To do mathematics, that is, in order to understand and to make contributions to this discipline, it is necessary to study its history. Mathematics is constantly building upon past discoveries. Those who wish to study a particular field of mathematics, whether it be statistics, abstract algebra, or continued fractions, will first need to study their field's past. In doing so, one is able to build upon past accomplishments rather than repeating them.
The origin of continued fractions is hard to pinpoint. This is due to the fact that we can find examples of these fractions throughout mathematics in the last 2000 years, but its true foundations were not laid until the late 1600's, early 1700's.
The origin of continued fractions is traditionally placed at the time of the creation of Euclid's Algorithm. Euclid's Algorithm, however, is used to find the greatest common denominator (gcd) of two numbers. However, by algebraically manipulating the algorithm, one can derive the simple continued fraction of the rational p/q as opposed to the gcd of p and q. (To see this, check out Theorem 1.) It is doubtful whether Euclid or his predecessors actually used this algorithm in such a manner. But due to its close relationship to continued fraction, the creation of Euclid's Algorithm signifies the initial development of continued fractions.
For more than a thousand years, any work that used continued fractions was restricted to specific examples. The Indian mathematician Aryabhata (d. 550 AD) used a continued fraction to solve a linear indeterminate equation. Rather than generalizing this method, his use of continued fractions is used solely in specific examples.
Throughout Greek and Arab mathematical writing, we can find examples and traces of continued fractions. But again, its use is limited to specifics.
Two men from the city of Bologna, Italy, Rafael Bombelli (b. c.1530) and Pietro Cataldi (1548-1626) also contributed to this field, albeit providing more examples. Bombelli expressed the square root of 13 as a repeating continued fraction. Cataldi did the same for the square root of 18. Besides these examples, however, neither mathematician investigated the properties of continued fractions.
Continued fractions became a field in its right through the work of John Wallis (1616-1703). In his book Arithemetica Infinitorium (1655), he developed and presented the identity
In his book Opera Mathematica (1695) Wallis laid some of the basic groundwork for continued fractions. He explained how to compute the nth convergent and discovered some of the now familiar properties of convergents. It was also in this work that the term "continued fraction" was first used.
The Dutch mathematician and astronomer Christiaan Huygens (1629-1695) was the first to demonstrate a practical application of continued fractions. He wrote a paper explaining how to use the convergents of a continued fraction to find the best rational approximations for gear ratios. These approximations enabled him to pick the gears with the correct number of teeth. His work was motivated impart by his desire to build a mechanical planetarium.
While the work of Wallis and Huygens began the work on continued fractions, the field of continued fractions began to flourish when Leonard Euler (1707-1783), Johan Heinrich Lambert (1728-1777), and Joseph Louis Lagrange (1736-1813) embraced the topic. Euler laid down much of the modern theory in his work De Fractionlous Continious (1737). He showed that every rational can be expressed as a terminating simple continued fraction. (See Theorem 1) He also provided an expression for e in continued fraction form. He used this expression to show that e and e2 are irrational. He also demonstrated how to go from a series to a continued fraction representation of the series, and conversely.
Lambert generalized Euler's work on e to show that both ex and tan x are irrational if x is rational. Lagrange used continued fractions to find the value of irrational roots. He also proved that a real root of a quadratic irrational is a periodic continued fraction.
The nineteenth century can probably be described as the golden age of continued fractions. As Claude Brezinski writes in History of Continued Fractions and Padre Approximations, "the nineteenth century can be said to be popular period for continued fractions." It was a time in which "the subject was known to every mathematician." As a result, there was an explosion of growth within this field. The theory concerning continued fractions was significantly developed, especially that concerning the convergents. (See Theorem 2 and Theorem 3 for more.) Also studied were continued fractions with complex variables as terms. Some of the more prominent mathematicians to make contributions to this field include Jacobi, Perron, Hermite, Gauss, Cauchy, and Stieljes. By the beginning of the 20th century, the discipline had greatly advanced from the initial work of Wallis.
Since the beginning of the 20th century continued fractions have made their appearances in other fields. To give an example of their versatility, a recent paper by Rob Corless examined the connection between continued fractions and chaos theory. Continued fractions have also been utilized within computer algorithms for computing rational approximations to real numbers, as well as solving indeterminate equations.
This brief sketch into the past of continued fractions is intended to provide an overview of the development of this field. Though its initial development seems to have to taken a long time, once started, the field and its analysis grew rapidly. Even today, the fact that continued fractions are still being used signify that the field is still far from being exhausted.