lemma 7.86 Terminating or repeating decimals are rational
open in the book ·
parts/02-mathematical-methods/05-real-analysis.tex:2540
· p. 237
Rests on
-
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
theorem A.40
Least-upper-bound property
¶
- depends_on definition 7.45 Series ¶ ↺
- proves proof ch:05-real-analysis@proof-27 ¶
-
depends_on
corollary A.46
Monotone convergence
¶
- proves proof ch:05-real-analysis@proof-57 ¶
Supports
-
depends_on
proposition 7.91
The golden ratio
¶
-
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 ¶
-
depends_on
proposition 7.92
Continued fraction and the Fibonacci ratios
¶
-
depends_on
proposition 7.85
Irrationality of $\pi$
¶
- 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 |
→ | Series | declared | parts/02-mathematical-methods/05-real-analysis.tex:2545 |
depends_on |
→ | Geometric series | declared | parts/02-mathematical-methods/05-real-analysis.tex:2545 |
depends_on |
← | The golden ratio | declared | parts/02-mathematical-methods/05-real-analysis.tex:2697 |
depends_on |
← | Irrationality of $\pi$ | declared | parts/02-mathematical-methods/05-real-analysis.tex:2472 |
proves |
← | ch:05-real-analysis@proof-57 | declared | parts/02-mathematical-methods/05-real-analysis.tex:2548 |