A219705 Decimal expansion of cos(log(2)).

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

Offset: 0

Author

Alonso del Arte, Nov 25 2012

Keywords

• nonn
• cons

Comments

In a letter to Christian Goldbach dated December 9, 1741, Leonhard Euler gave 10/13 as a rational approximation of this number.

Also, real part of 2^i. - Bruno Berselli, Dec 31 2012

The imaginary part of 2^i is A220085. - Robert G. Wilson v, Feb 04 2013

Examples

			0.76923890136...
		

References

• Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 309.
• W. Michael Kelley, The Humongous Book of Calculus Problems. New York: Alpha Books (Penguin Group) p. 233, Problem 15.22.

Links

• Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), 143 - 144.
• Elizabeth Volz, An English translation of portions of seven correspondences between Euler and Goldbach on Euler’s complex exponential paradox and special values of cosine, 2008
• Elizabeth Volz and Hieu D. Nguyen, Euler, Goldbach and exact values of trigonometric functions, 2008 preprint

Crossrefs

Cf. A002162, A021017, A220085 (imaginary part of 2^i).

Programs

Mathematica

RealDigits[Cos[Log[2]], 10, 105][[1]]

Maxima

fpprec:110; ev(bfloat(cos(log(2)))); /* Bruno Berselli, Dec 31 2012 */

PARI

cos(log(2)) \\ Charles R Greathouse IV, Nov 25 2012

Formula

cos(log(2)) = (2^i + 2^(-i))/2.

Extensions

a(43) ff. corrected by Georg Fischer, Apr 03 2020