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continued fraction

A representation of a real number in the form

continued fraction

which, mercifully for typesetters, can be written in compact notation as

x = [a0; a1, a2, a3, ...]

where the integers ai are called partial quotients. Although rarely encountered in school and even college math courses, continued fractions (CFs) provide one of the most powerful and revealing forms of numerical expression. Numbers whose decimal expansions look bland and unremarkable suddenly turn out, when unfolded as CFs, to possess extraordinary symmetries and patterns. CFs also offer a way of constructing rational approximations to irrational numbers and discovering the most irrational numbers.

CFs first appeared in the works of the Indian mathematician Aryabhata in the sixth century, who used them to solve linear equations. They reemerged in Europe in the 15th and 16th centuries and Fibonacci attempted to define them in a general way. The actual term "continued fraction" first appeared in 1653 in an edition of Arithmetica Infinitorum by John Wallis. Their properties were also studied by one of Wallis's English contemporaries, William Brouncker, who along with Wallis, was one of the founders of the Royal Society. At about the same time, in Holland, Christiaan Huygens made practical use of CFs in his designs of scientific instruments. Later, in the 18th and early 19th centuries, Carl Gauss and Leonhard Euler delved into many of their deeper properties.

CFs can be finite or infinite in length. Finite CFs are unique so long as we don't allow a quotient of 1in the final entry in the bracket; for example, we should write 1/2 as [0; 2] rather than as [0; 1, 1]. We can always eliminate a 1 from the last entry by adding to the previous entry. If CFs are finite in length then they can be evaluated level by level (starting at the bottom) and will reduce always to a rational fraction; for example, the CF [1; 3, 2, 4] = 40/31. However, CFs can be infinite in length, in which case they produce representations of irrational numbers. Here are the leading terms from a few notable examples of infinite CF:

     e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...]
     √2 = [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]
     √3 = [1; 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...]
     π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, ...]

These examples reveal a number of possibilities. All of the expansions have simple patterns except that for pi, which was first calculated by John Wallis in 1685 and has no obvious pattern at all. There also seems to be a preference for the quotients to be small numbers in these examples. The CF for e was first calculated by Roger Cotes, the Plumian Professor of Experimental Philosophy at Cambridge, in 1714.

If an infinite CF is chopped off after a finite number of steps then the result is a rational approximation to the original irrational. For example, in the case of π, chopping the CF at [3; 7] gives the familiar rational approximation for π of 22/7 = 3.1428571... Retaining two more terms leads to [3; 7, 15, 1] = 353/113 = 3.1415929..., which is an even better approximation to the true value of π (3.14159265...) and one that was known to the early Chinese. The more terms retained in the CF, the better the rational approximation becomes. In fact, the CF provides the best possible rational approximations to a general irrational number. Notice also that if a large number occurs in the expansion of quotients, then truncating the CF before that will produce an exceptionally good rational approximation. Most CF quotients are small numbers (1 or 2), so the appearance in the CF of π of a number as large as 292 so early in the expansion is unusual. It also leads to an extremely good rational approximation to π = [3; 7, 15, 1, 292] = 103993/33102.

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