Best Rational Approximations

Summary: Lagrange’s theorem (Add. I, Arts. 12, 14, 16): every principal convergent $p_n/q_n$ of the continued fraction of $a$ is closer to $a$ than any other rational $m/n$ with $n\le q_n$. Convergents are therefore the best possible rational approximations to a given denominator size.

Sources: additions-1, additions-2

Last updated: 2026-05-10

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Statement

Let $a\in \mathbb{R}$ have principal convergents $p_0/q_0,p_1/q_1,p_2/q_2,\dots$. Then for each $n$:

No rational $m/n^{\prime }$ with $n^{\prime }<q_n$ approximates $a$ as closely as $p_n/q_n$. More precisely, no fraction $m/n^{\prime }$ with $n^{\prime }\le q_n$ satisfies $\left|a-\frac{m}{n^{\prime }}\right|<\left|a-\frac{p_n}{q_n}\right|$ unless $m/n^{\prime }=p_n/q_n$ itself.

Equivalently: the principal convergents enumerate the record-holders in the sequence of best rational approximations of $a$ as the allowed denominator grows.

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Lagrange’s Proof (Add. I, Arts. 12 + 14)

Step 1 (Art. 12): Two consecutive convergents satisfy

$p_n{q}_{n-1}-p_{n-1}{q}_n=(-1{)}^{n-1},$

so the difference between them is $\pm 1/(q_n{q}_{n-1})$.

Step 2: Suppose a fraction $m/n^{\prime }$ lies between $p_{n-1}/q_{n-1}$ and $p_n/q_n$. Then

$\frac{m}{n^{\prime }}-\frac{p_{n-1}}{q_{n-1}}=\frac{m{q}_{n-1}-n^{\prime }{p}_{n-1}}{n^{\prime }{q}_{n-1}}$

is a positive multiple of $1/(n^{\prime }{q}_{n-1})$, so its magnitude is at least $1/(n^{\prime }{q}_{n-1})$. But the gap between the two convergents is $1/(q_n{q}_{n-1})$, so for $m/n^{\prime }$ to fit inside,

$\frac{1}{n^{\prime }{q}_{n-1}}\le \frac{1}{q_n{q}_{n-1}}⟹n^{\prime }\ge q_n.$

Hence between two consecutive convergents, no rational with denominator $<q_n$ exists.

Step 3 (Art. 14): The convergents alternate around $a$, so $a$ lies between every pair of consecutive convergents. Combining Step 2 with this alternation: every fraction $m/n^{\prime }$ with $n^{\prime }<q_n$ lies on the outer side of one of the two convergents bracketing $a$, hence further from $a$ than that convergent.

Repeating with the next pair excludes all $n^{\prime }<q_{n+1}$, etc.

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Strengthened Form (Add. I, Art. 14)

A sharper consequence, given the alternation: for any rational $m/n^{\prime }$ with $n^{\prime }\le q_n$,

$\left|a-\frac{m}{n^{\prime }}\right|\ge \frac{1}{n^{\prime }(q_n+n^{\prime })}.$

This is an unconditional lower bound on the approximation quality of any “non-convergent” rational, and explains why convergents are the canonical record-holders.

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Intermediate Fractions (Semi-Convergents) — Caveat (Add. I, Arts. 17–18)

The semi-convergents between two principal convergents share a partial version of the optimality property: each is the best approximation among all fractions with that denominator size and on the same side of $a$. Across both sides, however, a principal convergent on the opposite side can beat them — so semi-convergents are best-in-class only when one constrains the search to all-less or all-greater rationals.

This subtlety motivates the practical rule in Add. I, Art. 15: prefer all-positive partial quotients (always-below rounding) so that the principal convergents alternate around $a$ and bracket the truth from both sides.

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The $|p-aq|$ Form (Add. II, Arts. 23–27)

In the second Addition, Lagrange proves a stronger version of optimality, phrased not as $|a-p/q|$ but as $|p-aq|$:

Among all positive integer pairs $(p,q)$ with $p\le p_n$ and $q\le q_n$, the principal convergent $p_n/q_n$ uniquely minimizes $|p-aq|$.

The two forms differ by a factor of $q$, but the $|p-aq|$ statement is sharper:

• Strict comparison. $|p_n-a{q}_n|<|p^{\prime }-a{q}^{\prime }|$ holds for every smaller pair $(p^{\prime },q^{\prime })$, not just those with the same denominator size.
• Algorithmic. Solving Diophantine minimization problems like ${\min }_{p,q\in {\mathbb{Z}}^{+}}|A{p}^m+\dots +V{q}^m|$ (Add. II, Problem 2) reduces directly to scanning the convergents of each root of the associated polynomial.

Lagrange’s proof (Add. II, Art. 23): Suppose $(p,q)$ minimizes $|p-aq|$. Then there exist $r<p,s<q$ with $ps-qr=\pm 1$ (pigeonhole on residues of $q,2q,\dots ,pq$ mod $p$). Setting

$P=p-aq,R=r-as,$

the linear combination $y-az=(p-aq)t+(r-as)u=Pt+Ru$ shows that any pair $(y,z)$ smaller than $(p,q)$ writes as $y=pt+ru,z=qt+su$ with $t,u$ integers; hence $|y-az|\ge |Pt+Ru|>|P|$ unless $t=\pm 1,u=0$ (the trivial case). The descent on $(p,q)\to (r,s)\to (r^{\prime },s^{\prime })\to \dots$ is exactly the convergent recurrence, identifying optimal $(p,q)$ as principal convergents.

This is the form actually needed by Add. II, Problem 3 (binary-quadratic-forms) and Problems 2 and 4 — the $|a-p/q|$ form of Add. I would not suffice.

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Why “Lagrange’s Theorem”?

The result is sometimes attributed to Wallis or Huygens, since they identified the convergents and observed their good approximation behaviour. But the rigorous proof — combining the cross-product identity with the alternation — first appears in Lagrange’s Additions (this Chapter I) and in his contemporaneous Berlin Memoirs (1767–68).

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Calendar Application (Add. I, Art. 20)

The convergents to the year-length ratio $86400/20929$ are

$\frac{4}{1},\frac{29}{7},\frac{33}{8},\frac{128}{31},\frac{161}{39},\frac{2704}{655},\frac{2865}{694},\frac{5569}{1349},\frac{86400}{20929}.$

Lagrange uses optimality to argue that the Gregorian rule $400/97$ is suboptimal: 97 lies between the denominators $39$ and $655$ of two consecutive convergents, and by best-approximation $\frac{161}{39}$ already approximates the year length more accurately than any rational with denominator up to 655 — including $\frac{400}{97}$.

See calendar-approximations for the calendar-reform discussion.

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π Application (Add. I, Art. 21)

The convergent $\frac{3}{1}$ approximates $\pi$ better than any rational with denominator $<7$. $\frac{22}{7}$ beats every rational with denominator $\le 106$. $\frac{355}{113}$ beats every rational with denominator $\le 33,102$ — the next convergent’s denominator.

These statements are not heuristic: Lagrange’s theorem makes them rigorous.

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Related pages

• convergents
• continued-fractions
• semi-convergents
• calendar-approximations
• add1-continued-fractions
• add2-arithmetic-problems
• binary-quadratic-forms
• pell-equation