7の平方根
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".
For a family of good rational approximations, the square root of 7 can be expressed as the continued fraction
[ 2 ; 1 , 1 , 1 , 4 , 1 , 1 , 1 , 4 , ... ] = 2 + 1 1 + 1 1 + 1 1 + 1 4 + 1 1 + ... .