Continued fractions
Properties and invariant measure
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An irrational number can be represented in many ways. A less known representation is the one by continued fractions. Continued fractions give the best approximation of irrational numbers by rational numbers. Through the centuries many variants of continued fractions were used and invented. For example the Greeks used Euclid's divisor algorithm and Christiaan Huygens used continued fractions to construct his famous planetarium. The modern history started with Gauss who found the invariant measure of the regular continued fraction. After this many others, like famous probabilists Paul Lévy and ...
An irrational number can be represented in many ways. A less known representation is the one by continued fractions. Continued fractions give the best approximation of irrational numbers by rational numbers. Through the centuries many variants of continued fractions were used and invented. For example the Greeks used Euclid's divisor algorithm and Christiaan Huygens used continued fractions to construct his famous planetarium. The modern history started with Gauss who found the invariant measure of the regular continued fraction. After this many others, like famous probabilists Paul Lévy and Wolfgang Doeblin, contributed to what we now call the "metric theory of continued fractions". In this book we will take a look at some forms of continued fractions. Also, we will introduce a new continued fraction and study its properties.
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