Number Theory and its Applications

11 Continued Fractions

“If equations are trains threading the landscape of numbers, then no train stops at pi.”

— Richard Preston

11.1 Introduction

Being a natural object, continued fractions appear in many areas of Mathematics, sometimes in an unexpected way. The old name of continued fraction was “anthyphaeiretic ratios”, which the Dutch mathematician and astronomer in 1687 made the first practical application explaining how to use convergents to find the best rational approximations for ratios. To build a mechanical planetarium, motivated him to do so. Later on, many renowned mathematicians including Euler, Jacobi, Gauss and Cauchy get attracted to continued fractions. Continued fractions find its applications in some areas of contemporary ...