The Continued Fraction of the Circle Constant: π̂'s Algebraic Root and π's Famous Convergents¶

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There is a way to write any positive real number that is more revealing about its nature than its decimal expansion: the continued fraction. Every real number is built from a chain of integers

x = a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + ⋯)))

usually compressed into the notation [a₀; a₁, a₂, a₃, …]. Truncating the chain at any step yields the convergents — the best rational approximations a denominator of that size can offer. Because the two candidate circle constants differ in kind — one algebraic, one transcendental — their continued fractions behave differently, and that difference is worth reading carefully.

Two expansions, side by side¶

The two values open as follows (computed here to sixteen terms):

Golden Pi π̂ = 4/√φ = 3.1446055… → [3; 6, 1, 10, 1, 4, 2, 2, 1, 2, 5, 1, 4, 1, 4, 2, …]
Conventional Pi π = 3.14159265… → [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, …]

Both begin with the whole part 3, and both are approximated early by the same first non-trivial convergent:

22/7 = 3.142857… is a convergent of both constants. That is no accident — 22/7 is Archimedes' classical upper bound, and it is the best three-digit rational for either candidate. The two worlds agree on this early rung of the ladder, then climb apart.

π̂ is algebraic — it belongs to φ's field¶

The crucial, checkable fact about Golden Pi is that it is not a transcendental number. It is an algebraic number of degree 4 whose minimal polynomial is exact:

x⁴ + 16x² − 256 = 0

Substituting x = 4/√φ returns 0 to machine precision — the polynomial is satisfied identically. Solving it requires only square roots:

π̂ = 4/√φ = 4·√(2/(1+√5))

Every digit of π̂ therefore follows from a finite chain of square roots and rational operations on the golden ratio. That is exactly the recipe that makes a number constructible with compass and straightedge: π̂ lives in the tower of quadratic extensions that Euclidean construction is permitted to reach, in line with the site's constructibility argument for the circle constant.

Because π̂ is algebraic, its convergents are ordinary, well-behaved rationals, each stepping a little closer with modest partial quotients:

1283/408 = 3.14460784… — good to about 2 × 10⁻⁶
2827/899 = 3.14460512… — good to about 4 × 10⁻⁷

No startling integers appear in the early chain; the expansion is the quiet, structured one you would expect from a number tied to φ.

π is transcendental — and 292 makes the famous near-miss¶

Conventional π, by contrast, is transcendental (Lindemann, 1882), and its expansion shows it. The fourth partial quotient is the striking 292, which produces the celebrated convergent

355/113 = 3.141592920…   (error ≈ 2.7 × 10⁻⁷)

This is why 355/113 is one of the most famous fractions in mathematics: a denominator of only 113 yielding six correct figures of 3.14159… is an extraordinarily lucky contract. A continued fraction whose terms are large this early is a hallmark of an irrational number that is not governed by a simple algebraic law — the patternless digits of a transcendental.

The honest boundary: a lens, not an arbiter¶

None of this, by itself, decides which constant is "the" circle constant — and the honest position is to say so plainly. The continued fraction is a lens of the rational and analytic world. It converges to whatever real number you choose to define as the ratio of circumference to diameter:

In the constructed golden world, where 4/√φ is taken as exact, the expansion [3; 6, 1, 10, 1, 4, …] is simply the best-rational portrait of an algebraic, constructible number — and the circle and square share an exact algebraic bond.
In the analytic world, where the Leibniz, Machin, Wallis and Basel series are evaluated term by term, they pin the constant at 3.14159265…, and the expansion [3; 7, 15, 1, 292, …] with its 355/113 near-miss records that value.

Both books are kept. What the continued fraction does contribute is a classification: whichever constant one studies, the expansion exposes its algebraic or transcendental character for all to see. On the golden side it exposes a constructible number rooted in φ; on the analytic side, a transcendental one whose rational approximations are famously irregular. The two pictures are each internally consistent — and they are not the same picture.