semi-convergents

Semi-Convergents (Intermediate Fractions)

Summary: Between two consecutive principal convergents $p_{n-1}/q_{n-1}$ and $p_n/q_n$ — when the partial quotient $a_n>1$ — Lagrange inserts $a_n-1$ intermediate fractions by replacing $a_n$ in turn by $1,2,\dots ,a_n-1$. They form an arithmetic progression of numerators and denominators and share the optimality property within their monotone series.

Sources: additions-1

Last updated: 2026-05-10

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Construction (Add. I, Arts. 17–18)

Given consecutive convergents $A/A^{\prime }$ and $C/C^{\prime }$ with $C=\gamma B+A$, $C^{\prime }=\gamma B^{\prime }+A^{\prime }$, and partial quotient $\gamma >1$, the intermediate fractions are

$\frac{B+A}{B^{\prime }+A^{\prime }},\frac{2B+A}{2B^{\prime }+A^{\prime }},\frac{3B+A}{3B^{\prime }+A^{\prime }},\dots ,\frac{(\gamma -1)B+A}{(\gamma -1)B^{\prime }+A^{\prime }}.$

The full list interpolates from $A/A^{\prime }$ to $C/C^{\prime }$ via $\gamma -1$ steps. Numerators form the arithmetic progression $A,B+A,2B+A,\dots ,\gamma B+A=C$; denominators $A^{\prime },B^{\prime }+A^{\prime },2B^{\prime }+A^{\prime },\dots ,\gamma B^{\prime }+A^{\prime }=C^{\prime }$.

In modern notation, between $p_{n-1}/q_{n-1}$ and $p_n/q_n$ (with $a_n>1$), the intermediates are

$\frac{k\cdot p_{n-1}+p_{n-2}}{k\cdot q_{n-1}+q_{n-2}},k=1,2,\dots ,a_n-1.$

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Differences and Best-Approximation Within a Series

By Article 17 of the Additions, consecutive intermediates differ by

$\frac{(k+1)B+A}{(k+1)B^{\prime }+A^{\prime }}-\frac{kB+A}{k{B}^{\prime }+A^{\prime }}=\frac{1}{(k{B}^{\prime }+A^{\prime })((k+1)B^{\prime }+A^{\prime })}$

(using $B{A}^{\prime }-A{B}^{\prime }=1$). The differences shrink, the intermediates increase monotonically (or all decrease monotonically) toward the next principal convergent, and the same denominator-comparison argument as for principal convergents shows: no rational $m/n$ with $n$ less than the current denominator can fall between an intermediate and the principal convergent on the same side of $a$.

So intermediates are best approximations within their series (all-less or all-greater than $a$), but they are not best approximations across both series — a principal convergent on the opposite side may be closer.

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Two Complete Series (Add. I, Art. 18)

Combining the principal convergents with intermediates yields two complete monotone families converging to $a$:

Fractions increasing toward $a$ (less than $a$)

$\frac{A}{A^{\prime }},\frac{B+A}{B^{\prime }+A^{\prime }},\frac{2B+A}{2B^{\prime }+A^{\prime }},\frac{3B+A}{3B^{\prime }+A^{\prime }},\dots ,\frac{C}{C^{\prime }},\frac{D+C}{D^{\prime }+C^{\prime }},\frac{2D+C}{2D^{\prime }+C^{\prime }},\dots$

Fractions decreasing toward $a$ (greater than $a$)

$\frac{A+1}{1},\frac{2A+1}{2},\dots ,\frac{B}{B^{\prime }},\frac{C+B}{C^{\prime }+B^{\prime }},\frac{2C+B}{2C^{\prime }+B^{\prime }},\dots$

(The leading terms $\frac{kA+1}{k}$ before $B/B^{\prime }$ exist when $\beta >1$.)

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When the CF Terminates (Art. 18, end)

If $a$ is rational, the principal CF terminates at some $V/V^{\prime }$. One of the two monotone series ends at this $V/V^{\prime }$; the other series can be continued to infinity by treating the next “missing” partial quotient as $\infty$ — the corresponding intermediate fractions

$\frac{D+C}{D^{\prime }+C^{\prime }},\frac{2D+C}{2D^{\prime }+C^{\prime }},\frac{3D+C}{3D^{\prime }+C^{\prime }},\dots$

extend the truncated series indefinitely (Art. 18).

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Worked Example: $\pi$ (Add. I, Art. 21)

The principal convergents to $\pi$ begin $\frac{3}{1},\frac{22}{7},\frac{333}{106},\frac{355}{113},\dots$ with partial quotients $3,7,15,1,292,\dots$. The two monotone families are:

Less than $\pi$ (intermediates inserted): $\frac{3}{1},\frac{25}{8},\frac{47}{15},\frac{69}{22},\frac{91}{29},\frac{113}{36},\frac{135}{43},\frac{157}{50},\frac{173}{57},\dots ,\frac{333}{106},\frac{688}{219},\frac{1043}{332},\dots$

Greater than $\pi$: $\frac{4}{1},\frac{7}{2},\frac{10}{3},\frac{13}{4},\frac{16}{5},\frac{19}{6},\frac{22}{7},\frac{355}{113},\frac{104348}{33215},\dots$

(The decreasing series jumps from $\frac{22}{7}$ to $\frac{355}{113}$ because the partial quotient between them, $1$, allows no intermediates.)

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Worked Example: Calendar (Add. I, Art. 20)

For the solar-year ratio $86400/20929$ with partial quotients $4,7,1,3,1,16,1,1,15$, the principal convergents are listed in calendar-approximations. Between the 6th and 7th principal convergents (partial quotient $\gamma =1$), no intermediates can be inserted; but the 9th partial quotient is $15$, giving 14 intermediate fractions $\frac{2865+k\cdot 5569}{694+k\cdot 1349}$ for $k=1,\dots ,14$.

This is precisely how Lagrange enumerates all useful intercalation rules between, say, “1 day every 4 years” and “8 days every 33 years”.

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Related pages

• convergents
• continued-fractions
• best-rational-approximations
• calendar-approximations
• add1-continued-fractions