# Rational Approximations ## 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 $$ 0 \left| \alpha-\frac{p_n}{q_n} \right| $$ 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.