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[Gulnara]

After reading Keen's post: in a sense, there is no number 3, or any number existing, it is only the relational position, quantitative or qualitative among anything people can think up or operate with, that is given particular name. I think to assign place to 0.99 means simply to give it a shift in space, sort of. I can shift from 0.99 back to negative, -0.1, and I have 1 again, only in different, shifted , position. I can do similarly to tiniest fraction to keep 1 whole. Not every element can be divided into fractions, or tiniest fractions. Math is not to force the elements into endless division, math is to see what can and can not be done, I assume. What about mega-sized objects and constructs of this world? Can they be multiplied endlessly and actually answer this call of math? I doubt. Problem is, we have to come up with that next, largest number 1, that encompasses everything. We can not do that yet, we did not do that, we only called it abstractly Universe, or Multiverse. We, essentially, do not have that all-embracing 1, from where we'd even figure our own true size. We went as far as inventing infinity. That is clever, because then one does not have to find the mega-1. May be mega-1 does not exist and it's just tails and tails, all different length strings that end, rip, swirl, split, morph, chameleon into unknown substances and dimensions. Never ending quest of the Universe.

-- Updated Mon Feb 04, 2013 5:24 pm to add the following --

I think 0.0000...infinite....1 is basically equal to zero. The one is after infinite zeroes. That is, 0.000...the zeroes never end, infinite is infinite. A one after never ending zeroes? can such a number exist?

All depends on what one is talking about. Say, if I stumbled upon zero human DNA in an ancient gorilla bone, does it mean if I find a specimen older than that, it will not have human DNA? Of course, it does not mean that. It could mean that I will find part of DNA identical to humans in that older specimen, related to Gorilla, if it was animal that later branched out into prehuman species. so that piece of DNA can represent to me 1 prior to zero.

[enegue]

0.999... = 0.9 + 0.09 + 0.009 + ...

0.999... can thus be represented as a infinite geometric series (a + ar + ar^2 + ar^3, ...) first term, a = 0.9 and common ratio, r = 0.1. The sum of an infinite number of terms of such a series is given by the formula a/(1-r) = 0.9/(1-0.1) = 0.9/0.9 = 1

If you terminate the series short of infinity, you will not get a sum equal to 1. So, while 0.999... = 1 is true, 0.999...9 = 1 is false. Now, you have written "1 - 0.99… = 0.0…1", which is an invalid mathematical equation because you have an infinite number of digits on the left and a finite number of digits on the right.

It is true to say, 1 - 0.99...9 = 0.00...1, but since 0.99...9 = 1 is false, it is then false to conclude that 0.00...1 = 0.

There is no paradox, just a misunderstanding of how to represent an infinite sequence.

Cheers,
enegue

[MazerRackhem]

I think that Teh has done a good job of explaining away the paradox you suggest. As a Mathematician I would like to throw my hat into the ring for explaining how .9999... is exactly equal to 1. Perhaps the best way to explain this is to simultaneously show the error in the line of reasoning you have laid out (which on the face of it seems a very reasonable way to proceed at first thought.) Think of the problem you suggested:

1-.99999....=? We can find the solution to this problem by taking the series of partial subtractions and examining them as an infinite series. Then by taking the mathematical limit of that series we shall have our solution.

1-.9=.1, 1-.99=.01, etc. so that our infinite series is {.1, .01, .001, .0001, .00001, ...10^-n} where 'n' is the index number for the term in the series. Thus the answer to the problem 1-.999...=x is x= lim(n->infinity) [10^(-n)]=0.

Thus we find that sound and rigorous mathematics informs us that 1-.999...=0. Since in the algebra of real numbers all members of set of real numbers have Exactly one additive inverse (that is, for every X is an element of the reals there exists exactly one number Y is also an element of the reals such that X-Y=0) and we already know 1-1=0 we conclude that .999...=1 exactly. This is called closure under addition and is a necessary property of the set of all real numbers.

I apologize for any difficulties in reading my notation, this format does not lend well to mathematical formalism. In short summary: the paradox does not exist because no number .000...1 is ever produced (after a few moments of thinking one should realize why it never CAN be produced) thus all of the following work is undone. Since 1-.999... is exactly equal to 0 we see .999.. is exactly the same as 1. Put another way there is exactly 0 difference between 1 and .9999....

[Gulnara]

Ha-ha-ha. What is it, magician's persuasion that rabbit lives in a hat? Amusing, until someone pays you only 0.999 of what you earned, implying there is no reason to fight over the zero that is lacking.

[Teh]

Ha-ha-ha. What is it, magician's persuasion that rabbit lives in a hat? Amusing, until someone pays you only 0.999 of what you earned, implying there is no reason to fight over the zero that is lacking.

There is a difference between 0.999 and 0.999...

The difference is 0.001

[MazerRackhem]

All depends on what one is talking about. Say, if I stumbled upon zero human DNA in an ancient gorilla bone, does it mean if I find a specimen older than that, it will not have human DNA? Of course, it does not mean that. It could mean that I will find part of DNA identical to humans in that older specimen, related to Gorilla, if it was animal that later branched out into prehuman species. so that piece of DNA can represent to me 1 prior to zero.

It's fairly obvious to me that you don't understand the question being asked. Even more so from your wandering, as Reichenbach would say: "Picture language" in the large paragraph above, which for the sake of space I did not quote in its entirety. The question is a purely mathematical one with a straightforward and rigorous mathematical answer. Your talk of 'There is no number three' and this odd bit about human DNA in ancient gorillas is confused and far off topic.

Ha-ha-ha. What is it, magician's persuasion that rabbit lives in a hat? Amusing, until someone pays you only 0.999 of what you earned, implying there is no reason to fight over the zero that is lacking.

If you don't understand the question under discussion that is perfectly fine, but to fill up forum space mocking the efforts of those who do understand seems a rather juvenile waste.