The Golden Ratio

Acknowledgement: Pretty Golden Ratio picture sourced from Wikipedia at http://en.wikipedia.org/wiki/File:FakeRealLogSprial.svg

Once again, from "The Math Book" by Clifford Pickover ("Golden Ratio" - page 112, c.1509). The simple definition of one of the many interesting properties of the Golden Ratio was that:

(a + b)/b = a/b = phi (Golden Ratio) where b is the longer section.

I had primarily wanted to find out how the number 1.61... was derived.

In this case, Wikipedia had the answer:

http://en.wikipedia.org/wiki/Golden_ratio

I was aware that the Fibonacci series was a common underlying feature in nature and was glad to see this partly discussed while examining the Golden Ratio's feature in the same aspects of nature. Come to think of it, I believe I was taught (just the facts, not any deeper understanding) the relationship between the Fibonacci series and the Continued Fraction form for describing the Golden Ratio. Somehow that fact never really stuck ... maybe merely learning the facts just wasn't very fun for an undergrad with so many things (including girls?) on my mind.

Anyway, through this entry, I had also learned that the square-root of any non-square natural number is irrational. Funny how it has taken me so many years to realize this! The proof can be found here:

http://en.wikipedia.org/wiki/Quadratic_irrational

In any event, I think I am satisfied for now. This was pretty fun, I think I enjoyed the little adventure in Math (re-)exploration.