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#19
Sep206, 10:27 AM

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#20
Sep206, 11:27 AM

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#21
Sep206, 02:52 PM

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#22
Sep206, 07:59 PM

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[tex]0.9\sum_{n=0}^{\infty}\left(\frac{1}{10}\right)^n[/tex] converge to 1, and the sum is equal to 1, in the field of rational numbers which is not complete and has "holes". In fact, it is easy to show that any geometric series in which "a" and "r" are both rational converges, in the field of rational numbers, to a rational number. There is no need to bring real or irrational numbers into it. And the "apparently infinite divisible nature of the Universe" is just that "apparent". Have you never heard of atoms? The universe is not "infinitely divisible". 


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Sep206, 08:40 PM

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#24
Sep306, 03:48 AM

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#25
Sep306, 08:52 AM

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Representations of some real numbers by decimals is not unique, just like the fact that representations of some real numbers by fractions is not unique: 1/2, 2/4, etc. Surely that's not hard to grasp.



#26
Sep2606, 09:08 PM

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0.99…≠ 1
1. 10÷3 = 3+ 1 ÷3，so 10÷3 = 3 …1 is not correct. The values on each side of an equal sign means both values are strictly equal. 9÷3 = 3. it is right. It can be checked by direct computations (by times 3). 10÷3 =（9+1）÷3 =3 + 1÷3 is right now. It can be checked by direct computations (by times 3). So 10÷3 = 3…1 is not correct. It can not checked by direct computation. 1÷3 = 0.3… is not correct either. The right way is ： 1÷3=（0.9 +0.1）÷3 = 0.3+0.1÷3（≈ 0.3）. (1.1) =（0.99 +0.01）÷3 = 0.33 +0.01÷3（≈ 0.33）. (1.2) … =（11/10^n）÷3 +（1/10^n）÷3 （≈ 0. 3…3） (1.3) =（110/10^n+1）÷3+（10/10^n+1 ）÷3=（11/10^(n+1）÷3)+（1/10^(n+1 ）÷3 ) (1.4) =… In the division, because there is always a remainder of one, there will also always be a fraction of 3. So 1÷3 = 0.3… is not correct. Times 3, then 1≠0.9…. End. 


#27
Sep2606, 09:58 PM

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You are right; 1/3 is equal to 0.3 with a remainder of 1. That is, 1/3 = 0.3 + 0.1 / 3 But don't forget that 0.1 / 3 = 0.0333... 


#28
Sep2706, 02:16 AM

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Do you realize how many threes are in between the first 3 and the 1? In fact, the number you propose here isn't even possible as an element of the real numbers 


#29
Sep2706, 06:42 AM

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#30
Dec2806, 03:14 PM

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hi there, i have 17 years old, so don't put me with complicated math..
some days ago i have a discution with friends of me exactly about this subject. after some arguments, i acepted that 0.9(9)=1. i put myself thinking about it and i have a question about it: we have 2 lines(don't sure the traduction in inglish, but is a infinite number of points that are alined all in "front" of the other): A and B, they are perpendicular, their intercection is the point "p" and we start rotating B like the example above: image about it my question is: as 1=0.9(9)[or 0.9...], we can assume that, in the infinite, the point "p' "(a projection of "p") is at a infinite distance from point "p" right?? and can we assume that in infinite, the degree "b" is 90º? why not as 89.9(9)º=90º. right?? then, 2 paralen lines have, at least, 1 interception point... now, if 89.9(9)º=90º then in this image we can assume that degree "c" is too 89.9(9) right?? then there is another point(p'') that exist too, right??? and we assume 2 paralel lines have 2 intercection points... if not, where is the mistake?? thank you in advance, Regards, Littlepig 


#31
Dec2806, 04:56 PM

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What makes you think there is such a point, or two, at infinity? There isn't in euclidean geometry  parallel lines do not meet. There is no coordinate (x,y) with x,y real numbers where parallel lines meet.
If you wish to introduce points at infinity then you need projective geometry. 


#32
Dec2806, 05:21 PM

P: 100

but is that actually possible?? or is a terrible mistake saying it?? because basically, what i'm doing, is trying to separate 2 lines, but what happens is the more i try to separate then, the far way point "p' " is from p, however, it never separates, as lines are infinite, and the more degrees you rotate, the "faster" the point "p' " moves correct?? that's why i made such afirmation...in infinity, 89.9(9) is equal to 90º....so in infinite(paralelism) there is 2 intersections...which are infinity distants from "p". however, you can't say they don't exist, because otherwise you must assume that, rotating the line "b" you will make disapear point p, which actually don't apears to seems... regards, littlepig 


#33
Dec2806, 06:35 PM

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#34
Dec2806, 07:25 PM

P: 100

imagine you can't stop rotating, but you can't reach 90º...is like that...then, in the extrem, the point exists, and the degree is 89.9(9)º. that is like dividing 1 by 3 and then multiply by 3..0.9999(9)never ends...but you know that in the end, it is 1, don't know where is the end, but you know it exists...is the same, you don't have an ending, but you know, that in the end, there's a point... So, you don't need to ask where's the fixed distance, ask what happens to the degree when the fixed distance reaches to inf... 


#35
Dec2906, 05:10 AM

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Sigh. Take this part:
You are, as is almost always the problem. Using your intuition about what happens at every stage after a finite number of decimal places to assert something about the infinitely long decimal expansion. Your intuition is wrong. Infinity is not 'a really big real number'. It is not a real number. Parallel lines do not meet in the Euclidean plane. If I'm wrong (and I'm not), then feel free to write down the point of intersection: hint there is no such point as infinity on the Euclidean plane. The place to use points at infinity is projective geometry, and there need not be just one point at infinity. 


#36
Dec2906, 07:37 AM

P: 894

1 = .99999... and 89.999... = 90 but there is no point at infinity so what is the problem? 


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