222. Expression of a surd as a recurring continued fraction. This is the converse of the problem discussed in the last section, and shows that recurring continued fractions and quadratic equations are related in the same intimate way that terminating fractions and rational numbers (i.e. the roots of linear equations) are connected. We seek to express an irrational number, as, for instance, √2, as a continued fraction. This we may do

as follows.

Since 1 is the largest integer in √2 we may write

Since 2 is the largest integer in √2 + 1 we have

2+

2+

By continuing this process we continually get the denominator 2. Thus

√2=1+

1 1
2+2+

This process consists of the successive application of two operations, and affords the

RULE. Express the surd as the sum of two numbers the first of which is the largest integer that it contains.

Rationalize the numerator of the fraction whose numerator is the second of these numbers. Repeat these operations until a recurrence of denominators is observed.

be

This process may be applied to any surd, and a continued fraction which is recurring will always be obtained. We shall content ourselves with a statement of this fact without proof. If the surd is of the form a √, a continued fraction may derived for + √ and its sign changed. Since the real roots of any quadratic equation x2 + 2 α1x + α2 = 0 are surds of the form a± √b, where a and b are integers, it appears that the roots of any such equation may be expressed as recurring continued fractions. It can be shown that the real roots of the general quadratic equation aqx2 + а1x + α2 = 0 may also be so expressed.

EXERCISES

1. Express the following surds as recurrent continued fractions. (a) 2 + √13.

But since √13 - 3 is the same number that we have in (1), this fraction repeats from this point on, and we have

2. Express as a continued fraction the roots of the following equations.

223. Properties of convergents. The law of formation of convergents given in § 220 is valid whether the continued fraction is terminating or infinite. We should expect that in the case of an infinite fraction the successive convergents would give us an increasingly close approximation to the value of the fraction. This is indeed the fact, as we shall see.

THEOREM. The difference between the nth and (n+1)st con(−1)n+1 vergents is

In In + 1

We prove this theorem by complete induction.

Let the continued fraction be

We assume that the theorem holds for n = m, that is,

We must prove that it holds for n = m +1.

Then

am+29 m +1 + Im = Im +29

(1), § 220

am+2Pm+1+Pm = Pm+2°

Pm+1(am +29 m +1 + Im) — (am + 2Pm+1 + Pm) Im + 1

+1

=Pm+19m - PmLm+1

=

(PmLm+1
(m + 1 − Pm+19m)

= − (− 1)m+1 = (− 1)m+2.

by (1)

COROLLARY I. The difference between the successive convergents of a continued fraction with positive denominators approaches zero as a limit.

Since In = AnIn−1 + In−2, evidently an increases without limit when n is increased, since to obtain q, we add together positive numbers neither one of which can vanish.

Thus we can find a value of n large enough so that and

hence

-, will be smaller than any assigned number, which

1

In

In In + 1

1

is another way of stating that as n increases zero as a limit.

approaches

InIn+1

COROLLARY II. The even convergents decrease, while the odd convergents increase, as n increases.

We must show that

Pm+2 Pm

Im+2 Im

is negative or positive according as m is even or odd. Adding

By Corollary I, the denominator of the first fraction exceeds that of the second. Hence when m is odd the sum in the last member of the equation is positive, and when m is even the sum is negative.

We now see that any recurring fraction of the type considered in § 221 actually represents a number in the sense of § 74. We have seen that the successive odd convergents continually increase, while the even convergents continually decrease, until the difference between a pair of them is very small. Such sequences of numbers we have seen (§ 73 ff.) define real numbers.

224. Limit of error. We are now in a position to state a maximum value for the error made in taking any convergent of a continued fraction for the fraction itself.

THEOREM. The maximum limit of error in taking the nth 1 convergent for the continued fraction is less than

In In+1

Since by the theorem of the last section the value of the fraction is between any pair of consecutive convergents, it must differ from either of these convergents by less than they differ from each other, that is, by less than

1

In In + 1