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、${\displaystyle r^{2}=r\times r=7}$ をみたす実数rであり、冪根形式では^[1]${\displaystyle {\sqrt {7}}\,}$、 指数形式では${\displaystyle 7^{\frac {1}{2}}}$と表される。無理数かつ代数的数である。

The rectangle that bounds an equialateral triangle of side 2, or a regular hexagon of side 1, has size square root of 3 by square root of 4, with a diagonal of square root of 7.

A Logarex system Darmstadt slide rule with 7 and 6 on A and B scales, and square roots of 6 and of 7 on C and D scales, which can be read as slightly less than 2.45 and somewhat more than 2.64, respectively

最初の60桁の有効数字は

2.64575131106459059050161575363926042571025918308245018036833...^[2]

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。

${\displaystyle {\sqrt {7}}\,}$の小数表示100万桁以上が公開されている^[3]。

有理近似

Explanation of how to extract the square root of 7 to 7 places and more, from Hawney, 1797

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^[4] and 1852,^[5] 3 in 1835,^[6] 6 in 1808,^[7] and 7 in 1797.^[8] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".^[9]

For a family of good rational approximations, the square root of 7 can be expressed as the continued fraction

${\displaystyle [2;1,1,1,4,1,1,1,4,\ldots ]=2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+\dots }}}}}}}}}}.}$ 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 ${\displaystyle {\sqrt {7}}}$:

${\displaystyle {\frac {2}{1}},{\frac {3}{1}},{\frac {5}{2}},{\frac {8}{3}},{\frac {37}{14}},{\frac {45}{17}},{\frac {82}{31}},{\frac {127}{48}},{\frac {590}{223}},{\frac {717}{271}},\dots }$

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 ${\displaystyle {\sqrt {7}}}$; in other words, it is closer to ${\displaystyle {\sqrt {7}}}$ 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:

${\displaystyle {\frac {2}{1}}=2.0,\quad {\frac {3}{1}}=3.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {8}{3}}=2.66\dots ,\quad {\frac {37}{14}}=2.6429...,\quad {\frac {45}{17}}=2.64705...,\quad {\frac {82}{31}}=2.64516...,\quad {\frac {127}{48}}=2.645833...,\quad \ldots }$

Every fourth convergent, starting with 8/3, expressed as x/y, satisfies the Pell's equation^[10]

${\displaystyle x^{2}-7y^{2}=1.}$

When ${\displaystyle {\sqrt {7}}}$ is approximated with the Babylonian method, starting with x₁ = 3 and using x_n+1 = 1/2(x_n + 7/x_n), the nth approximant x_n is equal to the 2ⁿth convergent of the continued fraction:

${\displaystyle x_{1}=3,\quad x_{2}={\frac {8}{3}}=2.66...,\quad x_{3}={\frac {127}{48}}=2.6458...,\quad x_{4}={\frac {32257}{12192}}=2.645751312...,\quad x_{5}={\frac {2081028097}{786554688}}=2.645751311064591...,\quad \dots }$

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 ${\displaystyle x^{2}-7}$. The Newton's method update, ${\displaystyle x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),}$ is equal to ${\displaystyle (x_{n}+7/x_{n})/2}$ when ${\displaystyle f(x)=x^{2}-7}$. The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).

幾何学

Root rectangles illustrate a construction of the square root of 7 (the diagonal of the root-6 rectangle).

平面幾何学において、${\displaystyle {\sqrt {7}}\,}$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^[11][12][13]。

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${\displaystyle {\sqrt {7}}\,}$の対角線を持つ^[14]。

数学以外の分野

Scan of US dollar bill reverse with root 7 rectangle annotation

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${\displaystyle {\sqrt {7}}\,}$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）^[15]。

関連項目

• 平方根
• 2の平方根
• 3の平方根
• 5の平方根
• 6の平方根
• 10の平方根