Alphabeta Math

pi: the Equivalent Characterizations

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✓ 14 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 10 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

Circular arcs, circumference as arc length, and diameter

Definition

Riemann area between two continuous graphs and the disc as a vertically simple region

Theorem

Pi is equivalently the first sine zero, twice the first cosine zero, and half the least common period

Theorem

The arc length of a unit semicircle is pi

Theorem

Every circle has circumference 2 pi r and circumference-to-diameter ratio pi

Definition

Radian angle by unit-circle arc length

Theorem

Analytic sine and cosine agree with right-triangle ratios

Theorem

Inscribed regular-polygon perimeters increase to 2 pi, while circumscribed perimeters decrease to 2 pi

Theorem

A disc of radius r has Riemann area pi r squared; in particular the unit disc has area pi

Theorem

The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+...

Lemma

Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze

Theorem

Wallis's product: pi over two is the limit of the finite Wallis products

Corollary

The central binomial coefficient is asymptotic to 4ⁿ divided by the square root of pi n

Lemma

The finite Viete cosine product and its positive nested-radical factors

Theorem

Viete's nested-radical product: two over pi is the limit of the finite cosine products

Theorem

The zero, period, arc-length, polygonal, area, circumference, series, and product characterizations all give the same pi

Sources