graduate Posted November 24, 2011 Share Posted November 24, 2011 Why Math.pow(2, Infinity) => Infinity, but Math.pow(1, Infinity) => NaN ? I think 1 * 1 * 1 * 1... should be 1 Link to comment Share on other sites More sharing options...
Ingolme Posted November 24, 2011 Share Posted November 24, 2011 1 to the power of infinity is an indetermination in Calculus. When a value tends to 1, the result could be 0, 1 or infinity, depending on if it's one infinitesimal below 1, or one infinitesimal above 1. Link to comment Share on other sites More sharing options...
graduate Posted January 20, 2012 Author Share Posted January 20, 2012 Did not I make it clear in the expression that I mean "one" without any infinitesimal below / above, so the result should be "1"? Link to comment Share on other sites More sharing options...
Ingolme Posted January 20, 2012 Share Posted January 20, 2012 It's just how math works. When working with infinities you really can't know anything for certain. According to mathematicians, 1 is strictly equal to 0.9999... and it's also strictly equal to 1.000...001 so when tending to infinity it can't take a determined value (it's called an indetermination) A quick proof that 1 is strictly equal to 0.999...:1 / 3 = 0.333...0.333... * 3 = 0.999... Link to comment Share on other sites More sharing options...
ShadowMage Posted January 20, 2012 Share Posted January 20, 2012 A quick proof that 1 is strictly equal to 0.999...:1 / 3 = 0.333...0.333... * 3 = 0.999...I brought this up one time in my calculus class and we spent an entire class period working out this insane formula proving how .999... is equal to 1. :blink:I wasn't convinced... In fact, I still argue the point. EDIT: Speaking of formulas, we also worked through a formula proving that 0 is equal to 1. I really wish I could remember it, but it was fairly complicated too. Math is fun... Link to comment Share on other sites More sharing options...
Ingolme Posted January 20, 2012 Share Posted January 20, 2012 The kind of formulas that lead to 0 = 1 always involve some kind of mathematical fallacy. You can see where the 0 = 1 problem goes wrong in this article: http://en.wikipedia.org/wiki/Mathematical_fallacy 0.999 does actually equal 1Quotation from Wikipedia:http://en.wikipedia.org/wiki/0.999... The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. Nonetheless, some students question or reject it. Some can be persuaded by an appeal to authority from textbooks and teachers, or by arithmetic reasoning, to accept that the two are equal. Link to comment Share on other sites More sharing options...
graduate Posted January 21, 2012 Author Share Posted January 21, 2012 I want to study high math, can you recommend me a good text book for a novice? Link to comment Share on other sites More sharing options...
Ingolme Posted January 21, 2012 Share Posted January 21, 2012 You may want to start a career in some kind of scientific or technological degree, where they'll provide you with a lot of resources to learn math. If you're very interested in math, you could take a pure math career, but real math is so crazy that I'd stay away. I suggest that if you're still in school, ask your math teachers about a few things and try to learn what you can about it to see how much you actually like it. Link to comment Share on other sites More sharing options...
graduate Posted January 21, 2012 Author Share Posted January 21, 2012 I'm a grownup actually, but I always liked the mathematical overlook at the world. This book looks sound, but I think it's not for newbies.http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/ref=sr_1_7?ie=UTF8&qid=1327106003&sr=8-7 Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 The kind of formulas that lead to 0 = 1 always involve some kind of mathematical fallacy. You can see where the 0 = 1 problem goes wrong in this article: http://en.wikipedia....matical_fallacy Yeah, my teacher did explain the problem with it and I think the point of the excercise was to always be careful with your calculations because simple mistakes can lead to big problems. 0.999 does actually equal 1No it doesn't... (I'm one of those disbelievers Wikipedia was referring to.) I'm a grownup actually...A grownup, huh? I don't know of too many "grownups" who refer to themselves as a grownup... Link to comment Share on other sites More sharing options...
Ingolme Posted January 23, 2012 Share Posted January 23, 2012 No it doesn't... (I'm one of those disbelievers Wikipedia was referring to.)Rational numbers are numbers that can be represented with fractions. We know 0.999... is a rational number because you can always predict what the next decimal digit will be.For every rational number R, there exists a fraction a/b where a,b are integers (whole numbers) Can you find the numbers a and b such that 0.999... can be represented as a fraction that's different to 1? Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 Can you find the numbers a and b such that 0.999... can be represented as a fraction that's different to 1?No, I don't suppose I can. However, consider this: If a function has a limit of 1 (it's value approaches but never equals 1) Somewhere along the line, the function's value will have to equal 0.999... or 1.0...01 (depending on whether the value is greater or less than 1). If 0.999... = 1 = 1.0...01, then it would be correct to say that the function's value eventually does reach its limit. But this strictly defies the definition of a limit. Therefore, 0.999... != 1 and 1 != 1.0...01 Link to comment Share on other sites More sharing options...
Ingolme Posted January 23, 2012 Share Posted January 23, 2012 Take the limit of this infinite sum where n tends to infinity:∑9/10ⁿAt the point of infinity, this sum ends up being 1. When you say that a limit approaches, but does not equal, 1 you're imagining that infinity is just a really large number. If you do a really large, but finite, amount of operations, you will not reach 1; but if you actually do infinity operations you will get to 1. What, exactly, causes you to believe that they are different?In mathematics, you can't guide yourself by intuition. Algebraic proofs are necessary to see if an assumption is actually correct. Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 Take the limit of this infinite sum where n tends to infinity:∑9/10ⁿAt the point of infinity, this sum ends up being 1.No, it ends up being .999....9 + .09 + .009 + .0009 + .00009 + ... + .0...09 = .999... What, exactly, causes you to believe that they are different?The concept of infinity. It may not be representable, but it is still a real value. If it is a real value, then one can use infinity to show that 0.999... is (infintesimally) smaller than 1.1-(1/10^Infinity) = .999...If you actually take 1 divided by 10 to the Infinity power, you will get .0...01 Link to comment Share on other sites More sharing options...
Ingolme Posted January 23, 2012 Share Posted January 23, 2012 0.0...01 doesn't exist because infinity has no last digit.1/infinity is equal to 0. It has to be equal to 0 because there'%2 Link to comment Share on other sites More sharing options...
justsomeguy Posted January 23, 2012 Share Posted January 23, 2012 It's true that 0.999.. equals 1, but the thing to take away from that isn't some weird math proof, it's that our system of math is not perfect. That's really the only thing that is proven. You can imagine that 0.999.. and 1 are different numbers, but if that was the case then there would be a number between them. There isn't a number between them, so they're the same number. They might be represented differently, but they're the same number. The fact that they are represented as being different numbers shows that our math system is not perfect. Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 0.0...01 doesn't exist because infinity has no last digit.You're contradicting yourself:According to mathematicians, 1 is strictly equal to 0.9999... and it's also strictly equal to 1.000...001 1/infinity is equal to 0. It has to be equal to 0 because there's no other divisor you could use if you wanted to get 0 out of the division 1/x. An infinitesimal is 0.If you desire a certain result it is tempting to assume a lot of things. If you want 1/x to equal 0, it is tempting to assume that 1/infinity equals 0 when no other representable number will give you that result. In fact it does not. It equals an infinitely small decimal just above 0. There is no possible value of n or x that will make n/x = 0 true. Just to prove that infinity is a real value, what would you do with numbers such as these:.9...98.0...02.0...03.0...01234567?These numbers are real enough. There just happens to be an infinite number of digits prior to the last digits. They are all different and cannot be explicitly equal to any other number except themselves. Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 It's true that 0.999.. equals 1, but the thing to take away from that isn't some weird math proof, it's that our system of math is not perfect.Precisely my point. You can imagine that 0.999.. and 1 are different numbers, but if that was the case then there would be a number between them. There isn't a number between them, so they're the same number.This is perhaps the most convincing argument I've seen yet. However, there is a number between them. Lots in fact:0.9...9010.9...9020.9...903... Link to comment Share on other sites More sharing options...
justsomeguy Posted January 23, 2012 Share Posted January 23, 2012 No, there is not a number between repeating 9s and 1. 0.9...901 is less than repeating 9s, not between that and 1. Link to comment Share on other sites More sharing options...
Ingolme Posted January 23, 2012 Share Posted January 23, 2012 I am not contradicting myself. 0.000...01 is strictly equal to 0 so you could say it doesn't exist as its own number. That's why 1.000...01 is strictly equal to 1. The point here is that in calculus, 1^infinity cannot be determined because of the fact that 1 is an infinitely small range of values slightly below and above 1. Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 No, there is not a number between repeating 9s and 1. 0.9...901 is less than repeating 9s, not between that and 1.I suppose you are right on this one...However, this still supports my argument that the numbers are not equal. Infinity is a complex concept that cannot be fully understood. I can understand your arguments and the mathematical proofs. The problem with those proofs is that they're trying to bind infinite numbers within finite boundaries. Infinity is a value that does not fit well into the mathematical system. Mathematically, the numbers may be identical. Logically, they are not. Link to comment Share on other sites More sharing options...
Ingolme Posted January 23, 2012 Share Posted January 23, 2012 Hmm, you'll need to show the logical proof for that. Is it logic or just intuition that tells you that? Link to comment Share on other sites More sharing options...
ShadowMage Posted January 23, 2012 Share Posted January 23, 2012 A quote from the Wikipedia article you linked before: The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In most such number systems the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that fall infinitesimally short of 1.Logic may or may not play a part, but what this tells me is that it all boils down to interpretation. How do you interpret infinity? The common numbering system seems to neglect the idea of Infinity being a value and thus makes assumptions when dealing with infinite values. The quote above says that some systems, which do provide values for Infinity (through the use of infinitesimals), agree that .999... is not equal to 1. Infinity is simply beyond human comprehension. Let's just leave it at that. (Mainly because no amount of convincing is going to make me believe that .999... = 1 and we probably shouldn't wander too much farther off-topic) Link to comment Share on other sites More sharing options...
thescientist Posted January 24, 2012 Share Posted January 24, 2012 Link to comment Share on other sites More sharing options...
ShadowMage Posted January 24, 2012 Share Posted January 24, 2012 I'm a nerd and proud of it! Link to comment Share on other sites More sharing options...
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