Continued Fraction - Motivation and Notation

Motivation and Notation

Consider a typical rational number 415/93, which is around 4.4624. As a first approximation, start with 4, which is the integer part; 415/93 = 4 + 43/93. Note that the fractional part is the reciprocal of 93/43 which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal, to get a second approximation of 4 + 1/2 = 4.5; 93/43 = 2 + 7/43. The fractional part of 93/43 is the reciprocal of 43/7 which is about 6.1429. Use 6 as an approximation for this to get 2 + 1/6 as an approximation for 93/43 and 4 + 1/(2 + 1/6), about 4.4615, as the third approximation; 43/7 = 6 + 1/7. Finally, the fractional part of 43/7 is the reciprocal of 7, so its approximation in this scheme, 7, is exact (7/1 = 7 + 0/1) and produces the exact expression 4 + 1/(2 + 1/(6 + 1/7)) for 415/93. This expression is called the continued fraction representation of the number. Dropping some of the less essential parts of the expression 4 + 1 / (2 + 1 / (6 + 1 / 7)) gives the abbreviated notation 415/93=. Note that it is customary to replace only the first comma by a semicolon. Some older textbooks use all commas in the (n+1)-tuple, e.g. .

If the starting number is rational then this process exactly parallels the Euclidean algorithm. In particular, it must terminate and produce a finite continued fraction representation of the number. If the starting number is irrational then the process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are:

• √19 = . The pattern repeats indefinitely with a period of 6.
• e = (sequence A003417 in OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
• π = (sequence A001203 in OEIS). The terms in this representation are apparently random.

Continued fractions are, in some ways, more "mathematically natural" representations of real number than other representations such as decimal representations, and they have several desirable properties:

• The continued fraction representation for a rational number is finite and only rational numbers have finite representations. In contrast, the decimal representation of a rational number may be finite, for example 137/1600 = 0.085625, or infinite with a repeating cycle, for example 4/27 = 0.148148148148….
• Every rational number has an essentially unique continued fraction representation. Each rational can be represented in exactly two ways, since = . Usually the first, shorter one is chosen as the canonical representation.
• The continued fraction representation of an irrational number is unique.
• The real numbers whose continued fraction eventually repeats are precisely the quadratic irrationals. For example, the repeating continued fraction is the golden ratio, and the repeating continued fraction is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently random.
• The successive approximations generated in finding the continued fraction representation of a number, i.e. by truncating the continued fraction representation, are in a certain sense (described below) the "best possible".