Positive real number which when multiplied by itself gives 5
Square root of 5
The square root of 5 is the positive
real number that, when multiplied by itself, gives the prime number
5. 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. It can be denoted in
surd form as:
2.23606797749978969640917366873127623544061835961152572427089... (sequence A002163 in the
which can be rounded down to 2.236 to within 99.99% accuracy. The approximation 161/72 (≈ 2.23611) for the square root of five can be used. Despite having a
denominator of only 72, it differs from the correct value by less than 1/10,000 (approx. 4.3×10−5). As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.
(See the section below for their geometrical interpretation as decompositions of a rectangle.)
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 (or the product 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
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 Lucas number and the Fibonacci number is directly equal to the square root of :
Geometrically, corresponds to the
diagonal of a rectangle whose sides are of length
2, as is evident from the
Pythagorean theorem. Such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. This can be used to subdivide a square grid into a tilted square grid with five times as many squares, forming the basis for a
subdivision surface. Together with the algebraic relationship between and φ, this forms the basis for the geometrical construction of a
golden rectangle from a square, and for the construction of a
regularpentagon 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.
A rectangle with side proportions 1: is called a root-five rectangle and is part of the series of root rectangles, a subset of
dynamic rectangles, which are based on (= 1), , , (= 2), ... and successively constructed using the diagonal of the previous root rectangle, starting from a square. 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). It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between , φ and Φ mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length to both sides.
As such the computation of its value is important for generating
trigonometric tables. Since 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
and that is best possible, in the sense that for any larger constant than , there are some irrational numbers x for which only finitely many such approximations exist.
Closely related to this is the theorem that of any three consecutive
convergentspi/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.
^Ivrissimtzis, Ioannis P.; Dodgson, Neil A.; Sabin, Malcolm (2005), "-subdivision", in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A. (eds.), Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003, Mathematics and Visualization, Berlin: Springer, pp. 285–299,
^Chapman, Scott T.; Gotti, Felix; Gotti, Marly (2019), "How do elements really factor in ?", in Badawi, Ayman; Coykendall, Jim (eds.), Advances in Commutative Algebra: Dedicated to David F. Anderson, Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 171–195,
S2CID119142526, Most undergraduate level abstract algebra texts use as an example of an integral domain which is not a unique factorization domain