Continued Fractions

Summary: A continued fraction (CF) expresses a quantity $a$ as a nested sum $\alpha +1/(\beta +1/(\gamma +\cdots ))$, where $\alpha ,\beta ,\gamma ,\dots$ are integers. CFs terminate iff $a$ is rational, and provide the best possible rational approximations to irrational numbers.

Sources: additions-1, additions-2

Last updated: 2026-05-10

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Definition

In full generality, a continued fraction is

$\alpha +\frac{b}{\beta +\frac{c}{\gamma +\frac{d}{\delta +\cdots }}}$

with $\alpha ,\beta ,\gamma ,\dots$ and $b,c,d,\dots$ integers (positive or negative). Lagrange restricts attention to the unit-numerator form

$\alpha +\frac{1}{\beta +\frac{1}{\gamma +\frac{1}{\delta +\cdots }}}=[\alpha ;\beta ,\gamma ,\delta ,\dots ]$

since this is the only form of analytic utility (Add. I, Art. 1).

The integers $\alpha ,\beta ,\gamma ,\dots$ are the partial quotients.

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Generation by Iterated Integer-Part Extraction

Given any real $a$, set

$a_0=a,{\alpha }_n=\lfloor a_n\rfloor ,a_{n+1}=\frac{1}{a_n-{\alpha }_n}.$

Substitution gives $a=[{\alpha }_0;{\alpha }_1,{\alpha }_2,\dots ]$. The construction is uniquely determined by the choice “always take the integer below” (yielding all-positive partial quotients).

If at some stage $a_n-{\alpha }_n=0$, the process terminates. By Article 3 of Lagrange’s Addition I, this happens iff $a$ is rational.

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Termination Criterion

$a$	CF expansion
Rational $V/V^{\prime }$	Finite, last partial quotient appears explicitly
Irrational	Infinite, all partial quotients positive integers
Quadratic irrational	Infinite, eventually periodic — Lagrange proves this in Add. II, Art. 34

A CF terminating in unity (e.g. $[1;2,3,1]$) signals that the previous integer-part choice was not the nearest integer; the CF can be simplified by shortening: $[1;2,3,1]=[1;2,4]$ (Add. I, Art. 7).

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Reduction Rule for Vulgar Fractions (Add. I, Art. 4)

To express $A/B$ as a CF:

1. Divide $A$ by $B$ — quotient $\alpha$, remainder $C$.
2. Divide $B$ by $C$ — quotient $\beta$, remainder $D$.
3. Divide $C$ by $D$ — quotient $\gamma$, remainder $E$.
4. Continue until the remainder is 0.

This is precisely the Euclidean algorithm (ch1.3.7-greatest-common-divisor); the partial quotients of the CF are the successive quotients in the gcd computation. Hence reducing a fraction to a CF and computing $\gcd (A,B)$ are the same procedure, just with different outputs collected.

Example (Art. 5): For $1103/887$,

$1103=1\cdot 887+216,887=4\cdot 216+23,216=9\cdot 23+9,23=2\cdot 9+5,$ $9=1\cdot 5+4,5=1\cdot 4+1,4=4\cdot 1+0,$

so $1103/887=[1;4,9,2,1,1,4]$, and $\gcd (1103,887)=1$.

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Reduction of Decimals to CFs

For an irrational $a$ given to $N$ decimal places, both the truncated value and the value with the last digit increased by 1 are processed; the CF is taken only as far as the partial quotients agree (Add. I, Art. 8). Lagrange uses this on Ludolph’s 35-digit value of $\pi$ to extract the convergents up to $\frac{411557987}{131002976}$ and beyond.

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Sign Conventions (Arts. 6–7)

If integer parts are always taken below the value: all $\beta ,\gamma ,\dots >0$. If always above: all $\beta ,\gamma ,\dots <0$. Mixed choices give mixed signs.

The transformation

$\mu +\frac{1}{-\nu +\frac{1}{\pi +\cdots }}=\mu -1+\frac{1}{1+\frac{1}{\nu -1+\frac{1}{\pi +\cdots }}}$

flips a negative term back to positive at the cost of an inserted unit denominator; iterating produces an all-positive or all-negative form at will.

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Why Continued Fractions Matter

A. Best rational approximations. The truncations $[{\alpha }_0],[{\alpha }_0;{\alpha }_1],[{\alpha }_0;{\alpha }_1,{\alpha }_2],\dots$ — called convergents — are closer to $a$ than any other rational with smaller-or-equal denominator (see best-rational-approximations).

B. Diophantine algorithms. Solving $ax+by=c$ in integers, and $x^2-D{y}^2=1$ (the pell-equation), reduces in part to manipulating the CF of $a/b$ or $\sqrt{D}$.

C. Numerical root-finding. Roots of polynomial equations have CF expansions whose finitude detects rationality (Art. 9; cf. approximation-methods).

D. Calendar reform. The convergents to a year-length ratio give all “best” intercalation rules (see calendar-approximations).

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Historical

• Brouncker (~1655): first use; expressed $4/\pi$ as a CF (Add. I, Art. 2).
• Wallis (Arithmetica Infinitorum): proved Brouncker’s CF equivalent to his own infinite product, gave the reduction-to-vulgar-fraction method.
• Huygens (Opera Posthuma): discovered the deep approximation properties; applied them to the gear-tooth ratios of his planetary automaton.
• Euler (Petersburg Commentaries IX, XI; Berlin Memoirs 1767–68): extensive arithmetical theory.
• Lagrange (these Additions): synthesis and best-approximation proofs.

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Related pages

• convergents
• semi-convergents
• best-rational-approximations
• calendar-approximations
• add1-continued-fractions
• add2-arithmetic-problems
• periodicity-quadratic-irrationals
• binary-quadratic-forms
• square-root-continued-fractions
• ch1.3.7-greatest-common-divisor
• ch1.3.12-infinite-decimal-fractions
• approximation-methods