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Theodorus's Constant

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The name "Theodorus's constant" is used for at least two mathematical constants. The first Theodorus's constant is the elementary algebraic number sqrt(3), i.e., the square root of 3. It has decimal expansion

sqrt(3)=1.732050807...
(1)

(OEIS A002194) and is named after Theodorus, who proved that the square roots of the integers from 3 to 17 (excluding squares 4, 9, and 16) are irrational (Wells 1986, p. 34). The space diagonal of a unit cube has length sqrt(3).

sqrt(3) has continued fraction [1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001). In binary, it is represented by

sqrt(3)=1.1011101101100111101..._2
(2)

(OEIS A004547).

Another constant sometimes known as the constant of Theodorus is the slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point (x,y)=(1,0), given by

T=sum_(k=1)^(infty)1/((k+1)sqrt(k))
(3)
=sum_(k=1)^(infty)1/(k^(3/2)+k^(1/2))
(4)
=1/2-sum_(k=1)^(infty)(-1)^k[zeta(k+1/2)-1]
(5)
=1.8600250...
(6)

(OEIS A226317; Gautschi 1993; Gautschi 2010, pp. 1043, 1046; Finch 2019, p. 663), where zeta(z) is the Riemann zeta function.

See also

Irrational Number, Pythagoras's Constant, Square Root, Theodorus's Constant Digits, Theodorus Spiral

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References

Davis, P. J. Spirals from Theodorus to Chaos. Wellesley, MA: A K Peters, 1993.Finch, S. R. Mathematical Constants II. Cambridge, England: Cambridge University Press, 2019.Gautschi, W. "The Spiral of Theodorus, Special Functions, and Numerical Analysis." In Spirals from Theodorus to Chaos (Ed. P. J. Davis). Wellesley, MA: A K Peters, pp. 67-87, 1993.Gautschi, W. "The Spiral of Theodorus, Numerical Analysis, and Special Functions." J. Comput. Appl. Math. 235, 1042-1052, 2010. https://doi.org/10.1016/j.cam.2009.11.054.Jones, M. F. "Approximations to the Square Roots of the Primes Less than 100." Math. Comput. 22, 234-235, 1968.Sloane, N. J. A. Sequences A002194/M4326, A004547, A040001, and A226317 in "The On-Line Encyclopedia of Integer Sequences."Uhler, H. S. "Approximations Exceeding 1300 Decimals for sqrt(3), 1/sqrt(3), sin(pi/3), and Distribution of Digits in Them." Proc. Nat. Acad. Sci. USA 37, 443-447, 1951.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 34-35, 1986.

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Theodorus's Constant

Cite this as:

Weisstein, Eric W. "Theodorus's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TheodorussConstant.html

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