It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property.

This number appears in the fractional expression for the golden ratio.

[1] The first sixty significant digits of its decimal expansion are: which can be rounded down to 2.236 to within 99.99% accuracy.

Despite having a denominator of only 72, it differs from the correct value by less than ⁠1/10,000⁠ (approx.

As of January 2022, the numerical value in decimal of the square root of 5 has been computed to at least 2,250,000,000,000 digits.

[2] The square root of 5 can be expressed as the simple continued fraction The successive partial evaluations of the continued fraction, which are called its convergents, approach

than any rational number with a smaller denominator.

The convergents, expressed as ⁠x/y⁠, satisfy alternately the Pell's equations[3] When

is approximated with the Babylonian method, starting with x0 = 2 and using xn+1 = ⁠1/2⁠(xn + ⁠5/xn⁠), the nth approximant xn is equal to the 2nth convergent of the continued fraction: The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial

, the golden ratio and the conjugate of the golden ratio (Φ = −⁠1/φ⁠ = 1 − φ) is expressed in the following formulae: (See the section below for their geometrical interpretation as decompositions of a

then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio: The quotient of

and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the Lucas numbers:[5] The series of convergents to these values feature the series of Fibonacci numbers and the series of Lucas numbers as numerators and denominators, and vice versa, respectively: In fact, the limit of the quotient of the

corresponds to the diagonal of a rectangle whose sides are of length 1 and 2, as is evident from the Pythagorean theorem.

and φ, this forms the basis for the geometrical construction of a golden rectangle from a square, and for the construction of a regular pentagon given its side (since the side-to-diagonal ratio in a regular pentagon is φ).

Since two adjacent faces of a cube would unfold into a 1:2 rectangle, the ratio between the length of the cube's edge and the shortest distance from one of its vertices to the opposite one, when traversing the cube surface, is

By contrast, the shortest distance when traversing through the inside of the cube corresponds to the length of the cube diagonal, which is the square root of three times the edge.

... and successively constructed using the diagonal of the previous root rectangle, starting from a square.

[8] A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ).

[9] It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square.

, the square root of 5 appears extensively in the formulae for exact trigonometric constants, including in the sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.

[10] The simplest of these are As such, the computation of its value is important for generating trigonometric tables.

is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a dodecahedron.

[7] Hurwitz's theorem in Diophantine approximations states that every irrational number x can be approximated by infinitely many rational numbers ⁠m/n⁠ in lowest terms in such a way that and that

, there are some irrational numbers x for which only finitely many such approximations exist.

[11] Closely related to this is the theorem[12] that of any three consecutive convergents ⁠pi/qi⁠, ⁠pi+1/qi+1⁠, ⁠pi+2/qi+2⁠, of a number α, at least one of the three inequalities holds: And the

in the denominator is the best bound possible since the convergents of the golden ratio make the difference on the left-hand side arbitrarily close to the value on the right-hand side.

In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.

, was shown to be Euclidean, and hence a unique factorization domain, by Dedekind.

like any other quadratic field, is an abelian extension of the rational numbers.

The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination of roots of unity: The square root of 5 appears in various identities discovered by Srinivasa Ramanujan involving continued fractions.