Results 1 to 5 of 5

Math Help - How show continued fraction (65-sqrt(44))/37

  1. #1
    Newbie
    Joined
    Dec 2010
    From
    China
    Posts
    11

    How show continued fraction (65-sqrt(44))/37

    Please Some one give me the specific procedure . I'm confused because it's not the normal form . thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,738
    Thanks
    644
    Hello, wsc810!

    I have a solution, but it might not be acceptable.


    \text{Find the continued fraction for: }\:\dfrac{65 - \sqrt{44}}{37}

    I made two continued fractions.


    \displaystyle\text{We have: }\:\frac{65}{37} \;=\;1 + \frac{1}{1 + \frac{1}{3 +\frac{1}{9}}}

    . . \text{ which can be written: }\,\frac{65}{37} \:=\:[1,1,3,9]


    \displaystyle \text{We have: }\frac{\sqrt{44}}{37} \:=\:\frac{2}{37}\sqrt{11}

    \displaystyle \text{And: }\:\sqrt{11} \;=\;3 + \frac{1}{3 + \frac{1}{6 + \hdots}}}

    . . \text{which can be written: }\,\sqrt{11} \:=\:(\overline{3,3,6})


    \displaystyle \text{Therefore: }\:\frac{64-\sqrt{44}}{37} \;=\;[1,1,3,9] - \tfrac{2}{37}(\overline{3,3,6})

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Nov 2010
    Posts
    58
    Thanks
    18
    @wsc810 -- Your textbook should have a section that describes how to convert a general irrational number into a continued fraction. If so, just follow the instructions and apply them to your number. I'll give the steps needed to derive the first two partial quotients. You should be able to take it from there.

    Let \xi_0 = \frac{65 - \sqrt{44}}{37}. Then the first partial quotient, a_0, equals the integer part of \xi_0, or, using the floor function,

    \xi_0 = \lfloor\frac{65 - \sqrt{44}}{37}\rfloor

    which can easily be found with a calculator to equal 1. Thus, a_0 = 1.

    We now define \xi_1 := \frac{1}{\xi_0 - a_0} and a_1 := \lfloor\xi_1\rfloor. Therefore, since

    \xi_0 - a_0 = \frac{65 - \sqrt{44}}{37} - 1 = \frac{28 - \sqrt{44}}{37}

    and so

    \xi_1 = \frac{37}{28 - \sqrt{44}} = \frac{1036 + 37\sqrt{44}}{740}

    Thus,

    a_1 = \lfloor\xi_1\rfloor = 1

    which is again a simple calculation.

    So your continued fraction begins

    1 + \frac{1}{1 + \frac{1}{a_2 + \cdots}}

    You should be able to find the remaining partial quotients  a_2, a_3, \cdots by recursively defining

    a_i = \lfloor\xi_i\rfloor, \\\ \xi_{i+1} = \frac{1}{\xi_i - a_i}
    Last edited by Petek; March 10th 2011 at 02:50 AM. Reason: Typo
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Dec 2010
    From
    China
    Posts
    11
    Q_0=Q
    a_0=[(P a)/Q]
    P_0=P
    it's the starting value and a=Int(sqrt(d))
      P_{ n   1}=Q_n*a_n  - P_n
    Q_{n   1}=<br />
(d-P_{n   1}^2)/Q_n
    a_{n   1}=<br />
[(P_{n   1}    a)/Q_{n   1}]
    [ *] represents Int(*)
    a0=1
    P0=65
    Q0=37
    37*a0 - 65=-28 a0=1
    -20*a1-(-28)=8 a1=1
    1*a2 -28=6 a2=14
    why a2 is not right
    the actual vaule is 1 . Mathematica work out expression [1,1,1,2;1,2,1,1,1,12,1,1]
    Last edited by wsc810; March 10th 2011 at 04:27 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by wsc810 View Post
    Q_0=Q
    a_0=[(P + a)/Q]
    P_0=P
    it's the starting value and a=Int(sqrt(d))
    P_{ n + 1}=
    Q_n*a_n - P_n
    Q_{n + 1}=
    (d-P_{n + 1}^2)/Q_n
    a_{n + 1}=
    [(P_{n + 1} + a)/Q_{n + 1}]
    [ *] represents Int(*)
    a0=1
    P0=65
    Q0=37
    37*a0 - 65=-28 a0=1
    -20*a1-(-28)=8 a1=1
    1*a2 -28=6 a2=14
    why a2 is not right
    the actual vaule is 1 . Mathematica work out expression [1,1,1,2;1,2,1,1,1,12,1,1]

    A piece of advise: if you're going to write a lot or some rather involved mathematical

    expressions, you better take a peek at the forum's section "Math Resources", subsection

    "LaTeX help", and then re-write your stuff using LaTex, otherwise it comes out

    too messy and it might be that not many people will even take a look at it.

    For example, the following part of you message:

    P_{ n + 1}=
    Q_n*a_n - P_n
    Q_{n + 1}=
    (d-P_{n + 1}^2)/Q_n

    would come out, using LaTeX, as:

    \displaystyle{P_{n+1}=Q_na_n-P_n\,,\,\,Q_{n+1}=\frac{d-P_{n+1}^2}{Q_n}} ,and this is way clearer

    and thus much easier to read.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continued fraction.
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: January 27th 2011, 09:54 AM
  2. Value of a continued fraction.
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 11th 2009, 10:12 AM
  3. continued fraction help ?
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 10th 2008, 08:10 AM
  4. continued fraction
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: October 16th 2006, 09:11 PM
  5. continued fraction
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: February 27th 2006, 04:37 PM

Search Tags


/mathhelpforum @mathhelpforum