Square root of 7



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:
 * $$\sqrt{7}\,, $$

and in exponent form as:
 * $$7^\frac{1}{2}.$$

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:
 * 2.64575 13110  64590  59050  16157  53639  26042  57102  59183  08245  01803  6833....

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.

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 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


 * $$ [2; 1, 1, 1, 4, 1, 1, 1, 4,\ldots] = 2 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {4 + \cfrac 1 {1 + \dots}}}}}.$$

The successive partial evaluations of the continued fraction, which are called its convergents, approach $$\sqrt{7}$$:
 * $$\frac{2}{1}, \frac{3}{1}, \frac{5}{2}, \frac{8}{3}, \frac{37}{14}, \frac{45}{17}, \frac{82}{31}, \frac{127}{48}, \frac{590}{223}, \frac{717}{271}, \dots$$

Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257… , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271,  494,  765, 3554, 4319, 7873, 12192,….

Each convergent is a best rational approximation of $$\sqrt{7}$$; in other words, it is closer to $$\sqrt{7}$$ 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:
 * $$\frac{2}{1} = 2.0,\quad \frac{3}{1} = 3.0,\quad \frac{5}{2} = 2.5,\quad \frac{8}{3} = 2.66\dots,\quad \frac{37}{14} = 2.6429...,\quad \frac{45}{17} = 2.64705...,\quad \frac{82}{31} = 2.64516...,\quad \frac{127}{48} = 2.645833..., \quad \ldots$$

Every fourth convergent, starting with $8⁄3$, expressed as $x⁄y$, satisfies the Pell's equation
 * $$x^2 - 7y^2 = 1 .$$

When $$\sqrt{7}$$ is approximated with the Babylonian method, starting with $x_{1} = 3$ and using $x_{n+1} = 1⁄2(x_{n} + 7⁄x_{n})$, the $n$th approximant $x_{n}$ is equal to the $2^{n}$th convergent of the continued fraction:
 * $$ x_1 = 3, \quad x_2 = \frac{8}{3} = 2.66..., \quad x_3 = \frac{127}{48} =2.6458..., \quad x_4 = \frac{32257}{12192} = 2.645751312..., \quad x_5 = \frac{2081028097}{786554688} = 2.645751311064591..., \quad \dots$$

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 $$x^2-7$$. The Newton's method update, $$x_{n+1} = x_n - f(x_n)/f'(x_n),$$ is equal to $$(x_n + 7/x_n)/2$$ when $$f(x) = x^2 - 7$$. 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.

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

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

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.