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Math Help - continued fraction

  1. #1
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    continued fraction

    i have been shortly looking at the continued fraction (that Mr PerfectHacker seems to make publicity for) in mathworld enciclopeadia and i have not been conviced by that (divine, marvellous) representation of numbers:
    how do you add them divide them multiply them compare them? (for example)!

    i agree that (if its true) the representation of the square roots of two seems more periodic in this representation than in the decimal one but when i see the decimal representation of this number i have got some sort of an idea of what number this is and that seems to be a facility the continued fraction does not offer!

    What do you think?
    Last edited by SkyWatcher; February 26th 2006 at 10:42 AM.
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  2. #2
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    Quote Originally Posted by SkyWatcher
    i have been shortly looking at the continued fraction (that Mr PerfectHacker seems to make publicity for) in mathworld enciclopeadia and i have not been conviced by that (divine, marvellous) representation of numbers:
    how do you add them divide them multiply them compare them? (for example)!

    i agree that (if its true) the representation of the square roots of two seems more periodic in this representation than in the decimal one but when i see the decimal representation of this number i have got some sort of an idea of what number this is and that seems to be a facility the continued fraction does not offer!

    What do you think?
    There is no basic method for adding them or multiplying. For one thing countinued fractions sometimes a very elegant way to express numbers. For example the square roots irrationals have no repeating decimals but they have a repeating countinued fraction! Another place where countinued fractions are useful is when you want to approximate an irrational number by rationals. The beauty of this is that using countinued fractions you get the best possible fraction with that denominator as little as it can be. Further because,
    \left|x-\frac{p_n}{q_n}\right|<\frac{1}{q^2_n} the rate of convergence is extreme. Thus, as few convergents after the expansion and you have a super accurate approximation to 10 decimal places!
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  3. #3
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    I dont understand what the representation of a "suite" of numbers will have to do whith the velocity of the approximation of the "irrational" that "serie" of numbers is suppose to approximate(excuse my very bad english).
    whenever i made an approximation of a irrational number by a "serie" i have not use any decimal approximation of the numbers of the "serie" (the serial numbers) but letters and function.
    of course some series are very slow and some are quicker (but the real thing to consider is the speed of your computing machine when you want to get The approximation)
    So i think you should give a concrete exemple in order to conviced me!

    another thing i forgot to ask yesterday is "Is the continued fraction of a number is unique?"
    that i can check when i will have got time!
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  4. #4
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    Quote Originally Posted by SkyWatcher
    I dont understand what the representation of a "suite" of numbers will have to do whith the velocity of the approximation of the "irrational" that "serie" of numbers is suppose to approximate(excuse my very bad english).
    whenever i made an approximation of a irrational number by a "serie" i have not use any decimal approximation of the numbers of the "serie" (the serial numbers) but letters and function.
    of course some series are very slow and some are quicker (but the real thing to consider is the speed of your computing machine when you want to get The approximation)
    So i think you should give a concrete exemple in order to conviced me!

    another thing i forgot to ask yesterday is "Is the continued fraction of a number is unique?"
    that i can check when i will have got time!
    Let me answer your questions backwards.
    1)Uniqueness?
    We usually forbid the use of 1 in a finite countinued fraction for example,
    \frac{1}{1+\frac{1}{2+\frac{1}{3}}}=[0;1,2,3]
    Is clearly the same as,
    \frac{1}{1+\frac{1}{2+\frac{1}{2+\frac{1}{1}}}}=[0;1,2,2,1].
    Thus, by rules we forbid of using the last element to be one because of this. In case of infinite countinued fractions, yes!
    Rational: Then the countinued fraction is finite and is unique.
    Irrational: Then the countinued fraction is INfinite and is unique.
    Thus, two questions are answered; uniqueness and existence (namely that each number has a countinued fraction).

    Here is an example,
    Let us assume you wish to approximate \pi. Now you can say that \pi\approx 3.1415926... and from there stop at a specific decimal and express it as a fraction. The problem is that this is super-slow converging. But with countinued fraction it is much more accurate. Its countinued fraction starts out as
    [3;7,15,1,...]. Now if you stop at a certain point say two to the left you have \frac{333}{106} which is extremely accurate. Further, it can be shown that no other such fraction with a smaller denominator is a closer approximation.
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