A309949 Decimal expansion of the imaginary part of the square root of 1 + i.

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, 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, 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

Offset: 0

Author

Alonso del Arte, Aug 24 2019

Keywords

• nonn
• cons
• easy

Comments

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

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)).

Examples

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

Crossrefs

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

Programs

Maple

Digits := 120: Re(sqrt(-1 - I))*10^95:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 20 2019

Mathematica

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

PARI

imag(sqrt(1+I)) \\ Michel Marcus, Sep 16 2019

Formula

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

Equals sqrt((sqrt(2) - 1)/2) = A010767 * A182168. - Bernard Schott, Sep 16 2019

Equals Re(sqrt(-1 - i)). - Peter Luschny, Sep 20 2019

Equals Product_{k>=0} ((8*k - 1)*(8*k + 4))/((8*k - 2)*(8*k + 5)). - Antonio Graciá Llorente, Feb 24 2024