text: The maximum number of steps the Euclidean algorithm can take is asymptotically the natural log of n dividedby the natural log of this number.
full_question: The maximum number of steps the Euclidean algorithm can take is asymptotically the natural log of n dividedby the natural log of this number. The worst case of Borel's theorem on rational approximation is fornumbers in an equivalence class with this one. The continued fraction expansion of this number consists entirelyof ones, and it is equal to twice the cosine of 36 degrees. This number and its negative appear in Binet'sformula for the nth Fibonacci number. This number is the length of a regular pentagon's diagonal divided byits side length. For 10 points, identify this number, equal to 1 plus the square root of 5 over two, often denotedby the Greek letter phi. 
first_sentence: The maximum number of steps the Euclidean algorithm can take is asymptotically the natural log of n dividedby the natural log of this number.
raw_answer:  golden ratio [or golden mean; or obvious equivalents; or phi until it is read; or (1 + sqrt(5)) / 2 until it is read, prompt on 1.618033Â…] 