A135611 Decimal expansion of sqrt(2) + sqrt(3).

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

Offset: 1

Author

N. J. A. Sloane, Mar 03 2008

Keywords

• nonn
• cons

Comments

From Alexander R. Povolotsky, Mar 04 2008: (Start)

The value of sqrt(2) + sqrt(3) ~= 3.146264369941972342329135... is "close" to Pi. [See Borel 1926. - Charles R Greathouse IV, Apr 26 2014] We can get a better approximation by solving the equation: (2-x)^(1/(2+x)) + (3-x)^(1/(2+x)) = Pi.

Olivier Gérard finds that x is 0.00343476569746030039595770020414255107204742044644777... (End)

Another approximation to Pi is (203*sqrt(2)+ 197*sqrt(3))/200 = 3.1414968... - Alexander R. Povolotsky, Mar 22 2008

Shape of a sqrt(8)-extension rectangle; see A188640. - Clark Kimberling, Apr 13 2011

This number is irrational, as instinct would indicate. Niven (1961) gives a proof of irrationality that requires first proving that sqrt(6) is irrational. - Alonso del Arte, Dec 07 2012

An algebraic integer of degree 4: largest root of x^4 - 10*x^2 + 1. - Charles R Greathouse IV, Sep 13 2013

Karl Popper considers whether this approximation to Pi might have been known to Plato, or even conjectured to be exact. - Charles R Greathouse IV, Apr 26 2014

Examples

			3.14626436994197234232913506571557044551247712918732870...
		

References

• Emile Borel, Space and Time (1926).
• Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
• Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 44.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
• Burkard Polster, Irrational roots, Mathologer video (2018)
• Karl Popper, The Open Society and Its Enemies, 1962.
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A188640, A089078, A002193, A002194, A010464, A340616.

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018

Maple

evalf(add(sqrt(ithprime(i)), i=1..2), 118);  # Alois P. Heinz, Jun 13 2022

Mathematica

r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A135611 *)
ContinuedFraction[t, 120]  (* A089078 *)
RealDigits[Sqrt[2] + Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Oct 22 2016 *)

PARI

sqrt(2)+sqrt(3) \\ Charles R Greathouse IV, Sep 13 2013

Sage

numerical_approx(sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

Formula

Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - N. J. A. Sloane, Aug 27 2018

Equals 1/A340616. - Hugo Pfoertner, May 08 2024

Equals Product_{k>=0} (((4*k + 1)*(12*k + 11))/((4*k + 3)*(12*k + 1)))^(-1)^k. - Antonio Graciá Llorente, May 22 2024