509. The properties of continued fractions enable us to find two small integers whose ratio closely approximates to that of two incommensurable quantities, or to that of two quantities whose exact ratio can only be expressed by large integers.
Ex. Find a series of fractions approximating to 3.14159.
In the process of finding the greatest common measure of 14159 and 100000, the successive quotients are 7, 15, 1, 25, 1, 7, 4. Thus
The successive convergents are
This last convergent which precedes the large quotient 25 is a very hear approximation, the error being less than and therefore less than , or .000004.
510. Any convergent is nearer to the continued fraction than any other fraction whose denominator is less than that of the convergent.
Let x be the continued fraction, two consecutive convergents, a fraction whose denominator s is less than q_n.
If possible, let be nearer to x than {p_n}{q_n}, then must be nearer to x than [Art. 506]; and since x lies between and it follows that must lie between and .
Hence
that, is
that is, an integer less than a fraction; which is impossible. Therefore must be nearer to the continued fraction than.
