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You surely already know Pi, e, the golden ratio... each of them having weird and marvelous mathematical properties.
But this one is the most amazing of all. It really deserves to be considered as the first of all numbers:
https://www.youtube.com/watch?v=GFLkou8NvJo
Quote:
Originally Posted by https://en.wikipedia.org/wiki/Wau_(letter)
seems all other numbers can be derived from that one, using various mathematical operationsQuote:
Originally Posted by Renk
Joke or not, I'd never heard of it before, whereas my pewblic skool edekasion at least taught me that 0.999... = 1 =]
if we wanted to be precise, 0.999... can never reach 1, not even in infinity
so one could write: 0.999...≈1 or 0.999...≐1
I'm pretty sure he was making a floating point joke
No joke here. If 1/9 = 0.111... then 9/9 = 0.999..., but of course, any number divided by itself yields 1, therefore 0.999... = 1. Or at least that's what my math teacher told me, and my mind was blown at the time, but eventually it made sense. The trick is realizing that 0.999... is just an alternate representation of 9/9 and not a different number.
4,195,835 / 3,145,727 = 1.333739068902037589
There's a good floating point joke, one that cost millions of dollars.
I tnink you are refering to a kind of "asymptote's phenomena": A curve never reaches it's asymptote. It's true, but the point is here that 0.9999...(infinite expansion) is the asymptote itself, or more precisely a coding of the asymptote (the number 1).Quote:
Originally Posted by slikrapid
It's a pretty explanation. The sole point that remains: Is 0.1111111... x 9 really equals to 0.999999....? There are plenty of paradoxes when calculations involving an infinite number of terms are performed.Quote:
Originally Posted by anon
thats contradictory, 0.999... doesn't magically become 1, there is a difference involved, even if an infinitely small differenceQuote:
Originally Posted by Renk
hence the approximation symbolQuote:
Originally Posted by Renk
admit it, you used a calculator with the '=' symbol :PQuote:
Originally Posted by anon
other math teachers would say to leave the fraction as is, if it cannot be further reduced, so you would have a nice, clean calculation like this:Quote:
Originally Posted by anon
1/9*9=9/9=1
0.111...*9 does equal 0.999...Quote:
Originally Posted by Renk
but 0.111... does not equal 1/9, nor does 0.999... equal 1
the paradox here is assuming the ability to do precise calculations involving imprecise infinite numbers - so you start with an error (erroneous assumption) and keep making errors, just like IRLQuote:
Originally Posted by Renk
Mathematically it does: The "magic" here is the infinite. For every finite sequence of 9's, the number 0.999...999 is not egal to 1. But "0.999...." doesn't represents any finite sequence of 9. It represents (ie is a notation for) the limit value of such a (finite) sequence of 9 (more precisely, the limit value of the infinite sequence of finite sequences of 9). And this limit value is exactly 1. "1 = 0.9999..." in the same sense that "1/3 = 0.3333....." In fact, every decimal number (and only decimal numbers) has 2 decimal expansions: one expansion ending with 0 (named the "proper decimal representation", and an other expansion ending with an infinite sequences of 9 (named the improper one). Example: 0.25 = 0.2500000... = 0.24999999.... The two are equal to 1/4.Quote:
Originally Posted by slikrapid
This is related to the formal meaning of the sequence "0.11111...". Formally: Let Sn be the sum 10^(-1)+...+10^(-n): Sn=0,111..11 ("1" n times). The notation "0.1111..." doesn't represents any Sn's. It represents the limit value of the Sn's.Quote:
0.111...*9 does equal 0.999...
but 0.111... does not equal 1/9, nor does 0.999... equal 1
By direct calculation (geometrical sum) we have Sn= (10^(-1) - 10^(-n-1))/(1 - 10^(-1))
The limit value of 10^(-n-1) is 0. Then the limit value of the Sn's is (10^(-1) )/(1 - 10^(-1)), which is exactly 1/9.
We start with an error, and there always be an error (for every finite expansion), but these errors are going smaller and smaller and the limit value of theses errors is nothing else than 0, and so in considering limit value, we have equality.Quote:
the paradox here is assuming the ability to do precise calculations involving imprecise infinite numbers - so you start with an error (erroneous assumption) and keep making errors, just like IRL
Saying that "0.999...=1", or that "0.111... = 1/9", is exactly the same as saying that the limit value of, say, 1 + 1/n, is exactly 1.
No 1 + 1/n is exactly equal to 1. But the limit value is.
you don't understand infinity - the point is simple: 'it never ends', therefore it never gets equal to 1, because that would mean an end to infinityQuote:
Originally Posted by Renk
and you said it yourself: no finite sequence equals one, no matter how lengthy the sequence, therefore it would be illogical to assume some special behavior far far away in infinity
nothing special happens in numerical infinity, other than more of the same, so in the example above, you simply get an infinite number of nines, and not a zero or one, ever
such notation is flawed, because all these are different from each other and different from one:Quote:
It represents (ie is a notation for) the limit value
0.9 (0.1 difference)
0.99 (0.01 difference)
0.999 (0.001 difference)
0.999... (0.000...1 difference, infinitely small difference, but a difference nevertheless and a never-ending difference as well)
. <---inconvenient insurmountable gap for the sequence, the tiny elephant in the room
1 (no difference) <---limit value
so it would be inappropriate to use the same notation for the sequence and for its limit, which are two different things, as shown above
"Philosophically", you are contradicting yourself in saying that it would be "illogical" to "assume some special behavior" concerning infinite sequences, just after having stated that (by definition) infinite sequences, unlike the finite ones, "never end". Ending vs never ending is a big, big qualitative difference, rendering illogical to consider that infinite sequence could in no way have specific properties that the finite ones don't have.Quote:
you don't understand infinity - the point is simple: 'it never ends', therefore it never gets equal to 1, because that would mean an end to infinity
and you said it yourself: no finite sequence equals one, no matter how lengthy the sequence, therefore it would be illogical to assume some special behavior far far away in infinity
I your case, let me say that a=b if a-b=0. Let notice d=1-0.9999..... the distance between 1 and the (number represented by) infinite sequence 0.999.....
We have: 10*d = 10 -9,999... = 10-9 -0.9999... = 1-0.999.... So 10*d=d. So 9*d=0, and then d=0, which implies 1=0.9999......
The trick here is that I'm considering that 10*(1-0.999...) = 10- 10*0.999..., and that 10*0.999.... = 9.999... These properties are well known for finite sequences, but for the infinite ones???
Considering that these classic calculus rules remain true with infinite sequences can be seen as a daring gamble, and a questionable claim. But this can be mathematically derived (in a totally logical in rigorous way) from the very (rigorous) definition of the number represented by the infinite sequence 0.999.... . That's where the problem lies: What is the precise, exact, rigorous signification of "0.999...."???? Different choices can be made. One of these choices implies no contradiction, and lead to the conclusion that 0.9999.... = 1.
on the contrary, it is you who are assuming special behavior, wherein 0.999... somehow magically becomes 1, even though not one segment of the infinite sequence, no matter how lengthy, can confirm this assumptionQuote:
Originally Posted by Renk
again, it is you who are assuming a magical loss of difference in infinity, but as said before, there is an infinitely small difference between an infinite sequence and its finite limit, which in the above mentioned example would be:Quote:
Ending vs never ending is a big, big qualitative difference, rendering illogical to consider that infinite sequence could in no way have specific properties that the finite ones don't have.
1 - 0.999... = 0.0...1
it could be said that this infinite sequence is as close as you can get to the finite number 1 (its limit) - but still, the limit is never actually reached, as can be seen in the previous equationQuote:
What is the precise, exact, rigorous signification of "0.999...."????
in an infinite division of numbers between 0 and 1, the closest to number 1 is an infinite sequence 0.999...
on the other hand, in a finite division of numbers between 0 and 1, the closest to number 1 is:
0.9 if divided into 10 segments
0.99 if divided into 100 segments
0.999 if divided into 1000 segments
however, both in the finite and infinite division, there is always a segment left between the limiting number (limit) and its closest neighbor - how close the neighbor is depends on the division involved:
- infinite division produces an infinitely close neighbor
- finite division produces a finitely close neighbor
- no division produces no neighbor - this is the only case when the neighbor and the limit are the same number, as can be seen below for d=0
10*d=d only works for 0 (d=0):Quote:
So 10*d=d. So 9*d=0, and then d=0, which implies 1=0.9999......
for d=0 we have 10*0=0 OK
for d=0.1 we have 10*0.1=1<>0.1
for d=1 we have 10*1=10<>1
similarly, it doesn't work for d=0.999...
this can be used for practical purposes, to simplify calculations, but it is still only an approximation or simplification or assumed equality, not actual equality, hence it would be appropriate to use the approximation symbolQuote:
Different choices can be made. One of these choices implies no contradiction, and lead to the conclusion that 0.9999.... = 1.
It seems to me that assuming that infinite objects mandatory have to share the same properties than finite one's and nothing else is overly presumptuous.
It must not be the case, at least, it must not always be the case, or else we come to contradictions. This is the case with the development I presented above. the number d being the distance between 1 and the number 0.99999....., I conclude that 9*d=0. dividing by 9 gives d=0/9=0.
My argument only relies on some basic calculation rules, nothing more. If it is false, then at least one of these basic calculation rules fails in case of infinite sequences... Which itself consists in a singularity of an infinite thing. In other words, refusing that infinite objects can have very singular properties (comparative to finite ones) leads to the conclusion that these infinite things have in fact such very singular properties. It's not logically tenable.
For example, any (strict) subset of a finite subset has fewer elements than the whole set itself. The part is smaller than the whole. That's "finite common sense". But it's no more the case with infinite sets. For example the set of even (non negative integer) number 0, 2, 4.... does contain the same "number of elements" than the set of all non negative integer (even or odd). In that sense, in infinite case the part may be "as big" as the whole.
I have an other argument to try to convince you that 0.99999....=1. Suppose by contradiction that 0.9999...<1, ie d>0. Then there is an integer N, big enough, such that Nd>1, so d>1/N. Then, there is a (non negative integer) k such that N<10^k, and then 1/N >10^-k.
So we have d>10^(-k). But 10^(-k)= 1-0.9999....9999 (with exactly k 9's in the sequence).
So, we havbe 1-0.9999..... (infinite sequence) > 1-0.999999 (k numbers 9)
So we have: 0.999999 (k numbers 9) > 0.99999..... (infinite sequence), and that is a big contradiction, because we have:
0.999999 (k numbers 9)<0.9999999 (k+1 numbers 9) < 0.999999..... (infinite sequence).
So assuming that 0.9999 differs from 1 leads to a contradiction. Mathematically, this assumption is then false, and so 1=0.9999.....
similar properties, not 'the same' propertiesQuote:
Originally Posted by Renk
although, in another sense, 'the same' may apply as well: both appear in the mind as a result of thinking, therefore they necessarily have to share the same properties as all other things that appear that way
the solution of this equation is d=0Quote:
9*d=0
that means d=0, it does not mean d=0.0...1 or any other number
your calculation is erroneous, as already shown on several occasions - calculation rules did not fail hereQuote:
My argument only relies on some basic calculation rules, nothing more. If it is false, then at least one of these basic calculation rules fails in case of infinite sequences...
no, the part is still smaller than the whole, but nevertheless still infiniteQuote:
in infinite case the part may be "as big" as the whole.
it may be easier to think of them as 'countless': there are countless numbers between 0 and 1, but there are twice as much between 0 and 2
why complicate things further, when i already gave the correct value in the most simple way:Quote:
I have an other argument to try to convince you that 0.99999....=1
1 - 0.999... = 0.0...1
or written more elegantly:
1 - 0.9... = 0.0...1
Not always in case of infinite set: In your example, it is possible to match each number between 0 and 1 with an unique number between 0 and 2, and vice versa (mathematicians call that a bijection). In that case, you can mach e.g. every number x between 0 an 1 with the number 2*x, between 0 and 2, reciprocally you can match every number x between 0 and 2 with an unique number (in that case x/2) between 0 and 1. So there are exactly as many numbers in [0,1] than in [0,2] (and than in [0, 100000], and in fact than in the whole set R itself) This size (cardinality) is named aleph zero (I can't write it here).Quote:
no, the part is still smaller than the whole, but nevertheless still infinite
it may be easier to think of them as 'countless': there are countless numbers between 0 and 1, but there are twice as much between 0 and 2
Precisely. And as d=1-0.9999...., d=0 means 1=0.9999....Quote:
the solution of this equation is d=0
that means d=0, it does not mean d=0.0...1 or any other number
View differently, as we have 9*d=0, if d were non zero, then it would be possible to divide by d, leading to 9=0/d=0. Big contradiction again when assuming that 0.9999... is different from 1.
This calculation is only: 10*d = 10*(1-0.9999...) = 10*1 - 10*0.9999... = 10 - 9.9999... = 10 -9 -0.9999... = 1-0.9999 = d, so 10*d and 9*d=0. What is so erroneous in there?Quote:
your calculation is erroneous, as already shown on several occasions
This second argument I presented is not complicated at all. It's only based on logic and on one fundamental property of the set of all "real numbers" (this set happens to be "archimedian"). this second proof is even (in some sense) simpler, more elementary than the first, as almost no calculation rules are implied here, only very basic and fundamental facts about numbers.Quote:
why complicate things further
You are naming "0.0....1" what I named "d" (previously I defined d by the distance 1-0.9....). My choice is far more concise :smile:. Moreover, as the notation 0.0...1 may have no mathematical sense at all, I'm going to put it in quotes.Quote:
when i already gave the correct value in the most simple way:
1 - 0.999... = 0.0...1
or written more elegantly:
1 - 0.9... = 0.0...1
In any case, then again: 10*"0.0...1" = 10*(1-0.9...) = (see above) = 1-0.9... = "0.0...1".
So we have 10*"0.0...1" = "0.0...1", which leads to 9*"0.0...1"=0, and then, either "0.0...1"=0, either 9=0. What's your choice?
matching unique numbers is not the same as comparing themQuote:
Originally Posted by Renk
it is obvious that the infinite set between 0 and 1 is smaller than the set between 0 and 2, simply because you can find numbers in the latter that aren't available in the former, while the opposite does not apply
Quote:
(1) 9*d=0
sorry, but there is no d that satisfies both of these equations simultaneously, because the solutions are as follows:Quote:
(2) d=1-0.9999....
(1) d=0
(2) d=0.0...1
these two values are obviously different
if d were non zero, the equation would be invalid, only d=0 satisfies the equation - isn't that the goal of solving equations?Quote:
if d were non zero, then it would be possible to divide by d, leading to 9=0/d=0
simplified, the error is: 0.99 * 10 = 9.99, whereas it should be: 0.99 * 10 = 9.9, you have 'one too many nines' after multiplying by 10Quote:
10*d = 10*(1-0.9999...) = 10*1 - 10*0.9999... = 10 - 9.9999... = 10 -9 -0.9999... = 1-0.9999 = d, so 10*d and 9*d=0. What is so erroneous in there?
1*d = 0.0...1 <> 0
2*d = 0.0...2 <> 0
...
9*d = 0.0...9 <> 0
10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d
obviously, none of these equal zero and they are all different from each other
I compare the cardinality of the sets [0,1] and [0,2] (ie the quantity of numbers they are containing) in matching every number of the 1st set with every number of the second set, leading to conclude that these two sets have the same cardinality (each element of one of the set can be "married" to only one of the other set).Quote:
matching unique numbers is not the same as comparing them
When in some population you can marry each man with exactly a woman, and each woman with exactly one man, then in this population there are exactly as many men as women. This is even the very mathematical definition of "as many as".
"obviousness" is not an argument at all, particularly when considering infinite objects.Quote:
it is obvious that the infinite set between 0 and 1 is smaller than the set between 0 and 2
What you say means that [0,1] is a strict subset of [0,2]. That absolutely not implies that the cardinality of the second set is strictly greater than the cardinality of the first.Quote:
simply because you can find numbers in the latter that aren't available in the former, while the opposite does not apply
This reasoning assume you are considering a (finite) integer n. But it does no more work here because in your "0.0...1", your "n" in fact equals to "infinite", and "infinite"-1="infinite".Quote:
10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d
(what I just said is more intuitive than rigorous, due to the fact that your "0.0...1" is only a symbol, and as no clear mathematical meaning).
The fallacy in your reasoning is that you implicitly suppose that what you name "0.0...1" differs from 0. For you it's kind of an axiom. You consider from the outset that="0.0...1" can in no way be equal to 0. But preliminary admitting an assertion is not a good way to prove it.Quote:
sorry, but there is no d that satisfies both of these equations simultaneously, because the solutions are as follows:
(1) d=0
(2) d=0.0...1
For my part, I don't presuppose anything concerning the value of "0.0...1". I just consider this object and, in applying very basics calculus rules (or some reasoning involving basic properties of numbers) , I come to the conclusion that your "0.0...1" is in fact 0.
You can refuse to accept it, but, nevertheless, your "0.0...1" has the property that 10*"0.0...1" = "0.0....1", and then equals to zero.
Exactly. And as the equation is valid, d can not be non zero, ie d=0.Quote:
if d were non zero, the equation would be invalid
I don't say that 0.99*10 = 9.99, I say that 0.999...... *10 = 9,999.....Quote:
0.99 * 10 = 9.99, whereas it should be: 0.99 * 10 = 9.9
What is then the result of the operation 10*0.999..... for you?
All that leads me to think to a 3rd proof that 0.999.... =1:
We have:
1/3=0.333.....
and similarly:
2/3=2*(1/3)=2*0.333... =0.666......
And we have too: 3*(1/3)= 3*0.333....= 0.999....
But on the other hand:
3*(1/3) = 3/3=1.
Hence, 0.999..... =1.
This proof is analogous to the one given by anon previously:
1/9=0.1111.....
Then: 9*(1/9)=0,9999......
But in the same time, 9*(1/9)= 9/9=1.
So 0.999...=1.
The point is, 0.999.... doesn't look like 1, it even doesn't look like an integer at all but it is an integer, and even it is the number 1.
In fact, every decimal number has exactly 2 decimal representations. For example, the two decimal representations of 1/2 are 0.5 and 0.49999.....
You can visit any forum about mathematics, any wikipedia page about these subjects, you will find that 0.999... =1, that there are as many odd integers as there are integers, and that the set [0,1] has as many elements as the set [0,2], with arguments probably close to the ones I presented here.
It can even be proved that there are as many odd (or even) integers as there are decimal numbers (and that, despite the fact that eg the set [0,1] does contain only one even integer, and an infinite quantity of decimal numbers).
Mathematics are about what mathematical objects are, not about what they look like...
the matching works only for half of the numbers of the larger set: the smaller set [0,1] perfectly matches with the first half of the larger set [0,1], but the other 'half' of the larger set, namely <1,2] is left unmatched, which is obviousQuote:
Originally Posted by Renk
infinite objects are not immune to simple, obvious arguments about their natureQuote:
"obviousness" is not an argument at all, particularly when considering infinite objects.
'obvious' means: easily understood by an average open minded person
n is an infinite number of leading zeroesQuote:
This reasoning assume you are considering a (finite) integer n...."infinite"-1="infinite".
n-1 is 'one zero less than' an infinite number of leading zeroes
it is a way to label the 9th neighbor of d, the 10*d, where d=0.0...1
in a way, the infinitely small number is also finite, we just don't know how small it is, other than saying 'it is the smallest' - nevertheless, it also has its immediate neighbors which are all equally far away from each other (this smallest distance is an infinitely small constant of 0.0...1)
if we were to apply your idea of 0.0...1 = 0, the whole calculation of infinite numbers collapses to zero, but an infinitely small (non-zero) distance keeps it mathematically consistentQuote:
you implicitly suppose that what you name "0.0...1" differs from 0. For you it's kind of an axiom.
you missed the part where both equations (1) and (2) need to have a common solution, in order to both apply (to both be valid)Quote:
Exactly. And as the equation is valid, d can not be non zero, ie d=0.
[EDITED ANSWER:]Quote:
What is then the result of the operation 10*0.999..... for you?
the result would be:
0.999... -- n being the number of nines after the decimal point
10*0.999...=9.999...0 -- the result, n-1 being the number of nines after the decimal point
there are still 8 numbers between this result and your previous result:
9.999...0 -- the result, n-1 being the number of nines after the decimal point
9.999...1 -- a number in between, n-1 being the number of nines after the decimal point
...
9.999...8 -- a number in between, n-1 being the number of nines after the decimal point
9.999... -- your previous result, n being the number of nines after the decimal point
see, you just assumed thatQuote:
We have:
1/3=0.333.....
the problem with 1/3 is that it cannot be precisely represented with decimal numbers (10 doesn't divide with 3 nicely)
i can think of a third representation: 0.50...1Quote:
In fact, every decimal number has exactly 2 decimal representations. For example, the two decimal representations of 1/2 are 0.5 and 0.49999.....
but only one of these (0.5) is equal to 1/2, while others are either approximations or nearest neighbors
and using your logic, we can use numerous other representations, like: 0.50...2, 0.50...3, etc. or 0.49...8, 0.49...7 etc. simply because 1*d, 2*d, 3*d, etc. all equal zero - you see the absurdity of such reasoning?
that is just new-math, which has the same validity as (orwellian) new-speak - it is simply a sign of the times we live inQuote:
You can visit any forum about mathematics, any wikipedia page about these subjects, you will find that 0.999... =1
Mathematics are about what mathematical objects are, not about what they look like...
My my, I've caused quite a racket here... but we can all agree that zero is even, right? :troll:
Well, this section is named "Chill & Fun", probably for a reason, and maybe with hardcore mathematics we are going a little "off topic" here. But as a Distinguished Moderator allowed Himself to write a troll-like reply, maybe some rules have been provisionally relaxed. So let me answer, but for ther antepenultimate time, so as not to exaggerate. :cool:
"Obvious" doesn't work very well when arguing about infinity. What I spoke about concerning a smaller set as populated as some larger one, was an instance of mathematical results proved by G. Cantor (who by the way died insane). About the same subject, you can undertake some research about "Hilbert's paradox of the Grand Hotel"(*). You will see that common sense or obviousness don't mix well with infinity.Quote:
Originally Posted by slikrapid
But I prefer to focus on what has become this thread's very subject: The comparison of 1 and 0.999...999....
I said (and proved, in several ways) that 0.999....=1 (or equivalently, 1-0.9999...=0), you on the other hand refuse these evidences, and consider that 1-0.999...= 0.00....001 (which is absolutely not stupid at all, but is nevertheless absolutely wrong, except if you accept to consider that your "0.000...0001" is in fact equal to 0).
So before going any further, let me ask you a question.
Let us call d your 0.000....0001 (ie the difference between 1 and 0.999...). Then, how do you write d/10? d/100? d/1000? What are the decimal representations of these numbers?
And if n is any positive integer, what are the nth digits of your d, d/10, d/100, d/1000?
(*) Or even the Banach-Tarsky Paradox, which is even more disturbing .
so-called proofs that do not align with common logic have probably been improperly set up or defined or calculatedQuote:
Originally Posted by Renk
obviously, a fully occupied hotel (regardless of the number of rooms in the hotel) cannot accommodate any other guests if we want to keep one guest per room and without introducing other ways of driving the existing guests out of their roomsQuote:
Originally Posted by wiki
this does not change for infinite rooms occupied by infinite guests because whichever room you may choose, it will already be occupied by a guest, therefore no vacancy!
if the volumes were finite and known in the beginning, why would they somehow become unknown after chopping them up into a finite number of volumes?Quote:
Originally Posted by wiki
if it is impossible to define the volumes of the considered subsets, how can it be possible for them to become definable again simply by reassembling them?
child1: i can chop up a pea and re-ass-emble it into the Sun!
child2: you cannot do that!
child1: yes i can, because i said so! :P
the mad mathematician: yes i can, because my theorem said so!
the poor students of the mad mathematician: awww
well, i could also say that your claims have been disproven a number of times by using both ordinary logic and simple mathematicsQuote:
Originally Posted by Renk
there are no fractions of d simply because it already represents the smallest infinite numberQuote:
Let us call d your 0.000....0001 (ie the difference between 1 and 0.999...). Then, how do you write d/10? d/100? d/1000? What are the decimal representations of these numbers?
And if n is any positive integer, what are the nth digits of your d, d/10, d/100, d/1000?
Another way to express this paradox is as follows. Let consider an hotel with an infinite number of occupied rooms, all numbered from 0,1,2,etcQuote:
Originally Posted by slikrapid
Suddenly, an infinite number of customers, all numbered from 0,1,2 etc shows up.
Well, although all of the hotel's rooms are already occupied, it is possible to find room for every new guest without kicking out any old guests. (something impossible with a finite_number_of_rooms hotel). And the procedure is simple: For each former guest, if the room number he occupies is n, then he goes to room #2n. And for each new guest, if his number is for example n, then he goes to room #(2n+1). Everyone has a room. No one is thrown out. And this despite the fact that all the rooms were initially occupied...
Short answer: Because the usual mathematical 3 dimensional euclidian space is not exactly the same as it's physical counterpart. (and the "axiom of choice" does not have the same consequences in each of them).Quote:
if the volumes were finite and known in the beginning, why would they somehow become unknown after chopping them up into a finite number of volumes?
if it is impossible to define the volumes of the considered subsets, how can it be possible for them to become definable again simply by reassembling them?
Long answer: Read and learn the proofs:
http://web.mit.edu/andersk/Public/banach-tarski.pdf
https://hal.archives-ouvertes.fr/hal-01673378/document
In my opinion, the sole mean to fully comprehend this paradox (in such a way this paradox is no more a paradox, but something intuitive and natural) is to place it within the framework of the theory of Topos, invented by the mathematician A. Grothendieck.
Maybe Jesus, by multiplying the loaves and fishes from a single loaf and a single fish, has managed to implement this paradox in your physical reality, showing He was the incarnation of a purely mathematical Entity after all.
There is no such way as "it's true because I say it is true" in mathematics. Mathematics is not Politics. Something is mathematically true when it is formally, rigorously and logically deduced and proved, in a verifiable manner.Quote:
child1: i can chop up a pea and re-ass-emble it into the Sun!
child2: you cannot do that!
child1: yes i can, because i said so! :P
This reasoning is rooted on an unproven hypothesis such as "the smallest strictly positive number exists" (your d). But such a "number" doesn't exist (ie it is not definable in a way that does not give rise to any contradiction), because otherwise there would be a little room between 0 and d, and in that little room there would be the middle between 0 and d.Quote:
there are no fractions of d simply because it already represents the smallest infinite number
Believe it or not, there isn't complete agreement on the evenness of zero either (though I personally believe it is even, hence my comment).Quote:
Originally Posted by Renk
As for "off topic" in the Chill & Fun section, that's what you get when two distinguished members are allowed to participate. :wwink:
juggling mathematical expressions cannot substitute an error in logical thinkingQuote:
Originally Posted by Renk
no scientist fully understands properties of the physical space, therefore any scientific conclusion about it is necessarily erroneous - not to mention purely hypothetical imaginary paradoxesQuote:
Short answer: Because the usual mathematical 3 dimensional euclidian space is not exactly the same as it's physical counterpart. (and the "axiom of choice" does not have the same consequences in each of them).
events described in sacred scriptures are certainly sources of inspiration for scientists/theorists but due to their lack of Faith, they manage to misinterpret the events and thus go on making continuous errors, some of which may include creating nonsensical paradoxesQuote:
Maybe Jesus, by multiplying the loaves and fishes from a single loaf and a single fish, has managed to implement this paradox in your physical reality, showing He was the incarnation of a purely mathematical Entity after all.
Jesus did not multiply the loaves and fishes from a single loaf and a single fish, they were separate material manifestations from the surrounding material energy which is what this universe is made of - this energy can be transformed into any material shape or structure by able or worthy individuals, naturally only if God (aka God the Father) allows it to happen - as it is known, Jesus was (is) a saintly Son of the Father and therefore his wishes (for additional loaves and fishes) or Faith (in having a sufficient number of loaves and fishes) were pure and thus successful in any situation
so there is no need to manufacture some erroneous construction like 'incarnation of a purely mathematical Entity' even though God originally invented mathematics and logic and everything else you could (or couldn't) possibly think of
mathematics is handled by humans - humans may be considered political (homo politicus) and therefore there will be politics in the field of mathematicsQuote:
Mathematics is not Politics.
well, as shown before, the logical errors are pretty obvious for many of these so-called axioms or paradoxesQuote:
Something is mathematically true when it is formally, rigorously and logically deduced and proved, in a verifiable manner.
simple logic tell you that if you have a group of different numbers, there ought to be a smallest among them - and i am certain that if nonsensical paradoxes can be 'proven' mathematically, then this can also be proven just as easilyQuote:
This reasoning is rooted on an unproven hypothesis such as "the smallest strictly positive number exists" (your d). But such a "number" doesn't exist (ie it is not definable in a way that does not give rise to any contradiction), because otherwise there would be a little room between 0 and d, and in that little room there would be the middle between 0 and d.
that little room between 0 and d can be defined as 'the smallest non-divisible room' - of course, this can be disregarded and you can invent your own interpretation, where for example there is still some divisible room left inside to manufacture another paradox, say like 'The Paradox of Dividing the Non-divisible Room' - in this day and age (of corruption) it will probably earn you fame and rewards
zero behaves differently than all other even numbers and therefore ought to be given a special 'third' categoryQuote:
Originally Posted by anon
I knew that some people are considering 1 as a prime number (what it is not) but I wasn't aware of any controversy about evenness of 0. I just performed some search, showing there seems to exist some debate about that, but without finding the slightest argument in favour of the oddness of zero.Quote:
Originally Posted by anon
For me an integer number n is even if it is divisible by 2, or equivalently if there exists some integer p such that n = 2p. And an integer number n is odd if it is not even, or (equivalently as each in integer is either equal to 0 or to 1 mod 2), if there exists some integer p such that n = 2p+1.
Clearly, 0=2*0, with 0 being an integer number, so there exists an integer number p such that 0 = 2p, which qualifies 0 as an even number. On the other side, for every integer number p, 0 is not equal to 2p+1, showing that 0 in not an odd number. I can't see what is debatable in all that.
Except maybe if the people considering that 0 is not even, are refusing to consider 0 as an integer number in the first place???
Quote:
Originally Posted by slikrapid
Remember that, as stated on the 2nd page of the MIT paper to which I gave a link above (that you haven't studied, or even read), the Banach-Tarsky paradox is in fact a theorem, and not a formal intrinsic mathematical contradiction. It is called a "paradox" only because the result proved by Banach and Tarsky is "highly counter intuitive".
In this context, I said that the usual mathematical 3 dimensional euclidian space ("UM3dES") was not exactly the same as it's physical counterpart just because mathematically you can duplicate an object in 2 other objects each having the same volume as the former, which seems to be impossible in the physical world (reason why achieving something like this is considered as a miracle). But you are right, we don't know all about your physical world, and maybe (maybe!) doing this kind of thing is quite feasible in our physical world -which would then be less distant from the UM3dES than what I was saying.
But in either case, the result proved by Banach and Tarsky remains true, because rigorously logically and formally proved (if not, show me where are the errors in the paper I gave to you), showing that obviousness and "common sense" are not criteria of mathematical truth. Something can be mathematically true and completely defying common sense in the same time.
And so (maybe by manipulating what physicist name nowadays dark matter and black energy), He found a way to physically succeed in doing what Banach and Tarsky proved mathematically possible 2000 years later. But this part of my post was just a bit of humour. Let us not now enter into a theological/mystical/metaphysical debate.Quote:
Jesus did not multiply the loaves and fishes from a single loaf and a single fish, they were separate material manifestations from the surrounding material energy which is what this universe is made of - this energy can be transformed into any material shape or structure by able or worthy individuals, naturally only if God (aka God the Father) allows it to happen - as it is known, Jesus was (is) a saintly Son of the Father and therefore his wishes (for additional loaves and fishes) or Faith (in having a sufficient number of loaves and fishes) were pure and thus successful in any situation
Anyway, let me just say that I find this part of your post very beautiful and poetic. You also seem to have a deep Faith, deeply rooted in your soul, and I respect that.
Humans are not reductible to homo politicus (as they are not reductible to homo oeconomicus either).Quote:
mathematics is handled by humans - humans may be considered political (homo politicus) and therefore there will be politics in the field of mathematics
By the way, tell me what exactly is the political content (leftist? rightist? centrist?) of the following assertions:
a) "2 is an even integer"
b) "3 is a prime number"
c) "the sum of the 3 angle of any triangle is equal to 180° (or equivalently Pi radians)"
d) "The perimeter of a circle is its diameter times the number Pi"
e) etc etc etc...
You talk a lot and often as a mantra about errors of logic, but you haven't proved the existence of any, especially concerning Hilbert's Grand Hotel and the Banach Tarsky paradox. In fact, what you qualify as "logical" is only what your common sense considers as true. But by doing so, you make yourself deaf and blind to any radical novelty and you make it impossible for you to make any evolutionary leap. Just because something shocks your intuition or threatens the way you view and comprehend the world doesn't mean that this something is illogical or false. And just because you don't take the trouble to study a theorem that defies what you call common sense, doesn't mean this theorem is false either.Quote:
well, as shown before, the logical errors are pretty obvious for many of these so-called axioms or paradoxes. (...) simple logic tell you that if you have a group of different numbers, there ought to be a smallest among them - and i am certain that if nonsensical paradoxes can be 'proven' mathematically, then this can also be proven just as easily
No, because your "Paradox of Dividing the Non-divisible Room" is not proved.You tell about it, but you have not proved its existence. In fact, you are contradicting yourself: First you refuse to accept the fact that 0.999...=1 only because this assertion is not compatible with what you name common sense and logic, ultimately leading you to then have to admit the existence of a "number" that can't be divided by any other number than 1 and itself, and to admit the existence of an interval that has no middle point, what completely defies obviousness and common sense as well...Quote:
that little room between 0 and d can be defined as 'the smallest non-divisible room' - of course, this can be disregarded and you can invent your own interpretation, where for example there is still some divisible room left inside to manufacture another paradox, say like The Paradox of Dividing the Non-divisible Room
Moreover, all these completely not common sensing things that you have made yourself necessary to admit, are not proved in any way, unlike the Banach-Tarsky theorem. Unlike Hilbert's Grand Hotel problem, and unlike the identity of 0.9999... and 1.
And what you say is non proved, and even non provable, because rooted on the intrinsically self contradictory "number" d (=0.0000...1).
You already told me that this d admitted no fraction. So, explain me:
1) What is the smallest positive number divisible by 10 (let name it r)? How do you write it? What is the decimal representation of this number? And if n is any positive integer, what are the nth digit of this number?
2) Now let consider your d. What is the decimal representations of 10*d? Of 100*d? What are the decimal representations of these numbers? And if n is any positive integer, what are the nth digits of these numbers?
why would i bother reading such nonsense? to see how math can be used and abused?Quote:
Originally Posted by Renk
linguistically you can invent words without meaningful physical counterpartsQuote:
because mathematically you can duplicate an object in 2 other objects each having the same volume as the former, which seems to be impossible in the physical world (reason why achieving something like this is considered as a miracle)
artistically you can display things normally impossible in the physical world
philosophically you can establish imaginary concepts
historically you can invent false events
theoretically you can support erroneous concepts
on the contrary, in order to apply mathematical expressions properly, their usage and meaning needs to be understood: obviousness and common sense is certainly neededQuote:
showing that obviousness and "common sense" are not criteria of mathematical truth. Something can be mathematically true and completely defying common sense in the same time.
improper usage may lead to paradoxes
dark matter and dark energy are imaginary scientific concepts used as a crutch to support their flawed theories, as in: 'we are missing something in our calculations... lets call it the dark energy'Quote:
maybe by manipulating what physicist name nowadays dark matter and black energy)
mathematical models can be written in such a way to support a certain political agenda, even though the mathematical expressions used therein are politically neutralQuote:
tell me what exactly is the political content (leftist? rightist? centrist?) of the following assertions:
that is exactly the problem with mathematics (or mathematicians) nowadays: they are often divorced from logic and reality, playing like children, building their imaginary castles in the air, which is simply a sign of the times in the age of ignoranceQuote:
In fact, what you qualify as "logical" is only what your common sense considers as true. But by doing so, you make yourself deaf and blind to any radical novelty and you make it impossible for you to make any evolutionary leap. Just because something shocks your intuition or threatens the way you view and comprehend the world doesn't mean that this something is illogical or false. And just because you don't take the trouble to study a theorem that defies what you call common sense, doesn't mean this theorem is false either.
on the contrary, i have already shown on multiple occasions that 0.999 does not equal 1, be it mathematically or logically - seems to me you are the one constantly trying to find some way to support this ideaQuote:
In fact, you are contradicting yourself: First you refuse to accept the fact that 0.999...=1 only because this assertion is not compatible with what you name common sense and logic, ultimately leading you to then have to admit the existence of a "number" that can't be divided by any other number than 1 and itself, and to admit the existence of an interval that has no middle point, what completely defies obviousness and common sense as well...
what i did there is pointing out how generally by abusing mathematics no concept is safe from ending up perverted and how any perversion can be accepted as valid these days
this was already answered before:Quote:
1) What is the smallest positive number divisible by 10 (let name it r)? How do you write it? What is the decimal representation of this number?
2) Now let consider your d. What is the decimal representations of 10*d?
1*d = 0.0...1 <> 0
...
10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d
100*d = 0.0...100 <> 0 - this one has (n-2) leading zeroes, it is 100 times larger than dQuote:
Of 100*d?
in every one of these numbers: all digits are 0, except for one of them, which equals 1Quote:
what are the nth digits of these numbers?