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Continued Fraction Expansion of a Real Number





            The first steps in learning how to write out a continued fraction is to familiarize oneself with the process of performing certain processes on a rational numbers. Lets take the above fraction for example. By starting off with 790/479, we want to try to rewrite it in a continued fraction form. One of the processes that must be learned is the Euclidean Algorithm, which is introduced on a separate page located on the sidebar. The steps in the Euclidean Algorithm are used to find the greatest common denominator of two numbers as shown on the top left, but at the same time, we can use it to convert this rational number, 790/479, into a continued fraction. First we start off by dividing 790 by 479, and we find that 479 can only go into 790 one time. This process is shown with the blue coated one. After that process is completed, we add on the remainder which is 311, and now we have a complete equation that represents 790 in a N = Quotient + Remainder form. Now in the second step, we are left with 479. All we have to do is repeat the same steps as the previous one by representing 479 in N = Quotient + Remainder form. 311 once again only goes into 479 one time, so we write that out. The remainder left behind is 168 = 479 - 311. When repeating these steps over and over again by bringing down the quotients and remainders, we will eventually reach a zero as shown above. This indicates that the algorithm has been completed. The use of the Euclidean Algorithm is to find the greatest common denominator of the two numbers 790 and 479, and the number above the 0 is the greatest common denominator. Instead of using the algorithm that way, we take the blue coated numbers, and we can use those to convert 790/479 into a continued fraction.To provide a clearer picture on how the blue numbers above are used, we can take the general expression of a continued fraction that was previously shown on the home page:
 


The ao, a1, a2 ,a3, a4, ...  , an can now be replaced by the blue numbers [1, 1, 1, 1, 5, 1, 2, 1, 1, 3]. This algorithm can be used to convert any rational number to a continued fraction.


            For this next example, it will demonstrate the reverse process of turning an infinite continued fraction into an irrational number. The reason we must learn this process is so that we can later apply continued fractions to the discovery of the golden ratio (φ), a phenomenon that is still being studied for its unique and interesting properties.
                                         
                                     Let

                                    We can say that,
                                    where


                                    We can further simplify this expression to



                                    because y is self-similar, meaning that the expression y constantly repeats itself, so we can just write                              the rest of the fraction as y.

                             By performing basic algebra operations, the expression above can be simplified to



                                    
                                 and leads to the quadratic equation
                                   The next step is to solve the equation with the quadratic formula, where the discriminant will equal to                             60. y > 2 as shown two previous steps prior to the quadratic equation above. That means that one of                             the two roots of the quadratic equation cannot be y, therefore,



                                    Substitute y back into the equation x = 1 + 1/y. Finally,


                                    Now onto the Golden Ratioclick here

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