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Connecting Greek Ladders and Continued Fractions - History of Continued Fractions

Author(s): 
Kurt Herzinger (United States Air Force Academy) and Robert Wisner (New Mexico State University)

It is not clear when the notion of continued fractions was first realized. From [5] (pp. 28–29) we read that there are instances of ancient arithmetic which are suggestive of the notion of continued fractions without any formal development. For example, we have Euclid’s algorithm from the seventh book of his Elements (circa 300 B.C.E.). The algorithm computes the greatest common divisor of two integers but can be modified to produce a simple continued fraction for a rational number. For example, we can use Euclid's algorithm to compute \(\gcd(21,51)\) as follows:

\(51 = 2\cdot 21 +9\) \(\frac{51}{21} = 2 + \frac{9}{21}\)
\(21 = 2\cdot 9 + 3\) \(\frac{51}{21} = 2+\frac{9}{2\cdot 9 + 3} = 2+ \frac{1}{2+\frac{3}{9}}\)
\(9 = 3\cdot 3\) \(\frac{51}{21} = 2 + \frac{1}{2+\frac{1}{3}}\)

There is a reference to continued fractions in the works of the Indian mathematician Aryabhata (476–550 C.E.). In a work called the Aryabhatiya, we find one of the earliest attempts to produce a general solution to a linear indeterminate equation of the form \(by=ax+c\) where \(a,b,\) and \(c\) are integers. The technique demonstrated by Aryabhata is clearly related to continued fractions.

From 1202 C.E. we find in the Liber Abaci (The Book of Calculations) by Leonardo of Pisa, or Fibonacci, the symbol \(\frac{111}{345}\) which he used as an abbreviation for

\[\frac{1+\frac{1+\frac{1}{5}}{4}}{3} = \frac{1}{3}+\frac{1}{3\cdot 4}+\frac{1}{3\cdot 4\cdot 5}.\]

See section 15.2 of [2]. We might also write this as

\[\frac{1}{2+\frac{1}{3+\frac{1}{4}}}.\]

Most modern authorities agree that the theory of continued fractions began with Rafael Bombelli [5] (pp. 29–30). In his L'Algebra Opera (1572), Bombelli essentially proved that \(\sqrt{13}\) is the limit of the infinite continued fraction

\[3+\frac{4}{6+\frac{4}{6+\frac{4}{\ddots}}}\]

That is, \(\sqrt{13} = [3;4,6;4,6;4,6;\dots]\). However, in the notation of the time, Bombelli would have written this continued fraction as

\[\sqrt{13}=3+\frac{4}{6} _+ \frac{4}{6}_+\frac{4}{6}_+ \dots\]

We also find this notation used in modern texts such as [5].

Shortly after Bombelli, Pietro Antonio Cataldi expressed \(\sqrt{18}\) as \([4;2,8;2,8;2,8;\dots]\) and modified the notation as

\[\sqrt{18}=4\ \&\ \frac{2}{8.}\ \&\ \frac{2}{8.}\ \& \dots\]

William Brouncker (1620–1684) established this interesting identity:

\[\frac{4}{\pi} = 1+\frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{\ddots}}}}.\]

Brouncker made no further use of continued fractions. However his contemporary John Wallis, perhaps motivated by Brouncker's fraction, established many of the basic properties of convergents in his 1655 book Arithmetica Infinitorum. He also used the term "continued fraction" for the first time.

Christian Huygens (1629–1695) is credited with being the first to use continued fractions in a practical application. He used continued fractions for approximating gear ratios in the building of a mechanical planetarium.

The theory of continued fractions was developed further through the 18th and 19th centuries by Euler, Lambert, Lagrange, and many others. Continued fractions continue to play an important role today in number theory and Diophantine approximations.

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