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A202300
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Decimal expansion of the real root of x^3 + 2x^2 + 10x - 20.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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x = 1.36880810782137263522741433002132553954243554148753653...
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MATHEMATICA
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RealDigits[x /. FindRoot[x^3 + 2x^2 + 10x - 20 == 0, {x, 1.4}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Feb 27 2013 *)
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PROG
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(PARI) real(polroots(x^3+2*x^2+10*x-20)[1])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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