Square root of 7
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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: Template:Block indent
More than a million decimal digits of the square root of seven have been published.[1]
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[2] and 1852,[3] 3 in 1835,[4] 6 in 1808,[5] and 7 in 1797.[6] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".[7]
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.[8][9][10]
The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.[11]
Due to the Pythagorean theorem and Legendre's three-square theorem, 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). is the next smallest such number.[12]