proposition 7.85 Irrationality of $\pi$

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2468 · p. 236

Rests on

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 theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- proves proof ch:05-real-analysis@proof-57 ¶
depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
- depends_on equation 7.53 eq:ana-cos-sin-series ¶
- depends_on equation 5.3 eq:lin-binomial ¶
- depends_on proposition 7.49 Cauchy product ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.45 Series ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-30 ¶
- depends_on theorem 7.51 Termwise differentiation ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on theorem 7.8 Cauchy criterion ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶ ↺
    - depends_on theorem 7.8 Cauchy criterion ¶ ↺
    - proves proof ch:05-real-analysis@proof-29 ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - depends_on theorem 7.50 Power series; radius of convergence ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶ ↺
    - proves proof ch:05-real-analysis@proof-31 ¶
  - proves proof ch:05-real-analysis@proof-32 ¶
- proves proof ch:05-real-analysis@proof-46 ¶
depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
- depends_on definition 7.74 $\pi$ ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on lemma 7.73 $\cos 2 < 0$ ¶
    - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
    - depends_on proposition 7.52 Alternating series test ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-33 ¶
    - proves proof ch:05-real-analysis@proof-48 ¶
  - depends_on theorem 7.23 Intermediate value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on definition 7.20 Continuity at a point ¶ ↺
    - proves proof ch:05-real-analysis@proof-8 ¶
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
- depends_on equation 7.58 eq:ana-half-angle ¶
- 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 7.75 The first quadrant ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on definition 7.74 $\pi$ ¶ ↺
  - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶ ↺
  - depends_on theorem 7.23 Intermediate value theorem ¶ ↺
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
  - proves proof ch:05-real-analysis@proof-49 ¶
- depends_on lemma 7.72 Addition theorems ¶
  - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
  - depends_on equation 5.3 eq:lin-binomial ¶ ↺
  - depends_on proposition 7.49 Cauchy product ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-47 ¶
- proves proof ch:05-real-analysis@proof-51 ¶
depends_on proposition 7.48 Ratio test ¶ ↺
depends_on theorem 7.8 Cauchy criterion ¶ ↺
depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
- depends_on definition 7.41 Antiderivative ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶ ↺
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- proves proof ch:05-real-analysis@proof-25 ¶
proves proof ch:05-real-analysis@proof-56 ¶

Supports

depends_on remark 7.94 Three numbers, two kinds ¶
depends_on remark 7.87 Priority, and what irrationality does not give ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Terminating or repeating decimals are rational	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	→	Derivatives; the Pythagorean identity	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	→	Special values, periodicity, and the kernel	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	→	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	→	Cauchy criterion	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	→	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	←	Three numbers, two kinds	declared	parts/02-mathematical-methods/05-real-analysis.tex:2849
`depends_on`	←	Priority, and what irrationality does not give	declared	parts/02-mathematical-methods/05-real-analysis.tex:2589
`proves`	←	ch:05-real-analysis@proof-56	declared	parts/02-mathematical-methods/05-real-analysis.tex:2476