Golden ratio

Symbol: GoldenRatio —

\varphi

— The golden ratio (1.618...)

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("37f505"),
    SymbolDefinition(GoldenRatio, GoldenRatio, "The golden ratio (1.618...)"))

08fcaf

\varphi \in \left[1.6180339887498948482045868343656381177203091798058 \pm 3.72 \cdot 10^{-50}\right]

TeX:

\varphi \in \left[1.6180339887498948482045868343656381177203091798058 \pm 3.72 \cdot 10^{-50}\right]

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("08fcaf"),
    Formula(Element(GoldenRatio, RealBall(Decimal("1.6180339887498948482045868343656381177203091798058"), Decimal("3.72e-50")))))

77d2f8

\varphi = \frac{1 + \sqrt{5}}{2}

TeX:

\varphi = \frac{1 + \sqrt{5}}{2}

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("77d2f8"),
    Formula(Equal(GoldenRatio, Div(Add(1, Sqrt(5)), 2))))

e09458

\varphi \notin \mathbb{Q}

TeX:

\varphi \notin \mathbb{Q}

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("e09458"),
    Formula(NotElement(GoldenRatio, QQ)))

31f52c

\frac{1}{\varphi} = \varphi - 1

TeX:

\frac{1}{\varphi} = \varphi - 1

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("31f52c"),
    Formula(Equal(Div(1, GoldenRatio), Sub(GoldenRatio, 1))))

b464d3

{\varphi}^{2} - \varphi - 1 = 0

TeX:

{\varphi}^{2} - \varphi - 1 = 0

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("b464d3"),
    Formula(Equal(Sub(Sub(Pow(GoldenRatio, 2), GoldenRatio), 1), 0)))

d774fe

\left(\frac{a + b}{a} = \frac{a}{b}\right) \;\implies\; \left(\frac{a}{b} = \varphi\right)

Assumptions:

a \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)

TeX:

\left(\frac{a + b}{a} = \frac{a}{b}\right) \;\implies\; \left(\frac{a}{b} = \varphi\right)

a \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("d774fe"),
    Formula(Implies(Equal(Div(Add(a, b), a), Div(a, b)), Equal(Div(a, b), GoldenRatio))),
    Variables(a, b),
    Assumptions(And(Element(a, OpenInterval(0, Infinity)), Element(b, OpenInterval(0, Infinity)))))

77c324

\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} - x - 1\right] = \left\{\varphi, 1 - \varphi\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} - x - 1\right] = \left\{\varphi, 1 - \varphi\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("77c324"),
    Formula(Equal(Zeros(Sub(Sub(Pow(x, 2), x), 1), ForElement(x, CC)), Set(GoldenRatio, Sub(1, GoldenRatio)))))

6d2709

\varphi = 1 + \frac{1}{\varphi}

TeX:

\varphi = 1 + \frac{1}{\varphi}

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("6d2709"),
    Formula(Equal(GoldenRatio, Add(1, Div(1, GoldenRatio)))))

2e0596

\varphi = 1 + \frac{1}{1 + \frac{1}{\varphi}}

TeX:

\varphi = 1 + \frac{1}{1 + \frac{1}{\varphi}}

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("2e0596"),
    Formula(Equal(GoldenRatio, Add(1, Div(1, Add(1, Div(1, GoldenRatio)))))))

ebfcd8

\operatorname{spec}\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \left\{\varphi, 1 - \varphi\right\}

TeX:

\operatorname{spec}\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \left\{\varphi, 1 - \varphi\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)

Source code for this entry:

Entry(ID("ebfcd8"),
    Formula(Equal(Spectrum(Matrix2x2(1, 1, 1, 0)), Set(GoldenRatio, Sub(1, GoldenRatio)))))

98a765

\varphi = 2 \cos\!\left(\frac{\pi}{5}\right)

TeX:

\varphi = 2 \cos\!\left(\frac{\pi}{5}\right)

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("98a765"),
    Formula(Equal(GoldenRatio, Mul(2, Cos(Div(Pi, 5))))))

487e35

\varphi = 2 \sin\!\left(\frac{3 \pi}{10}\right)

TeX:

\varphi = 2 \sin\!\left(\frac{3 \pi}{10}\right)

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("487e35"),
    Formula(Equal(GoldenRatio, Mul(2, Sin(Div(Mul(3, Pi), 10))))))

fad16f

\varphi = 2 \sin\!\left(\frac{\pi}{10}\right) + 1

TeX:

\varphi = 2 \sin\!\left(\frac{\pi}{10}\right) + 1

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("fad16f"),
    Formula(Equal(GoldenRatio, Add(Mul(2, Sin(Div(Pi, 10))), 1))))

0cd1a4

{\varphi}^{n + 1} = {\varphi}^{n} + {\varphi}^{n - 1}

Assumptions:

n \in \mathbb{C}

TeX:

{\varphi}^{n + 1} = {\varphi}^{n} + {\varphi}^{n - 1}

n \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0cd1a4"),
    Formula(Equal(Pow(GoldenRatio, Add(n, 1)), Add(Pow(GoldenRatio, n), Pow(GoldenRatio, Sub(n, 1))))),
    Variables(n),
    Assumptions(Element(n, CC)))

6a11ce

{\varphi}^{n} = F_{n} \varphi + F_{n - 1}

Assumptions:

n \in \mathbb{Z}

TeX:

{\varphi}^{n} = F_{n} \varphi + F_{n - 1}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Fibonacci`	$F_{n}$	Fibonacci number
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("6a11ce"),
    Formula(Equal(Pow(GoldenRatio, n), Add(Mul(Fibonacci(n), GoldenRatio), Fibonacci(Sub(n, 1))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

2b6e60

\varphi = \lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}}

TeX:

\varphi = \lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}}

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Fibonacci`	$F_{n}$	Fibonacci number
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("2b6e60"),
    Formula(Equal(GoldenRatio, SequenceLimit(Div(Fibonacci(Add(n, 1)), Fibonacci(n)), For(n, Infinity)))))

e9a269

\varphi = \frac{1}{5} {\left(\frac{\eta(i)}{\eta\!\left(5 i\right)}\right)}^{2}

TeX:

\varphi = \frac{1}{5} {\left(\frac{\eta(i)}{\eta\!\left(5 i\right)}\right)}^{2}

Definitions:

Fungrim symbol	Notation	Short description
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Pow`	${a}^{b}$	Power
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("e9a269"),
    Formula(Equal(GoldenRatio, Mul(Div(1, 5), Pow(Div(DedekindEta(ConstI), DedekindEta(Mul(5, ConstI))), 2)))))

Definitions

Numerical value

Algebraic equations

Trigonometric formulas

Recurrence relations

Limit representations

Special function representations