A202300 Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20.

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

Offset: 1

Author

Charles R Greathouse IV, Jan 11 2012

Keywords

• nonn
• cons

Comments

There is a small typo in Posamentier & Lehmann (2007): this number is given as approximately 1.3688081075 rather than 1.3688081078, a mistake that can't be justified by rounding rather than truncating nor a loss of machine precision. - Alonso del Arte, Mar 24 2012

Perhaps the reason for the mistake is that the authors got the correct answer mixed up with Fibonacci's answer, which, though wrong, was very good for the time: 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6 = 1.36880810785322... But apparently they truncated at the first 5 and left out the 8 before that 5. - Alonso del Arte, Jun 09 2014

The complex roots are -1.68440405391... +- 3.43133135... * i. - Alonso del Arte, Jun 21 2014

Fibonacci calculated this constant to six sexagesimal digits and proved that it was neither rational nor a square root of a rational. - Charles R Greathouse IV, Oct 21 2022

Examples

			x = 1.36880810782137263522741433002132553954243554148753653...
		

References

• John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press (2006): 69-70.
• Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 63-64.
• Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007) p. 21.

Links

• Ezra Brown and Jason C. Brunson, Fibonacci's forgotten number (archived link)
• Stanislaw Glushkov, On approximation methods of Leonardo Fibonacci, Historia Mathematica 3 (1976), pp. 291-296.
• Wolfram|Alpha, real root of x^3 + 2x^2 + 10x - 20 = 0
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A159990, A243629, A244467.

Programs

Mathematica

RealDigits[x /. FindRoot[x^3 + 2x^2 + 10x - 20 == 0, {x, 1.4}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Feb 27 2013 *)

PARI

real(polroots(x^3+2*x^2+10*x-20)[1])

PARI

polrootsreal(x^3+2*x^2+10*x-20)[1] \\ Charles R Greathouse IV, Jan 05 2016

Formula

x = (2*sqrt(3930)/9 - 352/27)^(1/3) + (2*sqrt(3930)/9 + 352/27)^(1/3) - 2/3;

x = (1/3)*(-2 - 13 * 2^(2/3)/(176 + 3*sqrt(3930))^(1/3) + (2*(176 + 3*sqrt(3930)))^(1/3)).

The first formula comes from Posamentier & Lehmann (2007), the second from Wolfram|Alpha. - Alonso del Arte, Mar 24 2012