Irrationality The golden ratio is an
irrational number. Below are two short proofs of irrationality:
Contradiction from an expression in lowest terms , then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the positive integers have a lower bound, so cannot be rational. This is a
proof by infinite descent. Recall that: If we call the whole and the longer part , then the second statement above becomes To say that the golden ratio is rational means that is a fraction where and are integers. We may take to be in
lowest terms and and to be positive. But if is in lowest terms, then the equally valued is in still lower terms. That is a contradiction that follows from the assumption that is rational.
By irrationality of the square root of 5 Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the
closure of rational numbers under addition and multiplication. If is assumed to be rational, then , the
square root of , must also be rational. This is a contradiction, as the square roots of all non-
square natural numbers are irrational.
Minimal polynomial . The golden ratio's negative and reciprocal are the two roots of the quadratic polynomial . Since the golden ratio is a root of a polynomial with rational coefficients, it is an
algebraic number. Its
minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is x^2 - x - 1. This
quadratic polynomial has two
roots, and {{tmath|1=\textstyle -\varphi^{-1} }}. Because the
leading coefficient of this polynomial is 1, both roots are
algebraic integers. The golden ratio is also closely related to the polynomial , which has roots and {{tmath|\textstyle \varphi^{-1} }}. The golden ratio is a
fundamental unit of the
quadratic field {{tmath|\mathbb{Q}\bigl(\sqrt5~\!\bigr)}}, sometimes called the
golden field. In this field, any element can be written in the form , with rational coefficients and ; such a number has
norm . Other units, with norm , are the positive and negative powers of . The
quadratic integers in this field, which form a
ring, are all numbers of the form where and are integers. As the root of a quadratic polynomial, the golden ratio is a
constructible number.): \begin{align} \varphi^0 &= 1, \\[5mu] \varphi^1 &= 1.618033989\ldots \approx 2, \\[5mu] \varphi^2 &= 2.618033989\ldots \approx 3, \\[5mu] \varphi^3 &= 4.236067978\ldots \approx 4, \\[5mu] \varphi^4 &= 6.854101967\ldots \approx 7, \end{align} and so forth.
Golden spiral (red) and its approximation by quarter-circles (green), with overlaps shown in yellow whose radius grows by the golden ratio per of turn, surrounding nested golden isosceles triangles. This is a different spiral from the
golden spiral, which grows by the golden ratio per of turn. For a dodecahedron of side , the
radius of a circumscribed and inscribed sphere, and
midradius are (, , and , respectively): {{bi |left=1.6 |1=r_u = a\, \frac{\sqrt{3}\varphi}{2}, r_i = a\, \frac{\varphi^2}{2 \sqrt{3-\varphi}}, and r_m = a\, \frac{\varphi^2}{2}.}} While for an icosahedron of side , the radius of a circumscribed and inscribed sphere, and
midradius are: {{bi |left=1.6 |1=r_u = a\frac{\sqrt{\varphi \sqrt{5}}}{2}, r_i = a\frac{\varphi^2}{2 \sqrt{3}}, and r_m = a\frac{\varphi}{2}.}} The volume and surface area of the dodecahedron can be expressed in terms of : {{bi |left=1.6 |1=A_d = \frac{15\varphi}{\sqrt{3-\varphi}} and V_d = \frac{5\varphi^3}{6-2\varphi}.}} As well as for the icosahedron: {{bi|left=1.6|1=A_i = 20\frac{\varphi^{2}}{2} and V_i = \frac{5}{6}(1 + \varphi).}} . These geometric values can be calculated from their
Cartesian coordinates, which also can be given using formulas involving . The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are: (0,\pm1,\pm\varphi),\ (\pm1,\pm\varphi,0),\ (\pm\varphi,0,\pm1). Sets of three golden rectangles intersect
perpendicularly inside dodecahedra and icosahedra, forming
Borromean rings. In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. The three golden rectangles together contain all vertices of the icosahedron, or equivalently, intersect the centers of all of the dodecahedron's faces. A
cube can be
inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is times that of the dodecahedron's. In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in {{tmath|\textstyle \varphi \mathbin: \varphi^{2} }} ratio. On the other hand, the
octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's vertices touch the edges of an octahedron at points that divide its edges in golden ratio.
Other properties The golden ratio's
decimal expansion can be calculated via root-finding methods, such as
Newton's method or
Halley's method, on the equation or on (to compute first). The time needed to compute digits of the golden ratio using Newton's method is essentially , where is
the time complexity of multiplying two -digit numbers. This is considerably faster than known algorithms for pi| and e (mathematical constant)|. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers {{tmath|F_{25001} }} and {{tmath|F_{25000} }}, each over digits, yields over {{tmath|10{,}000}} significant digits of the golden ratio. The decimal expansion of the golden ratio has been calculated to an accuracy of twenty trillion ({{tmath|1=\textstyle 2 \times 10^{13} = 20{,}000{,}000{,}000{,}000}}) digits. In the
complex plane, the fifth
roots of unity {{tmath|1=\textstyle z = e^{2\pi k i/5} }} (for an integer ) satisfying are the vertices of a pentagon. They do not form a
ring of
quadratic integers, however the sum of any fifth root of unity and its
complex conjugate, ,
is a quadratic integer, an element of . Specifically, \begin{align} e^{0} + e^{-0} &= 2, \\[5mu] e^{2\pi i / 5} + e^{-2\pi i / 5} &= \varphi^{-1} = -1 + \varphi, \\[5mu] e^{4\pi i / 5} + e^{-4\pi i / 5} &= -\varphi. \end{align} This also holds for the remaining tenth roots of unity satisfying {{tmath|1=\textstyle z^{10} = 1}}, \begin{align} e^{\pi i} + e^{-\pi i} &= -2, \\[5mu] e^{\pi i / 5} + e^{-\pi i / 5} &= \varphi, \\[5mu] e^{3\pi i / 5} + e^{-3\pi i / 5} &= -\varphi^{-1} = 1 - \varphi. \end{align} For the
gamma function , the only solutions to the equation are and {{tmath|1=\textstyle z = -\varphi^{-1} }}. When the golden ratio is used as the base of a
numeral system (see
golden ratio base, sometimes dubbed
phinary or
-nary),
quadratic integers in the ring – that is, numbers of the form for and in – have
terminating representations, but rational fractions have non-terminating representations. The golden ratio also appears in
hyperbolic geometry, as the maximum distance from a point on one side of an
ideal triangle to the closer of the other two sides: this distance, the side length of the
equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is . The golden ratio appears in the theory of
modular functions as well. For |q| let R(q) = \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+{ \vphantom{1} \atop \ddots}}}}}. Then R(e^{-2\pi}) = \sqrt{\varphi\sqrt5}-\varphi ,\quad R(-e^{-\pi}) = \varphi^{-1}-\sqrt{2-\varphi^{-1}} and R(e^{-2\pi i/\tau})=\frac{1-\varphi R(e^{2\pi i\tau})}{\varphi+R(e^{2\pi i\tau})} where {{tmath|\operatorname{Im}\tau>0}} and {{tmath|\textstyle (e^z)^{1/5} }} in the continued fraction should be evaluated as {{tmath|\textstyle e^{z/5} }}. The function {{tmath|\textstyle \tau\mapsto R(e^{2\pi i\tau})}} is invariant under , a
congruence subgroup of the modular group. Also for
positive real numbers and such that \begin{align} \Bigl(\varphi+R{\bigl(e^{-2a}\bigr)}\Bigr)\Bigl(\varphi+R{\bigl(e^{-2b}\bigr)}\Bigr)&=\varphi\sqrt5, \\[5mu] \Bigl(\varphi^{-1}-R{\bigl({-e^{-a}}\bigr)}\Bigr)\Bigl(\varphi^{-1}-R{\bigl({-e^{-b}}\bigr)}\Bigr)&=\varphi^{-1}\sqrt5. \end{align} is a
Pisot–Vijayaraghavan number. ==Applications and observations==