At the SouthWest Fox speaker dinner we were talking about continued (or infinite) fractions.

I posed this question:

What does this expression evaluate to?

1 + 1

_________

1 + 1

________

1 + 1

_______

1 + 1

______

1 + 1

…

Clue: It’s adding some positive value to 1, so it must be larger than one. The thing it’s adding is the reciprocal of some number larger than 1 so….

The golden ratio : 1.618034

x = 1 + 1/x

==>

x = (1+sqrt(5))/2

==> 1.618034

1.618034 or -0.618034 (the golden ratio is an irrational number)

Raj:

Never assume that the fact that the expression contains sqrt(5) somewhere in it implies that both positive and negative roots are correct for the answer.

The answer must be positive in this case, and so you cannot take the negative root.

http://www.mlahanas.de/Greeks/GoldenSection.htm

It always helps if you have read "The Da Vinci Code"