Results 1 to 3 of 3

Math Help - Defining equation for golden ratio; quadratic fields

  1. #1
    Member
    Joined
    Jul 2008
    Posts
    78

    Defining equation for golden ratio; quadratic fields

    Definition: If \alpha is an irrational number in \mathbf{Q}\left(\sqrt{d}\right), then the equation ax^{2}+bx+c=0 is called the defining equation for \alpha if \alpha satisfies the equation and a, b, and c are integers, (a,b,c)=1, and a>0.

    Find a defining equation for the golden ratio \dfrac{1+\sqrt{5}}{2}.

    How do you approach this problem?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,
    Quote Originally Posted by Pn0yS0ld13r View Post
    Definition: If \alpha is an irrational number in \mathbf{Q}\left(\sqrt{d}\right), then the equation ax^{2}+bx+c=0 is called the defining equation for \alpha if \alpha satisfies the equation and a, b, and c are integers, (a,b,c)=1, and a>0.

    Find a defining equation for the golden ratio \dfrac{1+\sqrt{5}}{2}.

    How do you approach this problem?
    The solutions to such an equation are :

    x=\frac{-b {\color{red}\pm} \sqrt{b^2-4ac}}{2a}

    Since a and b are integers, the only way to get this \sqrt{5} is \sqrt{b^2-4ac}.
    So you can see that \frac{1{\color{red}-}\sqrt{5}}{2} will also be a root. (The reasoning may be similar to the ones involving complex roots).

    Develop \left(x-\frac{1+\sqrt{5}}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right) and identify a,b and c
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2008
    Posts
    78
    Thank you Moo.

    I can't believe I didn't get this before...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. About the golden ratio
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: July 3rd 2011, 03:33 AM
  2. [SOLVED] Golden Ratio
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: June 11th 2011, 07:00 PM
  3. Golden Ratios (Q Fields)
    Posted in the Number Theory Forum
    Replies: 10
    Last Post: July 14th 2010, 07:46 AM
  4. Golden ratio
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: August 17th 2008, 06:15 AM
  5. Golden Ratio equation
    Posted in the Algebra Forum
    Replies: 11
    Last Post: March 24th 2007, 06:12 PM

Search Tags


/mathhelpforum @mathhelpforum