Rational Approximations
Many important numbers are irrational:
Approximating Irrational Numbers
Many important numbers are irrational:
$$ \sqrt2,\qquad \pi,\qquad e. $$
Since irrational numbers cannot be written exactly as fractions, one seeks rational approximations
$$ \frac pq $$
that are close to the target number.
The central question is:
How well can irrational numbers be approximated by rational numbers?
Continued fractions provide the most systematic answer to this problem.
Measuring Approximation Error
Suppose $\alpha$ is a real number and
$$ \frac pq $$
is a rational approximation.
The approximation error is
$$ \left| \alpha-\frac pq \right|. $$
A good approximation has small error and relatively small denominator $q$.
For example,
$$ \pi\approx\frac{22}{7} $$
gives
$$ \left| \pi-\frac{22}{7} \right| \approx0.00126. $$
An even better approximation is
$$ \pi\approx\frac{355}{113}, $$
whose error is less than
$$ 3\times10^{-7}. $$
The denominator remains modest despite the high accuracy.
Dirichlet Approximation Theorem
A foundational result in Diophantine approximation is the following theorem.
Theorem. For every irrational number $\alpha$, there exist infinitely many rational numbers
$$ \frac pq $$
such that
$$ \left| \alpha-\frac pq \right| < \frac1{q^2}. $$
$$ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2} $$
This estimate is surprisingly strong. Random fractions generally do not approximate irrational numbers this well.
Continued fractions naturally produce approximations satisfying this inequality.
Convergents as Best Approximations
Let
$$ \alpha=[a_0;a_1,a_2,\dots] $$
be the continued fraction expansion of an irrational number.
Its convergents
$$ \frac{p_n}{q_n} $$
satisfy
$$ \left| \alpha-\frac{p_n}{q_n} \right| < \frac{1}{q_nq_{n+1}}. $$
Since
$$ q_{n+1}>q_n, $$
this implies
$$ \left| \alpha-\frac{p_n}{q_n} \right| < \frac1{q_n^2}. $$
Thus convergents automatically satisfy Dirichlet-quality bounds.
Moreover, convergents are best approximations in the following sense:
If
$$ 0<q<q_n, $$
then
$$ \left| \alpha-\frac pq \right|
\left| \alpha-\frac{p_n}{q_n} \right| $$
for every rational number $p/q$.
Hence no fraction with smaller denominator approximates $\alpha$ more accurately.
Example: Approximating $\sqrt2$
The continued fraction expansion is
$$ \sqrt2=[1;\overline2]. $$
Its convergents are
$$ 1, \frac32, \frac75, \frac{17}{12}, \frac{41}{29}, \dots $$
Now
$$ \sqrt2\approx1.414213562\dots $$
and
$$ \frac{99}{70}=1.414285714\dots $$
The error is
$$ \left| \sqrt2-\frac{99}{70} \right| \approx0.000072. $$
This accuracy is remarkable for such a small denominator.
Badly Approximable Numbers
Some irrational numbers are harder to approximate than others.
A number is called badly approximable if there exists a constant $c>0$ such that
$$ \left| \alpha-\frac pq \right|
\frac{c}{q^2} $$
for all rational numbers $p/q$.
Quadratic irrationals such as
$$ \sqrt2 $$
are badly approximable because their continued fraction coefficients remain bounded.
The golden ratio
$$ \varphi=\frac{1+\sqrt5}{2} $$
is the most badly approximable irrational number. Its continued fraction is
$$ [1;1,1,1,\dots]. $$
All partial quotients are as small as possible, forcing the slowest possible approximation improvement.
Very Good Approximations
Some numbers admit extraordinarily good rational approximations.
For example,
$$ \pi\approx\frac{355}{113} $$
is unusually accurate because of a large coefficient in the continued fraction expansion of $\pi$.
Numbers with exceptionally good approximations are connected to transcendence theory and irrationality measures.
For instance, Liouville numbers satisfy inequalities such as
$$ \left| \alpha-\frac pq \right| < \frac1{q^n} $$
for arbitrarily large $n$.
These numbers are transcendental.
Geometry of Approximation
Rational approximation can be interpreted geometrically.
The fraction
$$ \frac pq $$
corresponds to the lattice point
$$ (q,p) $$
in the plane.
Approximating $\alpha$ means finding lattice points close to the line
$$ y=\alpha x. $$
Thus Diophantine approximation becomes a problem about lattice geometry.
This viewpoint leads naturally to the geometry of numbers.
Farey Sequences
Farey sequences organize rational numbers by denominator size.
The Farey sequence of order $n$ consists of all reduced fractions between $0$ and $1$ whose denominators are at most $n$, arranged in increasing order.
Neighboring fractions
$$ \frac ab \quad\text{and}\quad \frac cd $$
satisfy
$$ bc-ad=1. $$
Farey sequences are closely connected with continued fractions, modular forms, and hyperbolic geometry.
Modern Perspective
Rational approximation lies at the intersection of:
- number theory,
- dynamical systems,
- geometry,
- harmonic analysis,
- ergodic theory.
The subject studies how arithmetic structure constrains approximation quality.
Questions about approximating real numbers eventually connect to:
- lattice reduction,
- modular surfaces,
- homogeneous dynamics,
- transcendence theory.
Thus the elementary problem of approximating irrational numbers leads naturally into deep areas of modern mathematics.