Convergents
Summary: The convergents of a continued fraction are the rational truncations at depth . They are computed by a two-term recurrence, are automatically in lowest terms, alternate around the target value, and obey sharp error bounds — making them the canonical best rational approximations.
Sources: additions-1
Last updated: 2026-05-10
Recurrence (Add. I, Art. 10)
For the continued fraction , define successive numerators and denominators by
A = \alpha, & A' = 1, \\ B = \beta A + 1, & B' = \beta, \\ C = \gamma B + A, & C' = \gamma B' + A', \\ D = \delta C + B, & D' = \delta C' + B', \\ \vdots & \vdots \end{array}$$ In modern notation, with partial quotients $a_0, a_1, a_2, \ldots$, $$p_{-1}=1, p_{-2}=0; \qquad p_n = a_n p_{n-1} + p_{n-2},$$ $$q_{-1}=0, q_{-2}=1; \qquad q_n = a_n q_{n-1} + q_{n-2}.$$ The $n$th **convergent** is $p_n/q_n$ — equivalently the value of the CF truncated after the $n$th partial quotient. The sequence $\frac{A}{A'}, \frac{B}{B'}, \frac{C}{C'}, \frac{D}{D'}, \ldots$ converges to $a$. If $a$ is rational $V/V'$, the sequence terminates exactly at $V/V'$; if irrational, the sequence is infinite (Add. I, Art. 10). --- ## The Cross-Product Identity (Add. I, Art. 12) Cross-multiplying consecutive convergents, $$BA' - AB' = 1, \quad CB' - BC' = -1, \quad DC' - CD' = 1, \quad ED' - DE' = -1, \ldots$$ i.e., $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$. Two corollaries follow immediately: 1. **Convergents are in lowest terms.** Any common divisor of $p_n$ and $q_n$ would divide $\pm 1$ — impossible. 2. **No rational $m/n$ with $n \le q_{n-1}$ lies strictly between two consecutive convergents.** (Foundation of the [[best-rational-approximations|best-approximation property]].) --- ## Alternation Around the Target (Add. I, Art. 14) When all partial quotients are positive (i.e., the CF is built by always-below integer-part rounding), the convergents are **alternately less than and greater than** $a$: $$\frac{p_0}{q_0} < a, \quad \frac{p_1}{q_1} > a, \quad \frac{p_2}{q_2} < a, \quad \frac{p_3}{q_3} > a, \ldots$$ Equivalently, $a$ always lies strictly between two consecutive convergents. Splitting into the two monotone series: - **Increasing series** (under $a$): $\frac{p_0}{q_0}, \frac{p_2}{q_2}, \frac{p_4}{q_4}, \ldots$ - **Decreasing series** (over $a$): $\frac{p_1}{q_1}, \frac{p_3}{q_3}, \frac{p_5}{q_5}, \ldots$ --- ## Error Bounds (Add. I, Art. 13) For consecutive convergents $C/C'$ and $D/D'$, $$\frac{1}{C'(D' + C')} \;<\; \left|\, a - \frac{C}{C'}\, \right| \;<\; \frac{1}{C' D'}.$$ In modern notation, $$\frac{1}{q_n(q_{n+1} + q_n)} \;<\; \left|\, a - \frac{p_n}{q_n}\, \right| \;<\; \frac{1}{q_n q_{n+1}}.$$ So the error decays roughly like $1/q_n^2$ — the **quadratic-decay** rate that characterises best rational approximations and lies at the heart of Diophantine approximation theory. **Worked example** (Add. I, Art. 21, $\pi$): The convergent $\frac{22}{7}$ has error $< \frac{1}{7 \cdot 106} \approx 0.00135$, and $> \frac{1}{7 \cdot 113} \approx 0.00126$. Both bounds are consistent with the actual error $|\pi - 22/7| \approx 0.00126$. --- ## Monotone Growth of Numerators and Denominators (Art. 11) If all $\alpha, \beta, \gamma, \ldots > 0$, then $A, B, C, D, \ldots$ and $A', B', C', D', \ldots$ are all positive and strictly increasing. This is essentially the recurrence: each new term equals (positive integer)·(previous) + (one before), which exceeds the previous. Even when some partial quotients are negative or equal to unity, Lagrange shows the denominators continue to grow, after possibly skipping a step. --- ## Computational Layout (Add. I, Art. 20) Lagrange organises the computation in a two-row table: write the partial quotients on top, then build the convergents underneath using the rule "next = (top above)·(previous) + (one before)": $$\begin{array}{c|cccccccc} \text{partial quotient} & \alpha & \beta & \gamma & \delta & \epsilon & \cdots \\ \hline \text{numerator} & \alpha & \beta\alpha+1 & \gamma B + \alpha & \delta C + B & \cdots \\ \text{denominator} & 1 & \beta & \gamma B' + 1 & \delta C' + B' & \cdots \end{array}$$ For the calendar example $86400/20929$ with partial quotients $4, 7, 1, 3, 1, 16, 1, 1, 15$, this gives $$\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}.$$ --- ## Related pages - [[continued-fractions]] - [[semi-convergents]] - [[best-rational-approximations]] - [[calendar-approximations]] - [[add1-continued-fractions]] - [[approximation-methods]]