Symbol: GoldenRatio — φ
— The golden ratio (1.618...)
Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
Source code for this entry:
Entry(ID("37f505"),
SymbolDefinition(GoldenRatio, GoldenRatio, "The golden ratio (1.618...)"))φ∈[1.6180339887498948482045868343656381177203091798058±3.72⋅10−50]
TeX:
\varphi \in \left[1.6180339887498948482045868343656381177203091798058 \pm 3.72 \cdot 10^{-50}\right]Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
Source code for this entry:
Entry(ID("08fcaf"),
Formula(Element(GoldenRatio, RealBall(Decimal("1.6180339887498948482045868343656381177203091798058"), Decimal("3.72e-50")))))φ=21+5
TeX:
\varphi = \frac{1 + \sqrt{5}}{2}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
| Sqrt | z
| Principal square root |
Source code for this entry:
Entry(ID("77d2f8"),
Formula(Equal(GoldenRatio, Div(Add(1, Sqrt(5)), 2))))φ∈/Q
TeX:
\varphi \notin \mathbb{Q}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
| QQ | Q
| Rational numbers |
Source code for this entry:
Entry(ID("e09458"),
Formula(NotElement(GoldenRatio, QQ)))φ1=φ−1
TeX:
\frac{1}{\varphi} = \varphi - 1Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
Source code for this entry:
Entry(ID("31f52c"),
Formula(Equal(Div(1, GoldenRatio), Sub(GoldenRatio, 1))))φ2−φ−1=0
TeX:
{\varphi}^{2} - \varphi - 1 = 0Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| Pow | ab
| Power |
| GoldenRatio | φ
| The golden ratio (1.618...) |
Source code for this entry:
Entry(ID("b464d3"),
Formula(Equal(Sub(Sub(Pow(GoldenRatio, 2), GoldenRatio), 1), 0)))(aa+b=ba)⟹(ba=φ)
Assumptions:a∈(0,∞)andb∈(0,∞)
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 | φ
| The golden ratio (1.618...) |
| OpenInterval | (a,b)
| Open interval |
| Infinity | ∞
| 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)))))x∈Czeros(x2−x−1)={φ,1−φ}
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 |
|---|
| Pow | ab
| Power |
| CC | C
| Complex numbers |
| GoldenRatio | φ
| The golden ratio (1.618...) |
Source code for this entry:
Entry(ID("77c324"),
Formula(Equal(Zeros(Sub(Sub(Pow(x, 2), x), 1), x, Element(x, CC)), Set(GoldenRatio, Sub(1, GoldenRatio)))))φ=1+φ1
TeX:
\varphi = 1 + \frac{1}{\varphi}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
Source code for this entry:
Entry(ID("6d2709"),
Formula(Equal(GoldenRatio, Add(1, Div(1, GoldenRatio)))))φ=1+1+φ11
TeX:
\varphi = 1 + \frac{1}{1 + \frac{1}{\varphi}}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| 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)))))))spec(1110)={φ,1−φ}
TeX:
\operatorname{spec}\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \left\{\varphi, 1 - \varphi\right\}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| Matrix2x2 | (acbd)
| Two by two matrix |
| GoldenRatio | φ
| 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)))))φ=2cos(5π)
TeX:
\varphi = 2 \cos\!\left(\frac{\pi}{5}\right)Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
| ConstPi | π
| The constant pi (3.14...) |
Source code for this entry:
Entry(ID("98a765"),
Formula(Equal(GoldenRatio, Mul(2, Cos(Div(ConstPi, 5))))))φ=2sin(103π)
TeX:
\varphi = 2 \sin\!\left(\frac{3 \pi}{10}\right)Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
| Sin | sin(z)
| Sine |
| ConstPi | π
| The constant pi (3.14...) |
Source code for this entry:
Entry(ID("487e35"),
Formula(Equal(GoldenRatio, Mul(2, Sin(Div(Mul(3, ConstPi), 10))))))φ=2sin(10π)+1
TeX:
\varphi = 2 \sin\!\left(\frac{\pi}{10}\right) + 1Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
| Sin | sin(z)
| Sine |
| ConstPi | π
| The constant pi (3.14...) |
Source code for this entry:
Entry(ID("fad16f"),
Formula(Equal(GoldenRatio, Add(Mul(2, Sin(Div(ConstPi, 10))), 1))))φn+1=φn+φn−1
Assumptions:n∈C
TeX:
{\varphi}^{n + 1} = {\varphi}^{n} + {\varphi}^{n - 1}
n \in \mathbb{C}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| Pow | ab
| Power |
| GoldenRatio | φ
| The golden ratio (1.618...) |
| CC | 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)))φ=n→∞limFnFn+1
TeX:
\varphi = \lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}}Definitions:
| Fungrim symbol | Notation | Short description |
|---|
| GoldenRatio | φ
| The golden ratio (1.618...) |
| SequenceLimit | limn→af(n)
| Limiting value of sequence |
| Infinity | ∞
| Positive infinity |
Source code for this entry:
Entry(ID("2b6e60"),
Formula(Equal(GoldenRatio, SequenceLimit(Div(Fibonacci(Add(n, 1)), Fibonacci(n)), n, Infinity))))