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.

The first sixty significant digits of its decimal expansion are: 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] 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 The successive partial evaluations of the continued fraction, which are called its convergents, approach

Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step: Every fourth convergent, starting with ⁠8/3⁠, expressed as ⁠x/y⁠, satisfies the Pell's equation[10] When

is approximated with the Babylonian method, starting with x1 = 3 and using xn+1 = ⁠1/2⁠(xn + ⁠7/xn⁠), the nth approximant xn is equal to the 2nth convergent of the continued fraction: All but the first of these satisfy the Pell's equation above.

The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).

[11][12][13] The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 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).

[15] 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.