7の平方根

7の平方根（ななのへいほうこん、）は、平方して7となる実数である。すなわち、formula_1
をみたす実数であり、冪根形式ではformula_2、
指数形式ではformula_3と表される。無理数かつ代数的数である。
最初の60桁の有効数字は
これは約99.99%の精度（1000分の1）以内で2.646に切り上げることができるが、正確な値とは約異なっている。（≈ 2.645833...）の方がより良い近似値である。分母がわずか48しかないにもかかわらず、正確な値とは（33000分の1）未満の差しかない。
formula_2の小数表示100万桁以上が公開されている。
有理近似.
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
The successive partial evaluations of the continued fraction, which are called its "convergents", approach formula_6:
Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257… , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,….
Each convergent is a best rational approximation of formula_6; in other words, it is closer to formula_6 than any rational with a smaller denominator. Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step:
Every fourth convergent, starting with , expressed as , satisfies the Pell's equation
When formula_6 is approximated with the Babylonian method, starting with and using , the th approximant is equal to the th convergent of the continued fraction:
All but the first of these satisfy the Pell's equation above.
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial formula_14. The Newton's method update, formula_15 is equal to formula_16 when formula_17. The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).
幾何学.
平面幾何学において、formula_2は一連の動的な長方形により、すなわち上図の長方形の最大の対角線として表される。
辺の長さが2の正三角形に外接する最小の長方形は長さformula_2の対角線を持つ。
数学以外の分野.
現行のアメリカ合衆国1ドル紙幣の裏にある大きな内箱は長さと幅の比がformula_2で、対角線の長さが6.0インチである（測定精度の範囲内で）。