Lagmental Vicfred

The Euclidean Algorithm Writes a Rational as a Continued Fraction by Vicfred

tags: mathematics, continued fractions, Diophantine approximation, Pell equations

Successive quotient and remainder steps become the partial quotients of a finite continued fraction. The point is to make the formal expression readable enough to audit line by line.

Statement

A continued fraction \([a_0;a_1,a_2,\ldots]\) produces convergents \(p_n/q_n\) with exceptional rational approximation. Quadratic irrationals are exactly the eventually periodic cases.

$$ a_0=\lfloor x\rfloor,\qquad x_{n+1}=\frac1{x_n-a_n} $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \frac ab=[a_0;a_1,\ldots,a_r] $$

Worked algebra

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \frac{415}{93}=4+\cfrac1{2+\cfrac1{6+\cfrac17}}=[4;2,6,7] $$

Conceptual compression

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad a_0=\lfloor x\rfloor,\qquad x_{n+1}=\frac1{x_n-a_n},\\[5pt] \mathsf{C}\;&:\quad \frac ab=[a_0;a_1,\ldots,a_r]. \end{aligned} $$

Caveat

Good approximation does not mean arbitrary denominator. The convergents are special because their determinants alternate between plus and minus one.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \frac ab=[a_0;a_1,\ldots,a_r] \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Tue 04 June 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.