square root of 3

The square root of 3MathworldPlanetmath, also known as Theodorus’s constant, is the number the square of which is equal to the integer 3. It is an irrational number, one of the first few to have been proved irrational. Theodorus of Cyrene proved that the square rootsMathworldPlanetmath of the integers 3, 5 to 8, 10 to 15 and 17 are all irrational. The decimal expansion of 3 is 1.7320508075688772935… (sequence http://www.research.att.com/ njas/sequences/A002194A002194 in Sloane’s OEIS). Its simple continued fractionMathworldPlanetmath is


repeating 1 and 2 periodically (Sloane’s http://www.research.att.com/ njas/sequences/A040001A040001).

Given a unit cube, the diagonal from the vertex joining three sides to the other vertex joining the three other sides is 3. Given a unit hexagon, the distance from one side to the parallel opposite side is 3. More generally, the ratio of the length of a side of a hexagon to the distance from that side to the opposing parallel side is 1:3, and the same ratio applies to the length of the side of a cube to the diagonal of that cube.


  • 1 M. F. Jones, “22900D approximations to the square roots of the primes less than 100”, Math. Comp 22 (1968): 234 - 235.
  • 2 H. S. Uhler, “Approximations exceeding 1300 decimals for 3, 13, sin(π3) and distribution of digits in them” Proc. Nat. Acad. Sci. U. S. A. 37 (1951): 443 - 447.
Title square root of 3
Canonical name SquareRootOf3
Date of creation 2013-03-22 17:29:46
Last modified on 2013-03-22 17:29:46
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A25
Synonym Theodorus’s constant
Synonym Theodorus’ constant