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Convergents

The convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages.

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Successive Rational Approximations

The convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages.

If

α=[a0;a1,a2,a3,… ], \alpha=[a_0;a_1,a_2,a_3,\dots],

then the nn-th convergent is

pnqn=[a0;a1,…,an]. \frac{p_n}{q_n} = [a_0;a_1,\dots,a_n].

Convergents are among the best possible rational approximations to α\alpha. They encode the arithmetic structure of the continued fraction expansion.

First Examples

Consider

2=[1;2‾]. \sqrt2=[1;\overline2].

Its convergents are

1,32,75,1712,4129,… 1, \frac32, \frac75, \frac{17}{12}, \frac{41}{29}, \dots

These fractions alternate around 2\sqrt2:

1<2<32, 1<\sqrt2<\frac32, 75<2<32, \frac75<\sqrt2<\frac32, 75<2<1712. \frac75<\sqrt2<\frac{17}{12}.

Each convergent improves the approximation.

For example,

9970=1.414285714… \frac{99}{70}=1.414285714\dots

differs from 2\sqrt2 by less than

10−4. 10^{-4}.

Recurrence Relations

Convergents satisfy elegant recursive formulas.

Define

p−2=0,p−1=1, p_{-2}=0,\qquad p_{-1}=1, q−2=1,q−1=0. q_{-2}=1,\qquad q_{-1}=0.

Then for n≥0n\ge0,

pn=anpn−1+pn−2, p_n=a_np_{n-1}+p_{n-2}, qn=anqn−1+qn−2. q_n=a_nq_{n-1}+q_{n-2}.

pn=anpn−1+pn−2 p_n=a_np_{n-1}+p_{n-2}

qn=anqn−1+qn−2 q_n=a_nq_{n-1}+q_{n-2}

These recurrences allow convergents to be computed efficiently.

Example

For

[1;2,2,2,… ], [1;2,2,2,\dots],

we have:

p0=1,q0=1, p_0=1,\qquad q_0=1, p1=2⋅1+1=3,q1=2⋅1+0=2, p_1=2\cdot1+1=3, \qquad q_1=2\cdot1+0=2, p2=2⋅3+1=7,q2=2⋅2+1=5. p_2=2\cdot3+1=7, \qquad q_2=2\cdot2+1=5.

Thus the convergents begin

1,32,75. 1,\frac32,\frac75.

Determinant Identity

Convergents satisfy the fundamental identity

pnqn−1−pn−1qn=(−1)n−1. p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}.

pnqn−1−pn−1qn=(−1)n−1 p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}

This has several important consequences.

First,

gcd⁡(pn,qn)=1. \gcd(p_n,q_n)=1.

Hence every convergent is already in lowest terms.

Second,

$$ \frac{p_n}{q_n}

\frac{p_{n-1}}{q_{n-1}}

\frac{(-1)^{n-1}}{q_nq_{n-1}}. $$

Thus consecutive convergents are extremely close when denominators are large.

Approximation Quality

Convergents approximate irrational numbers extraordinarily well.

If

pnqn \frac{p_n}{q_n}

is the nn-th convergent of α\alpha, then

∣α−pnqn∣<1qnqn+1. \left| \alpha-\frac{p_n}{q_n} \right| < \frac1{q_nq_{n+1}}.

In particular,

∣α−pnqn∣<1qn2. \left| \alpha-\frac{p_n}{q_n} \right| < \frac1{q_n^2}.

Thus the error decreases quadratically relative to the denominator.

This property explains why continued fractions are so effective for rational approximation.

Best Approximation Property

Convergents are optimal approximations among fractions with comparable denominator size.

Theorem. Let

pnqn \frac{p_n}{q_n}

be a convergent of an irrational number α\alpha. If

0<q<qn, 0<q<q_n,

then for every rational number p/qp/q,

$$ \left| \alpha-\frac pq \right|

\left| \alpha-\frac{p_n}{q_n} \right|. $$

Thus no fraction with smaller denominator approximates α\alpha more accurately.

This theorem makes convergents fundamental objects in Diophantine approximation.

Alternating Behavior

Convergents alternate around the irrational number.

If nn is even,

pnqn<α, \frac{p_n}{q_n}<\alpha,

while if nn is odd,

pnqn>α. \frac{p_n}{q_n}>\alpha.

Thus the convergents approach the limit from opposite sides.

This alternating structure reflects the recursive nature of continued fractions.

Connection with Pell Equations

Convergents of quadratic irrationals often produce solutions to Pell equations.

For example,

2=[1;2‾]. \sqrt2=[1;\overline2].

Its convergents satisfy

pn2−2qn2=±1. p_n^2-2q_n^2=\pm1.

Indeed,

32−2(22)=1, 3^2-2(2^2)=1, 72−2(52)=−1, 7^2-2(5^2)=-1, 172−2(122)=1. 17^2-2(12^2)=1.

Thus continued fractions generate infinitely many solutions of Pell equations.

Matrix Interpretation

Convergents admit a matrix formulation.

Define

M(a)=(a110). M(a)= \begin{pmatrix} a & 1\\ 1 & 0 \end{pmatrix}.

Then

$$ M(a_0)M(a_1)\cdots M(a_n)

\begin{pmatrix} p_n & p_{n-1}\ q_n & q_{n-1} \end{pmatrix}. $$

This representation connects continued fractions with linear algebra and group theory.

The determinant identity follows immediately because

det⁡M(a)=−1. \det M(a)=-1.

Arithmetic Significance

Convergents transform irrational numbers into structured sequences of rational approximations.

Unlike decimal approximations, convergents reflect intrinsic arithmetic properties of the number being approximated.

They reveal:

  • divisibility structure,
  • approximation quality,
  • periodicity,
  • solutions of Diophantine equations.

Thus convergents occupy a central role in classical and modern number theory.