16 Nov 08
Originally posted by sonhouseIt's a unique number, there is no other number with the same value.
Regular base 10 number, nothing tricky there. 1634, what makes it unusual?
On the other hand, every number are unique, so there is nothing more unique with this number than any other number, they are equally unique all of them.
Right?
17 Nov 08
Originally posted by FabianFnasWell, it has a certain property not many other numbers have. Got one correct reply by email already.
It's a unique number, there is no other number with the same value.
On the other hand, every number are unique, so there is nothing more unique with this number than any other number, they are equally unique all of them.
Right?
17 Nov 08
Statement (p): Every positive integer has an important property.
Proof by contradiction:
Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).
This contradicts the anti-statement (-p).
Therefore the statement (p) holds.
Conclusion: Every number
17 Nov 08
Originally posted by coquetteIsn't it funny, in Wolfram's column, he devotes an awful lot of time to the numbers that are of no interest to mathemeticians🙂
http://en.wikipedia.org/wiki/1634
http://www.stetson.edu/~efriedma/numbers.html
http://mathworld.wolfram.com/NarcissisticNumber.html
17 Nov 08
Originally posted by FabianFnasI've seen this in a few places now, and I still think it's great 🙂.
Statement (p): Every positive integer has an important property.
Proof by contradiction:
Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).
This contradicts the anti-statement (-p).
Therefore the statement (p) holds.
Conclusion: Every number
18 Nov 08
Well, 1634 has the same property as 153. It's the sum of it's digits' powers of number of digits.
1634 = 1^4 + 6^4 + 3^4 + 4^4 = 1 + 1296 + 81 + 256
153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27
However, 153 has some other properties, for example, 153 = 1! + 2! + 3! + 4! + 5! = 1 + 2 + 3 + ... + 15 + 16 + 17. However, the later two properties wouldn't be so special, if they didn't refer to the same number together with the aforementioned property.
19 Nov 08
1 edit
Originally posted by FabianFnasThat sounds great, but does not fit.
Statement (p): Every positive integer has an important property.
Proof by contradiction:
Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).
This contradicts the anti-statement (-p).
Therefore the statement (p) holds.
Conclusion: Every number
The first "number with nothing special about" is great, but nobody
is interested in the 2nd one.
Who was the 2nd man on the moon?
Who flew as 2nd man across the atlantic ocean?
Who was the 2nd fastest mouse in all of mexico?
Nobody cares. The winner takes it all.
20 Nov 08
Originally posted by afxYou missed the most important step in the proof.
That sounds great, but does not fit.
The first "number with nothing special about" is great, but nobody
is interested in the 2nd one.
Who was the 2nd man on the moon?
Who flew as 2nd man across the atlantic ocean?
Who was the 2nd fastest mouse in all of mexico?
Nobody cares. The winner takes it all.
20 Nov 08
Originally posted by FabianFnasi do love this, but i was looking forward to contributing a great wikipedia article i read once about mathematical paradoxes and systems of reference... but i can't seem to find it! pretty sure it was in reading about Hilbert/Russel/Godel, etc. if anyone finds it it's a great link.
Statement (p): Every positive integer has an important property.
Proof by contradiction:
Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).
This contradicts the anti-statement (-p).
Therefore the statement (p) holds.
Conclusion: Every number
that being said, the main issue with this proof (i think) is not only in its self-reference - the assumption that the lowest of a group is thereby important - but also in it's CHANGE of system of reference. this is similar to the (in)famous proof that "all numbers are interesting" in that we seek to prove that all numerical identities have a property that is defined only in language and has no definable mathematical meaning... and then in order to prove that all numbers share that property, we apply the linguistic notion of "importance" to a numerical property of an element of a list of numbers. it reminds me of the grammatical error of comparisons in the sentence: "Mick Jagger's voice is the better than all the singers in America." it is intuitively understood by the reader, but is incorrect and logically false to compare a voice to a group of people (instead it should be voice -> voice, or person -> people). similarly, a numerically defined concept, such as "lowest," should not be related to a linguistically defined concept like "important."
i.e. it is inappropriately (though naturally) assumed that the smallest number in a list is also thereby important. this of course speaks to each reader as an intuitive idea.. but if you disregard that assumption the paradox breaks down.
20 Nov 08
1 edit
Originally posted by TheMaster37No, I did not.
You missed the most important step in the proof.
The induction step only holds, if the 2nd "number with nothing special" is "special", because the first NWNS is special.
But thats not the case, because nobody is interested in the 2nd one.
The induction step here assumes, that the 2nd one is as interesting
as the first one.
ps: I know, what a proof is, especially an induction proof, I am a mathematician.
Edit: The problem with this funny "proof" is the funny precondition. Because "being of interest" is being a thing of human "feeling", which is not fixed. So all the rest is a thing of good vibrations