2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of . The remaining fractional part, , is the reciprocal of , and is around 6.1429. Use 6 as an approximation for this to obtain as an approximation for and , about 4.4615, as the third approximation; . Finally, the fractional part, , is the reciprocal of 7, so its approximation in this scheme, 7, is exact () and produces the exact expression for . The expression is called the continued fraction representation of . This can be represented by the abbreviated notation = [4; 2, 6, 7]. (It is customary to replace only the first comma by a semicolon.) Some older textbooks use all commas in the -tuple, for example, [4, 2, 6, 7]. 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: . The pattern repeats indefinitely with a period of 6. . The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle. . No pattern has ever been found in this representation. . The golden ratio, the irrational number that is the "most difficult" to approximate rationally . . The Euler–Mascheroni constant, which is expected but not known to be irrational, and whose continued fraction has no apparent pattern. 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: The continued fraction representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for example , or infinite with a repeating cycle, for example Every rational number has an essentially unique simple continued