# Convergents

## Successive Rational Approximations

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

If

$$
\alpha=[a_0;a_1,a_2,a_3,\dots],
$$

then the $n$-th convergent is

$$
\frac{p_n}{q_n} =
[a_0;a_1,\dots,a_n].
$$

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

## First Examples

Consider

$$
\sqrt2=[1;\overline2].
$$

Its convergents are

$$
1,
\frac32,
\frac75,
\frac{17}{12},
\frac{41}{29},
\dots
$$

These fractions alternate around $\sqrt2$:

$$
1<\sqrt2<\frac32,
$$

$$
\frac75<\sqrt2<\frac32,
$$

$$
\frac75<\sqrt2<\frac{17}{12}.
$$

Each convergent improves the approximation.

For example,

$$
\frac{99}{70}=1.414285714\dots
$$

differs from $\sqrt2$ by less than

$$
10^{-4}.
$$

## Recurrence Relations

Convergents satisfy elegant recursive formulas.

Define

$$
p_{-2}=0,\qquad p_{-1}=1,
$$

$$
q_{-2}=1,\qquad q_{-1}=0.
$$

Then for $n\ge0$,

$$
p_n=a_np_{n-1}+p_{n-2},
$$

$$
q_n=a_nq_{n-1}+q_{n-2}.
$$

$$
p_n=a_np_{n-1}+p_{n-2}
$$

$$
q_n=a_nq_{n-1}+q_{n-2}
$$

These recurrences allow convergents to be computed efficiently.

### Example

For

$$
[1;2,2,2,\dots],
$$

we have:

$$
p_0=1,\qquad q_0=1,
$$

$$
p_1=2\cdot1+1=3,
\qquad
q_1=2\cdot1+0=2,
$$

$$
p_2=2\cdot3+1=7,
\qquad
q_2=2\cdot2+1=5.
$$

Thus the convergents begin

$$
1,\frac32,\frac75.
$$

## Determinant Identity

Convergents satisfy the fundamental identity

$$
p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}.
$$

$$
p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}
$$

This has several important consequences.

First,

$$
\gcd(p_n,q_n)=1.
$$

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

$$
\frac{p_n}{q_n}
$$

is the $n$-th convergent of $\alpha$, then

$$
\left|
\alpha-\frac{p_n}{q_n}
\right|
<
\frac1{q_nq_{n+1}}.
$$

In particular,

$$
\left|
\alpha-\frac{p_n}{q_n}
\right|
<
\frac1{q_n^2}.
$$

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

$$
\frac{p_n}{q_n}
$$

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

$$
0<q<q_n,
$$

then for every rational number $p/q$,

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

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

This theorem makes convergents fundamental objects in Diophantine approximation.

## Alternating Behavior

Convergents alternate around the irrational number.

If $n$ is even,

$$
\frac{p_n}{q_n}<\alpha,
$$

while if $n$ is odd,

$$
\frac{p_n}{q_n}>\alpha.
$$

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,

$$
\sqrt2=[1;\overline2].
$$

Its convergents satisfy

$$
p_n^2-2q_n^2=\pm1.
$$

Indeed,

$$
3^2-2(2^2)=1,
$$

$$
7^2-2(5^2)=-1,
$$

$$
17^2-2(12^2)=1.
$$

Thus continued fractions generate infinitely many solutions of Pell equations.

## Matrix Interpretation

Convergents admit a matrix formulation.

Define

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

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

$$
\det M(a)=-1.
$$

## 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.