Skip to content

Rational Approximations

Many important numbers are irrational:

number-theory book chatgpt

Approximating Irrational Numbers

Many important numbers are irrational:

2,π,e. \sqrt2,\qquad \pi,\qquad e.

Since irrational numbers cannot be written exactly as fractions, one seeks rational approximations

pq \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 α\alpha is a real number and

pq \frac pq

is a rational approximation.

The approximation error is

∣α−pq∣. \left| \alpha-\frac pq \right|.

A good approximation has small error and relatively small denominator qq.

For example,

π≈227 \pi\approx\frac{22}{7}

gives

∣π−227∣≈0.00126. \left| \pi-\frac{22}{7} \right| \approx0.00126.

An even better approximation is

π≈355113, \pi\approx\frac{355}{113},

whose error is less than

3×10−7. 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 α\alpha, there exist infinitely many rational numbers

pq \frac pq

such that

∣α−pq∣<1q2. \left| \alpha-\frac pq \right| < \frac1{q^2}.

∣α−pq∣<1q2 \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

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

be the continued fraction expansion of an irrational number.

Its convergents

pnqn \frac{p_n}{q_n}

satisfy

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

Since

qn+1>qn, q_{n+1}>q_n,

this implies

∣α−pnqn∣<1qn2. \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<qn, 0<q<q_n,

then

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

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

for every rational number p/qp/q.

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

Example: Approximating 2\sqrt2

The continued fraction expansion is

2=[1;2‾]. \sqrt2=[1;\overline2].

Its convergents are

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

Now

2≈1.414213562… \sqrt2\approx1.414213562\dots

and

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

The error is

∣2−9970∣≈0.000072. \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>0c>0 such that

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

\frac{c}{q^2} $$

for all rational numbers p/qp/q.

Quadratic irrationals such as

2 \sqrt2

are badly approximable because their continued fraction coefficients remain bounded.

The golden ratio

φ=1+52 \varphi=\frac{1+\sqrt5}{2}

is the most badly approximable irrational number. Its continued fraction is

[1;1,1,1,… ]. [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,

π≈355113 \pi\approx\frac{355}{113}

is unusually accurate because of a large coefficient in the continued fraction expansion of π\pi.

Numbers with exceptionally good approximations are connected to transcendence theory and irrationality measures.

For instance, Liouville numbers satisfy inequalities such as

∣α−pq∣<1qn \left| \alpha-\frac pq \right| < \frac1{q^n}

for arbitrarily large nn.

These numbers are transcendental.

Geometry of Approximation

Rational approximation can be interpreted geometrically.

The fraction

pq \frac pq

corresponds to the lattice point

(q,p) (q,p)

in the plane.

Approximating α\alpha means finding lattice points close to the line

y=αx. 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 nn consists of all reduced fractions between 00 and 11 whose denominators are at most nn, arranged in increasing order.

Neighboring fractions

abandcd \frac ab \quad\text{and}\quad \frac cd

satisfy

bc−ad=1. 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.