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 Quote by saltydog Let me attempt a defense then: The sum: $$0.9\sum_{n=0}^{\infty}\frac{1}{10}^n$$ converges to 1
red alarm light comes on: that is 1. It's partial sums converge to 1.

 because between any two real number lies another real number (no holes). In this way the reals are "dense".
the rational numbers also have the property that 'between two rationals there is another rational', and they do not posses limits of sequences.

However, they are dense in the reals in the proper meaning of the word (a set is dense in itself tautologically, if the notion of denseness makes sense at all.)

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 Quote by matt grime I think saltydog is attempting to describe the 'continuum' property. Which nature does not necessarily follow, or use, at all, saltydog. Lots of parts of nature behave in a quantized manner. The rest just seems to belong in philosophy, not mathematics, though I have no idea what geometry has to do with any of this, nor have I ever come across the term 'linear geometry' before.
Very well Matt. I am struck by the similarities between the properties of non-linear systems and the geometry of math itself. Not non-linear geometry but the very geometry of mathematics itself: nested, fractal, and ergodic (the last property explaning why we can get to the same result from so many ways). But Philosophy it should be then.

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 However, they are dense in the reals in the proper meaning of the word (a set is dense in itself tautologically, if the notion of denseness makes sense at all.)
What saltydog stated was the definition of a "dense ordering" -- an order is dense iff for any two elements, you can find a third between them.

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 Quote by Hurkyl What saltydog stated was the definition of a "dense ordering" -- an order is dense iff for any two elements, you can find a third between them.
who knows what was meant. i focussed on the 'no holes' part.

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 Quote by saltydog Let me attempt a defense then: The sum: $$0.9\sum_{n=0}^{\infty}\frac{1}{10}^n$$ converges to 1 because between any two real number lies another real number (no holes). In this way the reals are "dense". My argument was not in regards to notation but to its relation to this property of the number system we create which bears a striking similarity to the apparently infinitely divisible nature of the Universe. Discussions about "0.99...=1" in my opinion reflect this beautiful connection between the geometry of real numbers and the geometry of nature.
That simply isn't true. The partial sums of
$$0.9\sum_{n=0}^{\infty}\left(\frac{1}{10}\right)^n$$
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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 Quote by matt grime I think saltydog is attempting to describe the 'continuum' property. Which nature does not necessarily follow, or use, at all, saltydog. Lots of parts of nature behave in a quantized manner. The rest just seems to belong in philosophy, not mathematics, though I have no idea what geometry has to do with any of this, nor have I ever come across the term 'linear geometry' before.
Blast you, matt! I wondered why this same topic showed up in "philosophy"- it's your fault! This isn't philosophy, it just bad mathematics- and mysticism.

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 Quote by HallsofIvy That simply isn't true. The partial sums of $$0.9\sum_{n=0}^{\infty}\left(\frac{1}{10}\right)^n$$ 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.
Very well Hall. I obviously don't have it then. Thanks.

 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".
And atoms are made of quarks and those of perhaps strings. But I do not in the least believe that is the end of it nor are super-clusters the end at the other extreme. Rather I suspect we encounter singularities which change the rules: "Infinitely divisible" is then a reflection of our limitations with understanding Nature.
 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.
 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) … =（1-1/10^n）÷3 +（1/10^n）÷3 （≈ 0. 3…3） (1.3) =（1-10/10^n+1）÷3+（10/10^n+1 ）÷3=（1-1/10^(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.

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 In the division, because there is always a remainder of one, there will also always be a fraction of 3.
There is only a remainder of 1 if you decide to compute finitely many digits.

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...

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 Quote by Changbai LI 0.99…≠ 1 1. 10÷3 = 3+ 1 ÷3，so 10÷3 = 3 …1 is not correct.
Do you mean to say that 1/3 isn't simply 3 repeating, but 3 repeating with a 1 at the end?

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

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 Quote by Changbai LI 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) … =（1-1/10^n）÷3 +（1/10^n）÷3 （≈ 0. 3…3） (1.3) =（1-10/10^n+1）÷3+（10/10^n+1 ）÷3=（1-1/10^(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.
Simple non-sense. The fact that every term in a sequence has a property (has a remainder when divided by 3) doesn't mean that the limit has that property. That's your logical error.
 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
 Recognitions: Homework Help Science Advisor 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.

 Quote by matt grime 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.
ok...don't know that...never heard about euclidean geometry and projective geometry...

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

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 Quote by Littlepig ok...don't know that...never heard about euclidean geometry and projective geometry... 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
Since infinity is just a concept and not a number, you cant say that there are points on the line at infinity. A point on a number line is always a finite distance from the zero point of the line or else it doesn't exist. Rotating the intersecting line so it is parallel and spaced from the number line doesn't make it any points disappear, it just moves the line so that all points thereon are a fixed distance from the number line. What is confusing about that?

 Quote by ramsey2879 Rotating the intersecting line so it is parallel and spaced from the number line doesn't make it any points disappear, it just moves the line so that all points thereon are a fixed distance from the number line. What is confusing about that?
no, my point is: there is no fixed point, the "fixed point" is the infinit, is the sucession, and the more you rotate, the far the point go, then, if you rotate till 89.9(9) degrees, the point is at inf distance from point origin but it is still there...
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...