a1, a2, a3, a4 ... are all _positive integers_, is called a _simple continued fraction_. In the case of this fraction a1, a2, a3, a4 ... are called the successive _partial quotients_. It is evident that, in this case, p1, p2, p3 ..., q1, q2, q3 ..., are two series of positive integers increasing without limit if the fraction does not terminate. b2 b3 b4 The general continued fraction a1 + -- -- -- ... is evidently a2 + a3 + a4 + equal, convergent by convergent, to the continued fraction [lambda]2b2 [lambda]2[lambda]3b3 [lambda]3[lambda]4b4 a1 + ----------- -------------------- -------------------- ..., [lambda]2a2 + [lambda]3a3 + [lambda]4a4 + where [lambda]2, [lambda]3, [lambda]4, ... are any quantities whatever, so that by choosing [lambda]2b2 = 1, [lambda]2[lambda]3b3 = 1, &c., it can be reduced to any equivalent continued fraction of the form 1 1 1 a1 + -- -- -- ... d2 + d3 + d4 + _Simple Continued Fractions._ 1. The simple continued fraction is both the most interesting and important kind of continued fraction. Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. A non-terminating simple continued fraction must be incommensurable. In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last 1 partial quotient, a_n as a_n - 1 + --. 1 The numerators and denominators of the successive convergents obey the law p_{n}q_{n-1} - p_{n-1}q_n = (-1)^n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are:-- The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property. Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent. Every convergent is nearer to the value of the whole fraction than any preceding convergent. Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent. The difference between the continued fraction and the n^{th} convergent 1 a_{n+2} is less than ------------, and greater than ------------. These limits q_{n}q_{n+1} q_{n}q_{n+2} may be replaced by the following, which, though not so close, are