proposition 7.91 The golden ratio

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2684 · p. 239

Rests on

depends_on definition 7.90 The golden ratio ¶
- depends_on definition 7.62 Real powers ¶
  - depends_on proposition 7.61 The logarithm ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on equation 7.42 eq:ana-exp-derivative ¶
    - depends_on lemma 7.60 Functional equation of the exponential ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.59 The real exponential ¶
      - depends_on equation 5.3 eq:lin-binomial ¶
      - depends_on proposition 7.49 Cauchy product ¶
      - proves proof ch:05-real-analysis@proof-38 ¶
    - depends_on proposition 7.32 Derivative of the inverse function ¶
      - depends_on definition 7.14 Inverse function ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on theorem 7.23 Intermediate value theorem ¶
      - proves proof ch:05-real-analysis@proof-16 ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on theorem 7.23 Intermediate value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-39 ¶
- depends_on proposition 7.63 Laws of real powers ¶
  - depends_on definition 7.62 Real powers ¶ ↺
  - depends_on equation 7.42 eq:ana-exp-derivative ¶ ↺
  - depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
  - depends_on proposition 7.31 Chain rule ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-15 ¶
  - depends_on proposition 7.61 The logarithm ¶ ↺
  - proves proof ch:05-real-analysis@proof-40 ¶
depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
- depends_on definition 7.45 Series ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
- depends_on proposition 7.46 Geometric series ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.45 Series ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- proves proof ch:05-real-analysis@proof-57 ¶
depends_on proposition 7.63 Laws of real powers ¶ ↺
depends_on proposition 7.89 Irrationality of square roots ¶
- depends_on definition 7.62 Real powers ¶ ↺
- depends_on lemma 7.65 The integer part ¶
  - depends_on axiom 3.33 Peano axioms ¶
  - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
    - depends_on definition A.36 Cut ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-32 ¶
  - depends_on lemma 7.64 Well-ordering of $\N$ ¶
    - depends_on axiom 3.33 Peano axioms ¶ ↺
    - depends_on axiom 3.34 Principle of induction ¶
    - proves proof ch:05-real-analysis@proof-41 ¶
  - proves proof ch:05-real-analysis@proof-42 ¶
- depends_on lemma 7.64 Well-ordering of $\N$ ¶ ↺
- depends_on proposition 7.63 Laws of real powers ¶ ↺
- proves proof ch:05-real-analysis@proof-58 ¶
proves proof ch:05-real-analysis@proof-59 ¶

Supports

depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
- depends_on remark 7.95 Where $\varphi$ is used ¶
depends_on proposition 125.2 Crystallographic restriction ¶
- depends_on remark 125.3 What the theorem does and does not forbid ¶
depends_on remark 7.95 Where $\varphi$ is used ¶ ↺
depends_on remark 7.94 Three numbers, two kinds ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	The golden ratio	declared	parts/02-mathematical-methods/05-real-analysis.tex:2697
`depends_on`	→	Terminating or repeating decimals are rational	declared	parts/02-mathematical-methods/05-real-analysis.tex:2697
`depends_on`	→	Laws of real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:2697
`depends_on`	→	Irrationality of square roots	declared	parts/02-mathematical-methods/05-real-analysis.tex:2697
`depends_on`	←	Continued fraction and the Fibonacci ratios	declared	parts/02-mathematical-methods/05-real-analysis.tex:2753
`depends_on`	←	Crystallographic restriction	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	←	Where $\varphi$ is used	declared	parts/02-mathematical-methods/05-real-analysis.tex:2876
`depends_on`	←	Three numbers, two kinds	declared	parts/02-mathematical-methods/05-real-analysis.tex:2849
`proves`	←	ch:05-real-analysis@proof-59	declared	parts/02-mathematical-methods/05-real-analysis.tex:2701