Alphabeta Mathpi: the Equivalent Characterizations: Examples and Counterexamples
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✓ 7 results · all verified · 4 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
One unit circle gives semicircle length pi, circumference 2 pi, diameter 2, and disc area pi
Example
Gregory-Leibniz partial sums bracket pi with an explicit remainder bound
Example
Wallis partial products are trapped by adjacent sine-power integrals
Example
The first Viete nested-radical products approximate two over pi
False statement
False: any positive zero of sine characterizes pi
False statement
False: circumference divided by radius equals pi
Counterexample