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Decimal expansion of the imaginary part of the square root of 1 + i.
3

%I #36 Feb 24 2024 17:27:39

%S 4,5,5,0,8,9,8,6,0,5,6,2,2,2,7,3,4,1,3,0,4,3,5,7,7,5,7,8,2,2,4,6,8,5,

%T 6,9,6,2,0,1,9,0,3,7,8,4,8,3,1,5,0,0,9,2,5,8,8,2,5,9,5,6,9,4,9,0,8,0,

%U 0,2,0,3,2,3,3,4,4,8,2,9,1,5,9,1,4,0,1,8,1,9,7,6,1,0,2

%N Decimal expansion of the imaginary part of the square root of 1 + i.

%C i is the imaginary unit such that i^2 = -1.

%C Multiplied by -1, this is the imaginary part of the square root of 1 - i. And also the real part of -sqrt(1 + i) - i + sqrt(1 + i)^3, which is a unit in Q(sqrt(1 + i)).

%F Equals sqrt(1/sqrt(2) - 1/2) = 2^(1/4) * sin(Pi/8).

%F Equals sqrt((sqrt(2) - 1)/2) = A010767 * A182168. - _Bernard Schott_, Sep 16 2019

%F Equals Re(sqrt(-1 - i)). - _Peter Luschny_, Sep 20 2019

%F Equals Product_{k>=0} ((8*k - 1)*(8*k + 4))/((8*k - 2)*(8*k + 5)). - _Antonio GraciĆ” Llorente_, Feb 24 2024

%e Im(sqrt(1 + i)) = 0.45508986056222734130435775782247...

%p Digits := 120: Re(sqrt(-1 - I))*10^95:

%p ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Sep 20 2019

%t RealDigits[Sqrt[1/Sqrt[2] - 1/2], 10, 100][[1]]

%o (PARI) imag(sqrt(1+I)) \\ _Michel Marcus_, Sep 16 2019

%Y Cf. A000108, A010767, A182168, A309948 (real part).

%K nonn,cons,easy

%O 0,1

%A _Alonso del Arte_, Aug 24 2019