## 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. =)

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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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.