Plane Geometry - Bruce E. Shapiro

SECTION 44. ESTIMATING π 247Define the notation{x} = x mod 1, the fractional part of x. Then{ n}∑{16 n 16 n−k ∞S p } =8k + p +∑ 16 n−k8k + pk=0k=n+1{ n{∑ 16 n−k , 8k + p }=8k + pk=0}∞∑ 16 n−k+8k + pDefines p (n) =k=n+1{ n∑ 16 n−k , 8k + p }k=0Then the n th hex-digit of π is given by8k + pd n = ⌊16 {4{s 1 (n)}2{s 4 (n)} − {s 5 (n)}−{s 6 (n)}}⌋where ⌊x⌋ denotes the greatest integer less than or equal to x. The followinggives a Mathematica implementation:s[p_, n_]:= Sum[Mod[16^(n-k), 8k+p]/(8k+p),{k,0,n}];r[p_, n_]:= Mod[s[p,n],1];hexit[n_]:= Floor[16.0*Mod[4r[1,n]-2r[4,n]-r[5,n]-r[6,n], 1]];Thenhexit/@Range[0,11]gives the first 12 hex figures to the right hand side of the radix point:{2, 4, 3, 15, 6, 10, 8, 8, 8, 5, 10, 3}which corresponds toπ − 3 ≈ .243F 6A88 85A3 16The correct value of in hex to 16 figures isπ ≈ 3.243F 6A88 85A3 08D3 16To convert the number back to decimal, divide each integer by its correspondingpower of 16.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.