Golden ratio properties (Challenge #10)

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. =)

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6 Responses to Golden ratio properties (Challenge #10)

  1. Pseudonym says:

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

  3. Jonathan says:

    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:

    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:

    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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