Continued Fractions

Abstract

This chapter explores fractional expressions which most people do not use or see in their everyday life. Two such fractions are the multi-decked expressions

$$\displaystyle{1 + \frac{1} {2 + \frac{1} {3+ \frac{1} {4+ \frac{1} {5} } } } }$$

and

$$\displaystyle{1 + \frac{1} {1 + \frac{1} {1+ \frac{1} {1+ \frac{1} {1+\cdots }}}}.}$$

Such a multi-layered fraction is a continued fraction, a term coined by Wallis. The Indian mathematician-astronomer Aryabhata (ca. 476–ca. 550) used continued fractions to solve the linear diophantine equation (LDE) $ax + by = c$, where a, b, c, x, and y are integers and (a, b) = 1. The Italian mathematician Rafael Bombelli (1526–1573) used continued fractions to approximate $\sqrt{13}$ in his L’Algebra Opera (1572). In 1613, Pietro Antonio Cataldi (1548–1626), another Italian mathematician, employed them for approximating square roots of numbers. The Dutch physicist and mathematician Christiaan Huygens (1629–1695) used them in the design of a mathematical model for planets (1703).