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  1. #71
    Senior Member nemo's Avatar
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    By the way, my plans for this Cantor Set thing is turning into a giant spectacle involving the both the Cantor Set and it's application in dynamical systems, tent maps, Lyapunov exponents, etc. etc.

    I finish finals this afternoon, so hopefully it'll be done sometime this weekend.

    I just can't help myself. =/
    You can't wait for inspiration. You have to go after it with a club. - Jack London

  2. #72
    Senior Member snegledmaca's Avatar
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    Quote Originally Posted by nemo View Post
    By definition, the decimal expansion of some number r is



    Where {a1, a2, a3, ...,} is some sequence. Notice that the set {a1, a2, a3, ...} is necessarily infinite. (Hence the number 1 can also be expanded as 1.000000...).

    The problem is that infinite sums are not actual sums.

    They are the limit of the sequence of partial sums -- in this case {a1/10, a1/10 + a2/100, a1/10 + a2/100 + a3/1000, ...}.

    So you can't just compare each of your xn's and yn's.
    But there are no sums here. The expressions are not numbers. Well not initially. Initially they could just as likely be 0.skzhnfhbd.... depending on what the digits are. I just took a pick from a pool of digits that happen to contain the numbers from 0 to 9. I could've pick them from the set that contains the alphabet for example. I only later "converted" them into numbers, well not really, merely said ok, these expressions I made are now numbers. How would this affect things?

    It's totally legitimate that the sequences {a1, a2, a3, ...} and {b1, b2, b3, ...} might not be equal, but the limits of the sequence of their partial sums are the same. In this case the set of an's and bn's are, {1, 0.0, 0.00, 0.000, ...} and {0, .9, .09, .009, ...}.
    Aha, I think I understand. So you are saying that these expressions merely represent numbers and are not actually those numbers. And the numbers they represent are the limits of their mathematical interpretations.

    Why is that? Why couldn't we just say that these are different numbers, why do we say they represent the number that is the limes of a sum that represents the expressions?

    Also, wouldn't this mean that there are more sequences {a1, a2, a3, ...} then there are real numbers? If more then one sequence can be joined to every real number and the sequences joined are all different?

    If you must provide a proof via set theory, I'd probably go after some argument using the suprema and infima of the respective sets of partial sums. That's really just an analysis argument, though.
    Since there are no sums involved that is not applicable. And it's not a must, it's for a personal project.

    Also, I'm not exactly sure what you mean by "the cardinality of the set." Which set are you talking about? R, or your sequences {x1/10, x2/100, ... xn/10^n, ...} and {y1/10, ... yn/10^n, ...}?
    My set {{x1, x2, x3, ... , xn, ...}} where xn is from {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9',}.

  3. #73
    Senior Member nemo's Avatar
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    Quote Originally Posted by snegledmaca View Post
    But there are no sums here. The expressions are not numbers. Well not initially. Initially they could just as likely be 0.skzhnfhbd.... depending on what the digits are. I just took a pick from a pool of digits that happen to contain the numbers from 0 to 9. I could've pick them from the set that contains the alphabet for example. I only later "converted" them into numbers, well not really, merely said ok, these expressions I made are now numbers. How would this affect things?
    The sums come from the definition of decimal expansion.

    That's what the notation is. When you put the decimal dot . you're putting shorthand for a sum. Instead of writing 3 + 1/10 + 4/100, we write 3.14. It's purely notation, and not meaningful on it's own right. Saying there's no sum involved is like saying there's no multiplication in a^n.

    I really don't see how you can avoid that.

    Aha, I think I understand. So you are saying that these expressions merely represent numbers and are not actually those numbers. And the numbers they represent are the limits of their mathematical interpretations.
    The sequences are different, but the limits of their partial sums are the same.

    But those limits are actually the number. Just because it's a limit doesn't mean it's any less "real" or anything.

    My point was mostly to remind you that when working with infinite sums, you can't expect them to have all the properties of addition, because you're not actually adding anything. That was part of your argument with the xn != yn thing.

    Why is that? Why couldn't we just say that these are different numbers, why do we say they represent the number that is the limes of a sum that represents the expressions?
    Because it doesn't make any sense to literally add infinitely many objects up. Furthermore, if you try to use some of the properties of addition on infinite sums (e.g. associativity), you can get silly results like 0 = 1. The "limit of the partial sums" definition avoids that.

    Also, wouldn't this mean that there are more sequences {a1, a2, a3, ...} then there are real numbers? If more then one sequence can be joined to every real number and the sequences joined are all different?

    Since there are no sums involved that is not applicable. And it's not a must, it's for a personal project.
    I think this is really what you're getting at -- you seem to know that 0.999... = 1.

    As to using sets to prove it -- you might try something equivalent but stated a different way, for instance prove that the set [0,1) has no greatest element.

    The other stuff I'll get to later -- finals, etc.
    You can't wait for inspiration. You have to go after it with a club. - Jack London

  4. #74

    Default Does the "196 Algorithm" terminate when run on 196?

    Take any number:

    Say: 12,
    reverse it, yielding 21.
    Add the two together 12+21=33

    a palindrome!

    What about 129?
    reversed: 921
    added together: 1050
    reverse that: 0501
    add together: 1551

    a palindrome!

    In general, take a number x, reverse the digits to get y. Then let z=x+y.
    If z is a palindrome, you're done.
    If not set x=z, and repeat the process till you get a palindrome.

    The above algorithm is called the 196 algorithm.

    Now I ask a simple question, let x=196. Will the algorithm ever terminate?

    After a few iterations, here are the values x takes on:
    196, 887, 1675, 7436, 13783, ....

    Incidentally, x=195 terminates with z=9339, I believe.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
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  5. #75

    Default Are there any odd perfect numbers?

    A perfect number is a number that is equal to the sum of its proper divisors.

    For example:
    6=1+2+3
    28=1+2+4+7+14

    Now, my question is simple, are there any odd perfect numbers?

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  6. #76

    Default Are all positive even numbers greater than or equal to 4 the sum of two primes?

    4=2+2
    6=3+3
    8=3+5
    10=5+5
    12=5+7
    14=7+7
    16=5+11
    18=7+11
    20=7+13
    22=11+11
    24=11+13
    .
    .
    .
    Are all positive even numbers, greater than or equal to 4, the sum of two primes?

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  7. #77

    Default Are there infinitely many "twin primes?"

    If there is a prime p, and another prime q=p+2, then the pair of primes p and q are "twin primes."

    Examples:
    5 and 7 are twin primes.
    11 and 13 are twin primes.
    17 and 19 are twin primes.
    29 and 31 are twin primes.

    Are there infinitely many "twin primes?"

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  8. #78
    Senior Member Flush's Avatar
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    0.999... - Wikipedia, the free encyclopedia...

    What do you guys think?
    01001001010011100100011001010000

  9. #79

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    Quote Originally Posted by Flush View Post
    nemo had explained earlier (sort-of) why 0.999...=1.

    The set that snegledmaca was describing is actually a countably infinite set (has the same cardinality as the natural numbers).

    When you start including the limits of such sets (which is what infinite decimal expansions are), then we can represent real numbers too.

    However, now we can have more than one decimal expansion represent the same number.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  10. #80
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    Oh no.

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