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Let's discuss Mathematics

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

  1. Robert FIN

    Robert FIN Monty n' Roll

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    Thank you. Awesome! The logic works now. Just didn't understand I had to factorize with the zero-points (in English??) method. I still got to think how to proof them that n(n-1)(n-2) is divisible by 3 but I think it will be an easier task, I'll ask if I have to. Thanks.
     
  2. warpus

    warpus In pork I trust

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    Actually instead of contradiction prove it using induction.

    Base case is simple.

    Assume true for n, prove for n+1 (induction step):

    Also let x = (n+2)

    We know that (n)(n-1)(n-2) is divisible by 3
    Therefore (x+2)(x+1)(x) is also divisible by 3

    Show that (x+3)(x+2)(x+1) is divisible by 3 to prove induction step

    = (x)(x+2)(x+1) + (3)(x+2)(x+1)

    We know that (x)(x+2)(x+1) is divisible by 3 as per our induction step assumption. (3)(x+2)(x+1) is also divisible by 3, obviously.

    Add two numbers divisible by 3 together and the result will also be divisible by 3. (This would have to be proved in some way but I'm lazy)

    Therefore (x+3)(x+2)(x+1) is divisible by 3
    Therefore (n+1)(n)(n-1) is divisible by 3

    Therefore the inductive step is proved.

    Therefore n(n-1)(n-2) is divisible by 3

    So of this notation is a bit off, you'll have to throw in k here and there and replace some of the n's I think, and expand on some of this, but you get the idea
     
  3. Kyriakos

    Kyriakos Alien spiral maker

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    Finished reading a book account of the proof offered in the early 90s for a^x + b^x = c^x not true for x>2 and a,b,c integers (ie so-called 'Fermat's last theorem').
    Book wasn't anything particularly good, (popularized account with general info, typically one chapter for each person of interest and his math used on this, and bio stuff). I liked the quest for unifying all math fields, cause i think it is obviously possible to do so (given all math is itself a subset in ways of human thought, and we use just that).
    I do wonder what kind of proof Fermat had in mind, though. Maybe going with geometry (infinite collapse, or huge number of sides polygon leading to circle) and probability (since raised powers are directly tied to probability anyway; i suppose this is why x^0 = 1, cause there is only one case of empty set).
     
  4. Samson

    Samson Deity

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    I think I read that book. I have always assumed he was just wrong.
     
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  5. Kyriakos

    Kyriakos Alien spiral maker

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    Me too. But i just assumed this for the book writer, not Fermat :D
     
  6. Ferocitus

    Ferocitus Deity

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    ll4tet.gif

    I wrote a memetic algorithm about 20 years ago that found this arrangement
    of hulls that minimises the wave-making resistance at one particular speed
    (technically, Froude number).
    It was surprising at first, but then very obvious after a little thought
    and some mathematical squiggling. I love that aspect of AI/nonlinear search.
     
  7. Kyriakos

    Kyriakos Alien spiral maker

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    ^Not commenting cause i don't know what that is.
    But i do love use of computers in graphic representation of math. It is something which allows for a (seemingly) very different source of input/impression, and thus may lead to furthering ideas on math itself (and not just consciously imo). :)
     
  8. Ferocitus

    Ferocitus Deity

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    gravmass.png bouvetwake.jpg bouvetclouds.jpg
    That one is simulating 4 ships travelling together and moving to the left
    at increasing speed. The ships each make waves themselves, but the wave
    patterns interfere. Sometimes the waves cancel at places behind the ship;
    sometimes they reinforce and the waves at that point are larger. At one speed,
    the waves almost disappear everywhere.
    Waves created by ships means that energy is being wasted, in the sense that
    it does not contribute to forward propulsion.

    We found ways of calculating the wave pattern made by ships that were up to
    100,000 times faster than was possible previously. That meant that I could
    then create a population of artificial lifeforms that had the mathematical
    behaviour of ships vis-a-vis waves. I then let that population evolve. By
    that I mean I kept the best ships, replaced the worse ones with random
    ships, and combined (or "mated") the best ships together. It's an extremely
    crude approximation of Darwinian evolution. After 1 to 100 hours of computer
    time, and many games of Civ while waiting, the population starts converging
    on a pattern that makes the smallest waves, i.e. it has the lowest wave drag.
    Without the mathematical work to devise fast accurate algorithms, it would have
    been a ridiculous exercise. Waiting a day or a few is Ok; waiting 100,000 to
    a million days is out of the question.
    For me it then all became a game, with a high score to beat (in that case, the
    lowest drag.)

    I also like creating patterns just for their own intrinsic beauty.
    The green one is just a collection of singularities (like tiny black holes and
    white holes) travelling just under the surface of an imaginary ocean.
    Grav-Mass is Richard Stallman's idea of celebrating Isaac Newton's birthday
    on the 25th of December.

    The grey picture is a simple wave pattern created by one singularity travelling
    towards the bottom left.

    It has some features that are similar to those in the bottom graphic, which
    is the pattern made by wind-driven clouds as they pass over a small island in
    the Indian Ocean.

    Of course, the real world is far more complicated than we could ever hope to
    simulate, but logic and mathematics (and physics) sometimes make you feel as
    if you are getting a glimpse into the structure of the universe and how it works.
    At another level, it's delusional, and that feeling of satisfaction is probably
    the same as when somebody with OCD lines up hundreds of matches on a table in
    nice straight lines. :)
     
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  9. Robert FIN

    Robert FIN Monty n' Roll

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    EDIT: I did the exam already so I don't need the answer anymore. They didn't ask it in the exam luckily :DDD

    I have the final exam of maths tomorrow. (high school)

    A question:

    How to count the radius when using integration to count volume? The formula is like: pi * (the integral of r^2 from starting point to ending point). And if the object spins around y or x axis, then the r is going to be just the function that is going to spin. But it changes somehow when the object spins around something like x=2 or y=-4 etc. In one example I read, there the object spins around x=3 so then they said that r=3-x and inserted that to the formula. How did they count the r, I don't understand the logic because they didn't explain.

    Again, sorry for bad English math terms. And thank you in advance.
     
    Last edited: Mar 22, 2017
  10. Danai Gurira

    Danai Gurira Chieftain

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    I have questions about my maths homework too as I am taking a summer courses and learning calculus. I am having problem with the derivatives of trigonometric functions. Here is the question:
    y = sin^4 (x)
    y = sin(x^4)
    Are the derivatives of these two the same?
    I need some help plz.
     
  11. Michkov

    Michkov Emperor

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    No they are not the same. How did you come up with the notion that the two derivatives would be the same?
     
  12. Petek

    Petek Alpha Centaurian Administrator Supporter

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    @Danal Gurira -- Hint: Study the examples in the link you provided and use the chain rule.
    @Michkov -- The question of whether or not the two derivatives are equal is the problem to be solved. Danal is not asserting that it's true.
     
  13. Michkov

    Michkov Emperor

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    Just trying to understand hangup is in the thought process. I could say apply the chain rule and see what happens, but that assumes she knows the chain rule and how to use it.
     
  14. AdrienIer

    AdrienIer Deity

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    Solution in spoilers in case you really can't solve it

    Spoiler :
    (Sin (x^4) )' = 4*(x^3)*cos (x^4) (it's (f (g (x)))' = g'(x)*f'(g (x)) with f = sin and g = x^4)
    (Sin (x)^4)' = cos (x)*4*(sin(x)^3) (this time f = x^4 and g = sin)

    So the derivatives of the two functions are different
     
    Last edited: Jul 8, 2017
  15. Petek

    Petek Alpha Centaurian Administrator Supporter

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    Please do not solve other member's homework for them. It is much more useful to provide hints. Thanks.
     
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  16. AdrienIer

    AdrienIer Deity

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    I'd say that it depends on at what stage the person needing help is. Sometimes you're at "I'm having a small problem with this can someone drop me a hint" and sometimes you're at "I'm at a complete loss I need to see it done properly once to get the basic principle". But you're right I should have at least spoilered it. (now that I have you may want to delete the quoted part of your message)
     
  17. Lohrenswald

    Lohrenswald 老仁森林

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    to have f(x) -> 0 for x -> positive infinity, A has to be 0, right?
    or is there something I'm missing?
     
  18. Olleus

    Olleus Deity

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    If a squared->0 faster than x->infinity you can also have A = B I believe
     
  19. AdrienIer

    AdrienIer Deity

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    If a>0 A=0. If a=0 A=B.
     
  20. Lohrenswald

    Lohrenswald 老仁森林

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    alright so, a chain mail or pyramid scheme kind of deal:

    we begin with one person, who "recruits" number a amount of people. Each of these a people are tasked with recruiting a more people. Also index the steps as n (wheather that first original guy's recruitment should be n=0 or n=1 I don't know (yet lol)). total number of people recruited is N.

    So, here's my lazy question lol: what does the function N(n) look like? It can't be as simple as a^n, because that doesn't factor in the people recruited before step n

    also, the obvious question is how many steps it takes to recruit all of the earth's population
     

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