| 1 | 1 | 1 | 1 | 1 | |||||
| 1 | + | 6 | + | 10 | + | 14 | + | 18 | + ..., |
may be used to approximate very rapidly to the value of e.
For the application of continued fractions to the problem “To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable,” the reader is referred to G. Chrystal’s Algebra, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the [calendar] (q.v.).
Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a + 1, put x = a + 1/y and form the equation in y; if the equation in y has a root between b and b + 1, put y = b + 1/z, and so on. Such a method is, however, too tedious, compared with such a method as Homer’s, to be of any practical value.
The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. If we suppose a/b to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq - bp = +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please. If we take aq-bp = +1 we have a general solution in integers of ax + by = c, viz. x = cq - bt, y = at - cp; if we take aq - bp = -1, we have x = bt - cq, y = cp - at.
An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G. Cantor in vol. 2 of the Acta Mathematica.
Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J. A. Serret’s Cours d’Algèbre Supérieure.
2. Recurring Simple Continued Fractions.—The infinite continued fraction
| a1 + | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||
| a2 | + | a3 | + ... + | an | + | b1 | + | b2 | + ... + | bn | + | b1 | + | b2 | + ... + | bn | + | b1 | + ..., |
where, after the nth partial quotient, the cycle of partial quotients b1, b2, ..., bn recur in the same order, is the type of a recurring simple continued fraction.