The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is an irrational algebraic number. The first thirty significant digits of its decimal expansion are:
More than a million decimal digits of the square root of seven have been published.
Rational approximations
The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773 and 1852, 3 in 1835, 6 in 1808, and 7 in 1797. An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".
Geometry
In plane geometry, the square root of 7 can be constructed via a sequence of dynamic rectangles, that is, as the largest diagonal of those rectangles illustrated here. The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7. Due to the Pythagorean theorem and Legendre's three-square theorem, 7 {\displaystyle {\sqrt {7}}} is the smallest square root of a natural number that cannot be the distance between any two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). 15 {\displaystyle {\sqrt {15}}} is the next smallest such number.
See also Square root Square root of 2 Square root of 3 Square root of 5 Square root of 6 Square root of 8
References