%I #23 Oct 27 2023 10:26:40

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

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

%U 4,4,0,4,6,0,6,3,0,9,4,2,0

%N Decimal expansion of the real root of x^3 + x - 500.

%C This appears in the solution of the fifteenth of thirty problems posed by Antonio Maria Fiore in the year 1535 to Niccolò Tartaglia. See the Alten et al. reference, p. 272.

%C See the Fauvel and Gray reference, p. 254, where this problem is translated as "A man sells a sapphire for 500 ducats, making a profit of the cube root of his capital. How much is this profit?" and the explanation given there is [x^3 + x = 500.] One assumes that the capital is the value of the sapphire before the selling. Note that the authors use 'profit of the cube root', whereas in the Alten et al. reference this is translated as 'Gewinn in der dritten Potenz'. But here the 'cube root' interpretation seems more plausible because then from c + c^(1/3) = 500 the capital c turns out to be 500 - x1, about 492.10 ducats, and the profit c^(1/3) is about 7.90 ducats, that is about 1.6 percent. (Not a big deal, though.) The Alten et al. version would give for the capital x1 about 7.10 ducats and a huge profit of about 492.10 ducats. One would like to read the original.

%D H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 272.

%D John Fauvel and Jeremy Gray (eds.), The History of Mathematics: A Reader, Macmillan Press, The Open University, 1988.

%H MacTutor History of Mathematics, <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Tartaglia.html">Nicolo Tartaglia</a>.

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

%F The real solution of the equation x^3 + x - 500 = 0 is x1 = (1/3)*((6750 + 3*sqrt(5062503))^(1/3) - (-6750 + 3*sqrt(5062503))^(1/3)).

%F 6750 = 2*(3*5)^3, 5062503 = 3*229*7369.

%F The two complex solutions are a + i*b and a - i*b with a = -x1/2 and b = i*sqrt(3)*y1/2, where y1 =

%F (1/3)*((6750 + 3*sqrt(5062503))^(1/3) + (-6750 + 3*sqrt(5062503))^(1/3)).

%e x1 = 7.895008285535911478048911606395731...

%e y1 = 7.979003018047682483096396690816339...

%t RealDigits[N[Solve[x^3 + x - 500==0, x][[1]][[1, 2]], 120]][[1]] (* _Vincenzo Librandi_, May 09 2015 *)

%o (PARI) polrootsreal(x^3 + x - 500)[1] \\ _Charles R Greathouse IV_, May 11 2015

%Y Cf. A257235, A257236.

%K nonn,easy,cons

%O 1,1

%A _Wolfdieter Lang_, May 08 2015