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

,

which can be found using the quadratic formula:

Closely related is its cousin, (phi-hat) . 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?

Extra points for especially slick proofs. =)

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

Expand the right-hand side and match up coefficients.

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

And so, using this property and property 3:

Because , we take the positive square root to prove property 4.

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

I like this one:

5.

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

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

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.

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