1. We have added a Gift Upgrades feature that allows you to gift an account upgrade to another member, just in time for the holiday season. You can see the gift option when going to the Account Upgrades screen, or on any user profile screen.
    Dismiss Notice

Let's discuss Mathematics

Discussion in 'Science & Technology' started by ParadigmShifter, Mar 16, 2009.

  1. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
  2. Atticus

    Atticus Deity Retired Moderator

    Joined:
    Aug 20, 2006
    Messages:
    3,666
    Location:
    Helsinki, Finland
    Idiotsopposite: What transcendetal numbers do you know? Can you reduce this question to knowledge of them? (Assuming that it is allowed, which I do assume, since otherwise this is quite hard).

    EDIT: Oops, didn't notice the page had changed.
     
  3. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    e and pi are the only transcendental numbers most people know about.

    Is that a hint, do you know how to show it Atticus? Or just speculative?

    EDIT: This looks like the correct theorem, which is an extension of one of the ones I posted links to earlier

    http://en.wikipedia.org/wiki/Baker's_theorem
     
  4. Atticus

    Atticus Deity Retired Moderator

    Joined:
    Aug 20, 2006
    Messages:
    3,666
    Location:
    Helsinki, Finland
    It's just speculative, but if Idiotsopposite isn't doing heavy algebra, it's well justified speculation ;)

    I'm 97% sure that assuming log m to be algebraic would make e also that.
     
  5. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    I think IdiotsOpposite is still at school. I'd look for a slightly easier problem in that case to investigate ;)

    Even the proof that e is transcendental is quite tricky, proof here: http://en.wikipedia.org/wiki/Transcendental_number

    EDIT: My analysis textbook has a proof that e is irrational, using just properties of power series, but that is a long way off showing it is transcendental.
     
  6. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    I used this method (it's called von Neumann normalization, or something), in a project of mine, building a physical random number generator. I believe I posted about it here...
     
  7. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Hmm, can't find it.

    Another interesting one:
    100 prisoners are locked in prison. In this prison, there are 100 identical rooms. In every room, there are 100 numbered boxes, each box containing the name of one prisoner in such a way that every prisoners name is in one box* in every room (the rooms are identical, so each name is in the same box in every room).
    The prison director proposes the following scheme: "Each prisoner will be led into a separate room and will be allowed to open 50 boxes. If every prisoner opens the box containing his own name, all prisoners will be released."
    Initially, the prisoners don't really like this scheme, each individual prisoners has a 50% chance of opening the correct box, so in total they should have a 1/2^100 chance of success. However, luckily, there is a mathematician in the room. He devises a clever scheme that greatly increases the chance of them winning (to more than 30%!) and tells all the prisoners.

    Note (Big hint!): His scheme would also work if all other prisoners were robots/computers that could follow a simple algorithm. However, the scheme can't work if the clever mathematician is only allowed to give each prisoner a list of boxnumbers he should open.


    *All prisoners have different names
     
  8. LulThyme

    LulThyme King

    Joined:
    Sep 15, 2004
    Messages:
    622
    When you asked this question, I thought you already knew this, so I didn't mention it, to avoid spoiling things, but this trick actually has a name:
    Neumann's coin trick
    in reference to the first known reference (by von Neumann).
     
  9. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Hey! I am as great as von Neumann then!

    This device will make me famous!


    Link to video.

    Dutchfire's problem sounds similar to one I heard before about lightbulbs. I'll have a drunken think about it after listening to the Big Black song.
     
  10. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Dutchfire, do the prisoners get to see the results of each box opening before opening another?
     
  11. IdiotsOpposite

    IdiotsOpposite Boom, headshot.

    Joined:
    Mar 10, 2007
    Messages:
    2,718
    Location:
    America.
    Well, I know that ln(2) and ln(3) are transcendental, and I believe that a integer multiple of a transcendental number is also transcendental. This means that if the natural log of all primes is transcendental, then the natural log of all composite numbers is transcendental, at least so I believe. (Because, for example, ln(4)=ln(2^2)=2ln(2) ) Although... if a and b are transcendental, is a+b also transcendental? I think that's rather necessary.

    And hey, I believe that any good mathematician can prove anything he wants if he has the will (and it's correct.)
     
  12. Mise

    Mise isle of lucy

    Joined:
    Apr 13, 2004
    Messages:
    28,622
    Location:
    London, UK
    I don't even see how that would help?!
     
  13. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Me neither, but it helps to get all the info beforehand ;) Especially when a bit tipsy.
     
  14. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Yeah, x=1 is the only rational number such that log(x) is algebraic, but if you know that the proof is trivial ;)

    if a and b are transcendental, a+b might not be. b = -a, b = -a + k, where k is algebraic, etc.

    EDIT: You are correct that any algebraic rational multiple of a transcendental number is also transcendental, however. That is easy to show.

    EDIT2: Might be true for algebraic multiples too, but that would need a proof.
     
  15. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Yes .
     
  16. IdiotsOpposite

    IdiotsOpposite Boom, headshot.

    Joined:
    Mar 10, 2007
    Messages:
    2,718
    Location:
    America.
    Damn. I was hoping that I could prove numbers like log(6) were transcendental by manner of them being log(2)+log(3).
     
  17. dutchfire

    dutchfire Deity Retired Moderator

    Joined:
    Jan 5, 2006
    Messages:
    14,106
    Location:
    -
    Gödel disagrees
     
  18. Mise

    Mise isle of lucy

    Joined:
    Apr 13, 2004
    Messages:
    28,622
    Location:
    London, UK
    I'm assuming the solution would have them all open each box with an equal probability, rather than simply opening at random?
     
  19. ParadigmShifter

    ParadigmShifter Random Nonsense Generator

    Joined:
    Apr 4, 2007
    Messages:
    21,810
    Location:
    Liverpool, home of Everton FC
    Might be true if the numbers aren't rational multiples of each other. Needs a proof though...

    EDIT: But the proof is probably covered by one of the 3 theorems I posted earlier. I think one of those proves the result you are after though.

    EDIT2: When I say log I mean log to the base e of course.
     
  20. IdiotsOpposite

    IdiotsOpposite Boom, headshot.

    Joined:
    Mar 10, 2007
    Messages:
    2,718
    Location:
    America.
    I think that for two transcendental numbers a and b, if b-a is not algebraic (or 0) then a+b is transcendental. But that would also need a proof.
     

Share This Page