Square root of 7

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

${\displaystyle [2;1,1,1,4,1,1,1,4,\ldots ]=2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+\dots }}}}}}}}}}.}$ (sequence A010121 in the OEIS)

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…(sequence A041008 in the OEIS) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequence A041009 in the OEIS).

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).

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

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.[11][12][13]

The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.[14]

Due to the Pythagorean theorem and Legendre's three-square theorem, ${\displaystyle {\sqrt {7}}}$ 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). ${\displaystyle {\sqrt {15}}}$ is the next smallest such number.[15]

Scan of US dollar bill reverse with root 7 rectangle annotation

On the reverse of the current US one-dollar bill, the "large inner box" has a length-to-width ratio of the square root of 7, and a diagonal of 6.0 inches, to within measurement accuracy.[16]

• Square root
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6