Square root of 7

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

Thesquare root of 7 is the positivereal number that, when multiplied by itself, gives theprime number7. It is more precisely called theprincipal 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 insurd form as:^[1]

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

and in exponent form as:

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

It is anirrationalalgebraic number. The first sixty significant digits of itsdecimal 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 about⁠1/4,000⁠. The approximation⁠127/48⁠ (≈ 2.645833...) is better: despite having adenominator of only 48, it differs from the correct value by less than⁠1/12,000⁠, or less than one part in 33,000.

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

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 byNewton'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 thecontinued fraction

${\displaystyle [2;1,1,1,4,1,1,1,4,\dots ]=2+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\dots }}}}}.}$ (sequenceA010121 in theOEIS)

The successive partial evaluations of the continued fraction, which are called itsconvergents, 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…(sequenceA041008 in theOEIS) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequenceA041009 in theOEIS).

Each convergent is abest 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:

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

Every fourth convergent, starting with⁠8/3⁠, expressed as⁠x/y⁠, satisfies thePell's equation^[10]

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

When${\displaystyle \sqrt{7}}$ is approximated with theBabylonian method, starting withx₁ = 3 and usingx_n+1 =⁠1/2⁠(x_n +⁠7/x_n⁠), thenth approximantx_n is equal to the2ⁿth convergent of the continued fraction:

${\displaystyle x_1=3,x_2=\frac{8}{3}=\mathrm{2.66...},x_3=\frac{127}{48}=\mathrm{2.6458...},x_4=\frac{32257}{12192}=\mathrm{2.645751312...},x_5=\frac{2081028097}{786554688}=\mathrm{2.645751311064591...},\dots }$

All but the first of these satisfy the Pell's equation above.

The Babylonian method is equivalent toNewton's method for root finding applied to the polynomial${\displaystyle x^2-7}$. The Newton's method update,${\displaystyle x_{n+1}=x_n-f(x_n)/f^{\prime }(x_n),}$ is equal to${\displaystyle (x_n+7/x_n)/2}$ when${\displaystyle f(x)=x^2-7}$. The method thereforeconverges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).

Inplane geometry, the square root of 7 can be constructed via a sequence ofdynamic rectangles, that is, as the largest diagonal of those rectangles illustrated here.^[11][12][13]

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

Due to thePythagorean theorem andLegendre's three-square theorem,${\displaystyle \sqrt{7}}$ is the smallest square root of anatural number that cannot be the distance between any two points of a cubicinteger lattice (or equivalently, the length of thespace diagonal of arectangular cuboid with integer side lengths).${\displaystyle \sqrt{15}}$ is the next smallest such number.^[15]

On the reverse of thecurrent 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

• Mathematical constants
• Quadratic irrational numbers

• Articles with short description
• Short description matches Wikidata