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Thread: Interesting mathematical statements

  1. #51
    what is that?It is Cowpea RealiseNothing's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by glittergal96 View Post
    Classic one.

    For arbitrary polynomial equations of degree 1,2,3,4 over the complex numbers, we can find the solutions in terms of the coefficients, the basic operations of arithmetic, and radicals (taking roots of some degree).

    For degree 5 and greater, one can prove that this is not generally possible!
    Pretty pumped to do this in galois theory this sem (apparently it's related to the group from algebra?).

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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    Missed and fixed.

    Next bizarre truth



    Supposedly first proven by Bernoulli.

    Here is the Polylogarithmic evaluation of the infinite sum.









    Idk how Bernoulli did it, but you can easily evaluate such sums (replace 3 with any positive integer in fact) by repeatedly applying the operator (z d/dz) to the geometric series, which we already know how to sum. Then just chuck in 1/2.

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    Re: Interesting mathematical statements

    @realisenothing

    Yep, solvability of a polynomial equation is related to a certain group theoretic property. The symmetric group S_n has this property iff n < 5.

    (And A_60 fails to have this property as it is structurally pretty much the same as S_5.)
    Last edited by glittergal96; 25 Dec 2015 at 9:55 AM.

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    Re: Interesting mathematical statements

    So, to be more explicit about the geometric series approach to that summation:



    where



    is the geometric series in the unit disk (which is easy to differentiate).

    This gives us the rational function expression for S.

  5. #55
    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    An Identity that comes from the proof of Euler's Partition Theorem:



    In other words...



    Don't ask me about the radius of convergence, I have no idea.
    kawaiipotato likes this.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  6. #56
    Loquacious One turntaker's Avatar
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    Re: Interesting mathematical statements

    Its Christmas why r u guys doing maths.
    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    Re: Interesting mathematical statements

    Radius of convergence will just be 1.

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    Re: Interesting mathematical statements

    Quote Originally Posted by turntaker View Post
    Its Christmas why r u guys doing maths.
    if you think this is surprising, then you're in for a shocker if you knew me IRL.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  9. #59
    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    if you think this is surprising, then you're in for a shocker if you knew me IRL.
    Life isn't all about maths though...
    -------------------------------------
    Brahmagupta's formula:


  10. #60
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    Life isn't all about maths though...
    -------------------------------------
    Brahmagupta's formula:

    No. Almost all of it is though. :PPPPPP

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    A must know for the Ext 2 student:

    Fundamental Theorem of Algebra

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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    This has appeared elsewhere on this forum before:

    Expressions for the Golden Ratio



  13. #63
    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    If you square the first 9 numbers with only 1's in them:

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  14. #64
    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements







    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  15. #65
    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Very famous result on what e actually is:



    Still famous but not as famous result on what e is, esp amongst HSC students:


  16. #66
    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    Very famous result on what e actually is:



    Still famous but not as famous result on what e is, esp amongst HSC students:





    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Tbh isn't the first statement really just a Taylor series?

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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    Tbh isn't the first statement really just a Taylor series?
    Yeah, but sometimes it's actually more convenient to define functions by their power series representation and then prove other things about them.

  19. #69
    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by InteGrand View Post
    Yeah, but sometimes it's actually more convenient to define functions by their power series representation and then prove other things about them.








    The first and second statements are nearly equivalent down to the family of solutions for any differential equation.

    The third statement is somewhat convoluted.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  20. #70
    Taking a break! dan964's Avatar
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    Re: Interesting mathematical statements

    987654321 is divisible by 9
    987654312 is divisible by 8 (this particular one gives 123456789 btw)
    987654213 is divisible by 7
    987653214 is divisible by 6
    987643215 is divisible by 5
    987543216 is divisible by 4
    986543217 is divisible by 3
    976543218 is divisible by 2
    876543219 is divisible by 1

    0123456789 * 2 = 246913578 (a permutation of 0123456789)
    And likewise up to 100; excluding multiples of 3

  21. #71
    HSC Hipster Soulful's Avatar
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    Re: Interesting mathematical statements

    There is approximately a 50% chance of two people sharing the same birthday in a room of 23 people


    ~ First in Drama 2015 ~

  22. #72
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    Re: Interesting mathematical statements

    Quote Originally Posted by Soulful View Post
    There is approximately a 50% chance of two people sharing the same birthday in a room of 23 people
    This is, of course, assuming that every birthday date is identically probable, and that leap years do not exist, and that birthdays are treated as truly random. Under these conditions, a group of 70 people have a 99.9% chance of two people sharing a birthday.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  23. #73
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    Re: Interesting mathematical statements

    Certainly not a pure mathematician, but Cauchy's Integral Formula stands out for me.

    Then I moved to stats.

  24. #74
    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by AAEldar View Post
    Certainly not a pure mathematician, but Cauchy's Integral Formula stands out for me.

    Then I moved to stats.
    That's not a bad thing, there are unsolved problems in stats applicable to the real world such as p-values, error values, correlation vs. causation, interpolation of data, etc.
    Most of these are relevant to the sciences, as statistical methods are required for collecting any information. Even the medical fields and the humanities require it.

    But I digress.



    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  25. #75
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    Re: Interesting mathematical statements

    There is a lot of highly theoretical mathematics here.
    Does anyone have more real-world mathematical statements?
    (The birthday problem was a good one)

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