around some thought experiences, but mostly on sets of numbers that span infinitely. I have never made an argument that sets cannot span infinitely, so I will assume he misunderstood my position. The second link clarifys my point quite nicely, that infinity does NOT end. Here is a quote conveniently close to the top: "And then, dreadfully, "does it end?" And at last, the awful truth: it NEVER ends." It appears my opponent wishes to disprove the right hand left hand concept in the second link. Here is the page again: http://www.mathacademy.com... His disproof states "[the] theory is flawed because when the decimals reach infinity they would stop". This is an extraordinary claim to say that one can reach infinity and that it ends, or simply even has an end. This is built on the discussion where he attempted to define 'infinite' as 'finite' and then dismiss it as unimportant/funny. Without proving this point, his disproof is lacking in every way. Continuing his charade of extraordinary and unverified claims, he states that 1/3 != .333 repeating, because of a flaw in our based 10 number system. Again, this has yet to be demonstrated OR cited. He extends upon this conceived notion by trying to put a remainder after the infinite digits. Simply put, something WITHOUT AN END cannot have something AFTER it. He seems to be suggesting a number in the form of ".9999...1", which applies to the emotional response to the infinitude dilemma, rather than a logical mathematically derived form. His next argument falls under the realm of limits which you can learn about in any Introductory Calculus class. He states that going to the infinite decimal place (once again dependent on his claim you can reach it) an infinitely small gap appears between the graph separating the number. The next claim that .9999 repeating would have a slightly different y value than the y value for 1 is equally illogical. The two numbers are one in the same and can be written in two different notations, and are not separate on the number line. Thus no conflict in x nor y values would arise. Note: When I mentioned the burden of proof, I was referring to disproving the math I support. I agree that calling it burden of refutation might be more generally obvious. :D ADDITIONAL PROOFS OF .9999 REPEATING My opponent attacked the validity of 1/3 = .3333 repeating earlier in conjunction with all limits relevant to this discussion. Therefore, I will upload the mathematical limit notation for such limits with the original example of .9999 repeating: http://i237.photobucket.com... For clarification, the 10^n power represents the growing decimal places of continuing nines. A converging sums proof is ideal for our discussion of approaching infinity. I should also recap my proof that there is no number between .999 repeating and 1. No number between means the difference is zero. A difference of zero means it is the same number. (a - a = 0) REVISIT I clumsily forgot to adress one of your problems which did too depend on a finite infinity. :P _______________________________________________________________________________________________ <> <> <> <> <> <> <> <> <> <> <> <> <> YOUR POST WAS THIS: <> <> <> <> <> <> <> <> <> <> <> Let x = .999... 10x = 9.999... (multiplying RHS and LHS by 10) 10x - x = 9.999... - x (subtracting x from both sides) 9x = 9 x = 1 .999... = 1 But the problem with this that if infinity did equal a finite number then the nines would have to stop a t some point. Now to use an example ( In this example I will say that infinity equals to 5 because I don't feel like writing a lot of 9's) Let x = .99999 (.9 to the infinite digit) 10x = 9.9999 10x – x = 9.9999 - .99999 9x = 8.00001 Thus .99999…. wouldn't't equal 1 ______________________________________________________________________________________________ <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> In addition to the dependence on a 'finite' infinity, 9.9999 - .99999 = 8.99991 (Not 8.00001) A growning number of nines before the '1' recedes into the argument where I mentioned the ludicrous form of ".999...1". It's in the middle of my post. I await your response. PS. VOTE PRO yaya! XD