CONTINUED FRACTIONS, PELL’S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in. Continued fractions have been studied for over two thousand years, with one of the first recorded studies being that of Euclid around 300 BC (in his book Elements.

Continued fractions Next: Up: Previous: Continued fractions The history of can be traced back to an algorithm of Euclid. Let us recall this algorithm. Suppose we would like to find the greatest common divisor of numbers 75 and 33.

The last non-zero remainder, 3 in our case, is the greatest common divisor of 75 and 33. There is no evidence though that Greeks knew about the connection between the left column and the right column above. The first continued fraction was used in 1572 by Bombelli to approximate. The first infinite continued fraction appears in 1659 in the work of Lord Brouncker to expand.

Soul Position 8 Million Stories Zip on this page. It is systematic development of the theory starting in 1737 that showed the value of the notion for both number theory and analysis. A torrent of results followed.

In 18th and 19th centuries everybody who was anybody in mathematics contributed. If the number is rational the continued fraction terminates like for. If the number is irrational the continued fraction goes on forever. For example, for the irrational number we can execute the Euclidean algorithm, in essence looking for the greatest common divisor of and. The algorithm will never terminate since the two numbers are incommensurate. Concluding The esthetic beauty of continued fractions may go some ways towards justifying the significance of some numbers from algebra or geometry.

The continued fraction expansion would suggest that the number has some significance. In fact, this number is none other than the If we terminate the infinite continued fraction for the irrational number at the th step we will obtain a rational approximation to. The rational number is called the th convergent for. For example, the first 4 convergents to numbers and are The name convergent comes from the fact that convergents do converge to the number. For example, Here is the graph for. We see that convergents alternately lie above and below the exact value of. Here is the graph for.

We see the same alternating pattern of approximation. In fact, this is true in general for any number. The speed of convergence of continued fractions to a number they represent varies from number to number (but it is always very very fast). Here is a comparison between the convergence errors for (blue) and (red). The continued fraction expansions have many remarkable properties. We will be interested mainly in its approximating power relevant for the design of a good calendar system. It turns out that the convergents for the irrational number have superior approximating properties.

The following definition makes it precise what we mean by a good approximation. Dennison Case Company Serial Numbers. Definition 1 The fraction is called a good approximation for if for any and any integer we have. The good approximations occur when q=2, 5, 12 and 29. Bonsai & Suiseki Magazine Pdf. The next good approximation occurs when q=70.

The good approximations occur at q=7, 106 and 113. The next good approximation does not occur before q=33,102. Observe that the numbers are exactly the denominators in the convergents for and respectively. This is not an accident and holds in general for all convergents and for all numbers. We state it precisely and unambiguously in the form of a Theorem. T HEOREM 1 Every convergent is a good approximation (in the sense of Definition 1) for and conversely, every good approximation to is one of the numbers for some.