Square root of 7

Square root of 7

Rationality	Irrational
Representations
Decimal	2.645751311064590590..._10
Algebraic form	${\displaystyle {\sqrt {7}}}$
Continued fraction	${\displaystyle 2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}$

The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in surd form as:^[1]

${\displaystyle {\sqrt {7}}\,,}$

and in exponent form as:

${\displaystyle 7^{\frac {1}{2}}.}$

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.64575131106459059050161575363926042571025918308245018036833....^[2]

which can be rounded up to 2.646 to within about 99.99% accuracy (about 1 part in 10000); that is, it differs from the correct value by about 1/4,000. The approximation 127/48 (≈ 2.645833...) is better: despite having a denominator of only 48, it differs from the correct value by less than 1/12,000, or less than one part in 33,000.

More than a million decimal digits of the square root of seven have been published.^[3]

Rational approximations

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).

Geometry

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]

Outside of mathematics

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]

See also

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

References

This page was last edited on 23 December 2023, at 04:57