The error in taking the last of these is less than and is
therefore less than or , and still less than .00001. Thus the seventh convergent gives the value to at least four places of decimals.
512. Every periodic continued fraction is equal to one of the roots of a quadratic equation of which the coefficients are rational.
Let x denote the continued fraction, and y the periodic part, and suppose that
,
and ,
where are positive integers.
Let , be the convergents to x corresponding to the quotients h, k respectively; then since y is the complete quotient, we have ; whence y = .
Let be the convergents to y corresponding to the quotients u, v respectively; then y= .
Substituting for y in terms of x and simplifying, we obtain a quadratic of which the coefficients are rational.
The equation , which gives the value of y, has its roots real and of opposite signs; if the positive value of y be substituted in x = , on rationalizing the denominator the value of x is of the form , where A, B, C are integers, B being positive since the value of y is real.
