structural:reformatting
Motivation and notation

Consider, for example, the 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. 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 obtain a second approximation of 4 + 1 / 2 = 4.5. Now, 93 / 43 = 2 + 7 / 43;the remaining fractional part, 7 / 43, is the reciprocal of 43 / 7, and 43 / 7 is around 6.1429. Use 6 as an approximation for this to obtain 2 + 1 / 6 as an approximation for 93 / 43 and 4 + 1 / 2 + 1 / 6, about 4.4615, as the third approximation. Further, 43 / 7 = 6 + 1 / 7. Finally, the fractional part, 1 / 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.

The expression 4 + 1 / 2 + 1 / 6 + 1 / 7 is called the continued fraction representation of 415 / 93. This can be represented by the abbreviated notation 415 / 93 =. (It is customary to replace only the first comma by a semicolon to indicate that the preceding number is the whole part.) Some older textbooks use all commas in the (n + 1) -tuple, for example,.

If the starting number is rational, then this process exactly parallels the Euclidean algorithm applied to the numerator and denominator of the number. In particular, it must terminate and produce a finite continued fraction representation of the number. The sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm. 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:

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