7の平方根

7の平方根（ななのへいほうこん、英: square root of 7）は、平方して7となる実数である。すなわち、 r 2 = r × r = 7 {\displaystyle r^{2}=r\times r=7} をみたす実数rであり、冪根形式では 7 {\displaystyle {\sqrt {7}}\,} 、 指数形式では 7 1 2 {\displaystyle 7^{\frac {1}{2}}} 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 2.64575131106459059050161575363926042571025918308245018036833... 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1/4,000異なっている。127/48（≈ 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1/12,000（33000分の1）未満の差しかない。 7 {\displaystyle {\sqrt {7}}\,} の小数表示100万桁以上が公開されている。 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 + … . {\displaystyle [2;1,1,1,4,1,1,1,4,\ldots ]=2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+\dots }}}}}}}}}}.} オンライン整数列大辞典の数列 A010121 The successive partial evaluations of the continued fraction, which are called its convergents, approach 7 {\displaystyle {\sqrt {7}}} : 2 1 , 3 1 , 5 2 , 8 3 , 37 14 , 45 17 , 82 31 , 127 48 , 590 223 , 717 271 , … {\displaystyle {\frac {2}{1}},{\frac {3}{1}},{\frac {5}{2}},{\frac {8}{3}},{\frac {37}{14}},{\frac {45}{17}},{\frac {82}{31}},{\frac {127}{48}},{\frac {590}{223}},{\frac {717}{271}},\dots } Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…オンライン整数列大辞典の数列 A041008 , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…オンライン整数列大辞典の数列 A041009. Each convergent is a best rational approximation of 7 {\displaystyle {\sqrt {7}}} ; in other words, it is closer to 7 {\displaystyle {\sqrt {7}}} 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: 2 1 = 2.0 , 3 1 = 3.0 , 5 2 = 2.5 , 8 3 = 2.66 … , 37 14 = 2.6429... , 45 17 = 2.64705... , 82 31 = 2.64516... , 127 48 = 2.645833... , … {\displaystyle {\frac {2}{1}}=2.0,\quad {\frac {3}{1}}=3.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {8}{3}}=2.66\dots ,\quad {\frac {37}{14}}=2.6429...,\quad {\frac {45}{17}}=2.64705...,\quad {\frac {82}{31}}=2.64516...,\quad {\frac {127}{48}}=2.645833...,\quad \ldots } Every fourth convergent, starting with 8/3, expressed as x/y, satisfies the Pell's equation x 2 − 7 y 2 = 1. {\displaystyle x^{2}-7y^{2}=1.} When 7 {\displaystyle {\sqrt {7}}} is approximated with the Babylonian method, starting with x1 = 3 and using xn+1 = 1/2(xn + 7/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction: x 1 = 3 , x 2 = 8 3 = 2.66... , x 3 = 127 48 = 2.6458... , x 4 = 32257 12192 = 2.645751312... , x 5 = 2081028097 786554688 = 2.645751311064591... , … {\displaystyle x_{1}=3,\quad x_{2}={\frac {8}{3}}=2.66...,\quad x_{3}={\frac {127}{48}}=2.6458...,\quad x_{4}={\frac {32257}{12192}}=2.645751312...,\quad x_{5}={\frac {2081028097}{786554688}}=2.645751311064591...,\quad \dots } 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 x 2 − 7 {\displaystyle x^{2}-7} . The Newton's method update, x n + 1 = x n − f ( x n ) / f ′ ( x n ) , {\displaystyle x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),} is equal to ( x n + 7 / x n ) / 2 {\displaystyle (x_{n}+7/x_{n})/2} when f ( x ) = x 2 − 7 {\displaystyle f(x)=x^{2}-7} . The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps). 平面幾何学において、 7 {\displaystyle {\sqrt {7}}\,} 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 7 {\displaystyle {\sqrt {7}}\,} の対角線を持つ。 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 7 {\displaystyle {\sqrt {7}}\,} 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）。 平方根 2の平方根 3の平方根 5の平方根 6の平方根（英語版）