ALGEBRA PRACTICE WITH CLIL Continued fraction A «continued fraction is an expression of the form: b1 a 1 + ___________ b2 a 2 + _______ a3 + . . . So, if b1 = b2 = ...= bn = 1 such expression is called «arithmetical continued fraction : 1 a 1 + ___________ 1 a 2 + _______ 1 a 3 + ___ ... In addition to their peculiarity, it is interesting to highlight that each terminating arithmetical continued fraction represents a rational number and vice versa. m r Truly, a rational number can be represented in the form __ = q + __ where q is the quotient and r is the remainder. n n Thus, _m_ = q + _r_ = q + _1_. n n _n_ r n It can be developed as a continued fraction by repeating the process on the __ fraction until the remainder will r be 0. Here is an example to help you gain a better understanding of the procedure. 67 Imagine we want to develop the rational number ___ in the form of an arithmetical continued fraction. 29 1 1 1 1 1 9 1 67 ___ = 2 + ___ = 2 + ___ = 2 + _____ = 2 + _____ = 2 + _________ = 2 + _________ = 2 + _________ 29 2 1 1 1 1 29 29 ___ 3 + __ 3 + __ 3 + _____ 3 + _____ 3 + _____ 1_ 1_ 1 9 9 _9_ _ _ 4+ 4+ 4 + __ 2 2 2 2 __ 1 Hence: 67 1 ___ = 2 + _________. 1 29 3 + _____ 1 4 + __ 2 67 Sometimes, it can be also written such as ___ = (2; 3; 4; 2) where the sequence of the first addends of the 29 subsequent fractions is indicated in the parentheses. Now, try to write some examples of rational numbers in the form of an arithmetical continued fraction and vice versa. Exercises [ ] 32 1. ___ 4. (44; 2; 1; 1; 1; 3) 271 2. ____ 5. (2; 1; 1; 2) 152 3. ____ 6. (1; 2; 3; 4 ; 5) 7 16 215 86