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

1 1
simpler, viz. ------- and ------------------ .
q^{2}_n q_n(q_n + q_{n+1})

Every simple continued fraction must converge to a definite limit; for
its value lies between that of the first and second convergents and,
since

p_n p_{n-1} 1 p_n p_{n-1}
--- ~ ------- = ------------, Lt. ----- = Lt. -------,
q_n q_{n-1} q_{n}q_{n-1} q_n q_{n-1}

so that its value cannot oscillate.

The chief practical use of the simple continued fraction is that by
means of it we can obtain rat