Golden ratio properties (Challenge #10)

Posted on September 6, 2007 by Brent

Remember the golden ratio, \varphi (phi)? It’s the positive solution to the equation

x^2 - x - 1 = 0,

which can be found using the quadratic formula:

\displaystyle \varphi = \frac{1 + \sqrt{5}}{2}.

Closely related is its cousin, \hat{\varphi} (phi-hat) = \frac{1 - \sqrt{5}}{2}. As we’ll see, these famous constants actually relate to Fibonacci numbers in some amazing ways. But first, we’ll need a few properties of these numbers. Can you show why each of the following is true?

  1. \varphi^2 = \varphi + 1
  2. \varphi + \hat{\varphi} = 1
  3. \varphi \hat{\varphi} = -1
  4. \varphi - \hat{\varphi} = \sqrt{5}

Extra points for especially slick proofs. =)

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
This entry was posted in challenges, famous numbers, fibonacci, golden ratio, number theory, proof. Bookmark the permalink.

6 Responses to Golden ratio properties (Challenge #10)

  1. Pseudonym says:
    September 7, 2007 at 2:53 am

    Well property 1 is obvious. For properties 2 and 3:

    x^2 - x - 1 = (x - \phi)(x - \hat{\phi})

    Expand the right-hand side and match up coefficients.

    For point 4, first we show, using property 2:

    \phi^2 + \hat{\phi}^2 = (\phi + 1) + (\hat{\phi} + 1) = 3

    And so, using this property and property 3:

    (\phi - \hat{\phi})^2 = \phi^2 - 2\phi\hat{\phi} + \hat{\phi}^2 = 3 + 2 = 5

    Because \phi > \hat{\phi}, we take the positive square root to prove property 4.

  2. Steve Gilberg says:
    September 7, 2007 at 11:38 am

    Too bad I don’t know how to insert formula symbols in comments.

  3. Jonathan says:
    September 7, 2007 at 10:46 pm

    I like this one:
    5. \frac{1}{\phi} = \phi - 1

    I’ve done them many times before, but I’ll have a go again, on the side, since they are so much fun!

  4. Jonathan says:
    September 9, 2007 at 4:50 pm

    Let me try adding my property 5 again. The only LaTeX I know is what I use in wordpress, so this may be tricky:

    \frac{1}{\varphi} = \varphi – 1

    I will be amazed and delighted if it works

  5. Brent says:
    September 11, 2007 at 4:22 pm

    Jonathan: I use wordpress, but running it myself instead of on wordpress.com; on this blog you can surround stuff with tex and /tex in square brackets to get LaTeX. Yes, your property 5 is a nice one too!

    Pseudonym: to get the phi-hat symbol in LaTeX, use \hat{\phi}, I don’t think \^ works in math mode.

  6. Pingback: Golden powers | The Math Less Traveled

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