The square root of 5 is the positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
. It can be denoted in
surd form as:
:
It is an
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
. The first sixty significant digits of its
decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, ...
are:
: .
which can be rounded down to 2.236 to within 99.99% accuracy. The approximation (≈ 2.23611) for the square root of five can be used. Despite having a
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
of only 72, it differs from the correct value by less than (approx. ). As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.
Rational approximations
The square root of 5 can be expressed as the
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
:
The successive partial evaluations of the continued fraction, which are called its
''convergents'', approach
:
:
Their numerators are 2, 9, 38, 161, … , and their denominators are 1, 4, 17, 72, … .
Each of these is a
best rational approximation
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer p ...
of
; in other words, it is closer to
than any rational with a smaller denominator.
The convergents, expressed as , satisfy alternately the
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
s
:
When
is approximated with the
Babylonian method, starting with and using , the th approximant is equal to the th convergent of the continued fraction:
:
The Babylonian method is equivalent to
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
for root finding applied to the polynomial
. The Newton's method update,
, is equal to
when
. The method therefore
converges quadratically.
Relation to the golden ratio and Fibonacci numbers
The
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
is the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
of
1 and
. The
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
ic relationship between
, the golden ratio and the
conjugate of the golden ratio () is expressed in the following formulae:
:
(See the section below for their geometrical interpretation as decompositions of a
rectangle.)
then naturally figures in the closed form expression for the
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s, a formula which is usually written in terms of the golden ratio:
:
The quotient of
and (or the product of
and ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the
Lucas number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci n ...
s:
:
The series of convergents to these values feature the series of Fibonacci numbers and the series of
Lucas number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci n ...
s 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
:
:
Geometry
Geometrically,
corresponds to the
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
of a
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
whose sides are of length
1 and
2, as is evident from the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
. Such a rectangle can be obtained by halving a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, 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
In the field of 3D computer graphics, a subdivision surface (commonly shortened to SubD surface) is a curved surface represented by the specification of a coarser polygon mesh and produced by a recursive algorithmic method. The curved surface, ...
. Together with the algebraic relationship between
and , this forms the basis for the geometrical construction of a
golden rectangle
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.
Golden rectangles exhibit a special form of self-similarity ...
from a square, and for the construction of a regular
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be sim ...
given its side (since the side-to-diagonal ratio in a regular pentagon is ).
Since two adjacent faces of a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
would unfold into a 1:2 rectangle, the ratio between the length of the cube's
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed ...
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
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...
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 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 ), or into two golden rectangles of different sizes (of dimensions and ). It can also be decomposed as the union of two equal golden rectangles (of dimensions ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between
, and 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.
Trigonometry
Like
and
, 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. The simplest of these are
:
As such the computation of its value is important for
generating trigonometric tables
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tab ...
. 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
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
.
Diophantine approximations
Hurwitz's theorem in
Diophantine approximations states that every
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
can be approximated by infinitely many
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s in
lowest terms in such a way that
:
and that
is best possible, in the sense that for any larger constant than
, there are some irrational numbers for which only finitely many such approximations exist.
Closely related to this is the theorem
[ that of any three consecutive convergents , , , of a number , 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
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
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.
Algebra
The ring