Results 1 to 7 of 7

Math Help - Fractions

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    14

    Red face Fractions

    Continued fractions:
    X= a+b y=
    Last edited by miz.perfect84; August 11th 2009 at 11:14 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    AMI
    AMI is offline
    Junior Member AMI's Avatar
    Joined
    Jun 2009
    Posts
    40
    a) x-a=a+\frac{1}{1+\frac{1}{b_1+\frac{1}{b_2+\dots}}}-a=\frac{1}{1+\frac{1}{[b_1,b_2,\dots]}}=\frac{1}{1+\frac{1}{\beta}}=\frac{\beta}{\beta+  1} \Rightarrow(\beta+1)(x-a)=\beta (1)
    y-a=a+\frac{1}{1+b_1+\frac{1}{b_2+\frac{1}{b_3+\dots  }}}-a=\frac{1}{1+\beta} \Rightarrow(\beta+1)(y-a)=1 (2)
    (1) + (2) \Rightarrow x+y=2a+1\Rightarrow -x=y-(2a+1) \stackrel{2a+1\in\mathbb{Z}}{=} [a-(2a+1),1+b_1,b_2,b_3,\dots]=[-a-1,1+b_1,b_2,b_3,\dots] and -y=x-(2a+1)=\dots
    b) Apply a):
    x=[1,1,2,2,3,3,\dots],y_0:=[1,1+2,2,3,3,\dots] \stackrel{\text{a)}}{\Longrightarrow}-x=y_0-(2\times1+1)=y_0-3=\dots.
    y=[1,2,3,4,\dots],x_0:=[1,1,2-1,3,4,\dots]\Rightarrow-y=x_0-(2\times1+1)=\dots
    c) .. I think I don't understand the notation you are using..
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2009
    Posts
    14

    continued fraction

    the continued fraction of x, y], where x and y are positive integers.
    Last edited by miz.perfect84; August 11th 2009 at 11:14 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    AMI
    AMI is offline
    Junior Member AMI's Avatar
    Joined
    Jun 2009
    Posts
    40
    Hmm.. I still don't understand. You mean the problem asks to put -[a,b,a,b,a,b,a,\dots] in the form of a continued fraction?!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jun 2009
    Posts
    14

    yes

    where a and b are positive integers, for example - [1, 2, 1,2,1,2...]
    Last edited by miz.perfect84; August 11th 2009 at 11:16 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jun 2009
    Posts
    14

    error

    sorry the example was supposed to be -[1,2, 1, 2, 1, 2 ....]
    Last edited by miz.perfect84; August 11th 2009 at 11:15 AM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    AMI
    AMI is offline
    Junior Member AMI's Avatar
    Joined
    Jun 2009
    Posts
    40
    Well, in this case, just apply again a):
    y:=[a,b,a,b,\dots]=[a,1+(b-1),a,b,\dots]
    x:=[a,1,b-1,a,b,\dots]
    \Rightarrow-y=x-(2a+1)=[-a-1,1,b-1,a,b,\dots]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: April 28th 2010, 10:53 AM
  2. SAT fractions
    Posted in the Algebra Forum
    Replies: 4
    Last Post: December 1st 2009, 04:40 PM
  3. Simplifying Fractions over fractions
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 28th 2009, 11:20 AM
  4. Algebraic fractions with 3 or more fractions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: November 19th 2008, 03:52 AM
  5. simplifing fractions over fractions
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 16th 2007, 02:57 PM

Search Tags


/mathhelpforum @mathhelpforum