Thursday, September 22, 2016
Entry 3
Ramanujan gave the beautiful continued fractions (with the second simplified by this author) 51/4√ϕ−ϕ=e−2π/51+e−2π1+e−4π1+e−6π1+⋱ √51+ϕ−1(−ϕ5+√ϕ10+1)1/5−ϕ=e−2π/√51+e−2π√51+e−4π√51+e−6π√51+⋱ (−ϕ5+√ϕ10+1)1/5=e−2π/(5√5)1+e−2π/√51+e−4π/√51+e−6π/√51+⋱with the golden ratio ϕ=1+√52. In 1913, the British mathematician G.H. Hardy, after reading the letter Ramanujan sent to him (which included examples of these extraordinary continued fractions), remarked, “…the [theorems] defeated me completely; I had never seen anything in the least like them before.” He would have been even more amazed had he known that these were connected to geometry.
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