simple continued fraction for |- !Step !RealNumber !Integerpart !Fractionalpart !Simplified !Reciprocalof |- !1 | | | | | |- !2 | | | | | |- !3 | | | | | |- !4 | | | | |colspan="2"|STOP |- | colspan="7" style="text-align:right;" | Continued fraction form for = |} Notations The integers , etc., are called the coefficients or terms of the continued fraction. One can abbreviate the continued fraction in the notation of Carl Friedrich Gauss or as , or in the notation of Pringsheim as or in another related notation as Sometimes angle brackets are used, like this: The semicolon in the square and angle bracket notations is sometimes replaced by a comma. One may also define infinite simple continued fractions as limits: This limit exists for any choice of and positive integers . Finite continued fractions Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and the other coefficients are positive integers. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore always greater than 1, if present. In symbols: . . Reciprocals The continued fraction representations of a positive rational number and its reciprocal are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively. In other words, the numbers represented by and are reciprocals. For instance if is an integer and then and . If then and . The last number that generates the remainder of the continued fraction is the same for both and its reciprocal. For example, and . Infinite continued fractions and convergents Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction. An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction. The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number