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Unformatted text preview: A SHORT INTRODUCTION TO CONTINUED FRACTIONS P. GUERZHOY Abstract. In this talk we introduce continued fractions, prove their basic properties and apply these properties to solve a practical problem. We also state without proof some further properties of continued fractions and provide a brief overview of some facts in this connection. The talk is elementary; it is aimed at undergraduate mathematics majors and mathematics graduate students. 1. Basic Notations In general, a (simple) continued fraction is an expression of the form a + 1 a 1 + 1 a 2 + . . . , where the letters a , a 1 , a 2 , . . . denote independent variables, and may be interpreted as one wants (e.g. real or complex numbers, functions, etc.). This expression has precise sense if the number of terms is finite, and may have no meaning for an infinite number of terms. In this talk we only discuss the simplest classical setting. The letters a 1 , a 2 , . . . denote positive integers. The letter a denotes an integer. The following standard notation is very convenient. Notation. We write [ a ; a 1 , a 2 , . . . , a n ] = a + 1 a 1 + 1 a 2 + . . . + 1 a n if the number of terms is finite, and [ a ; a 1 , a 2 , . . . ] = a + 1 a 1 + 1 a 2 + . . . for an infinite number of terms. Still, in the case of infinite number of terms a certain amount of work must be carried out in order to make the above formula meaningful. At the same time, for the finite number of terms the formula makes sense. Example 1. [- 2; 1 , 3 , 5] =- 2+1 / (1+1 / (3+1 / 5)) =- 2+1 / (1+5 / 16) =- 2+1 / (21 / 16) =- 2+16 / 21 =- 26 / 21 Notation. For a finite continued fraction [ a ; a 1 , a 2 , . . . , a n ] and a positive integer k ≤ n , the k-th remainder is defined as the continued fraction r k = [ a k ; a k +1 , a k +2 , . . . , a n ] . 1 2 P. GUERZHOY Similarly, for an infinite continued fraction [ a ; a 1 , a 2 , . . . ] and a positive integer k , the k-th remainder is defined as the continued fraction r k = [ a k ; a k +1 , a k +2 , . . . ] . Thus, at least in the case of a finite continued fraction, α = [ a ; a 1 , a 2 , . . . , a n ] = a + 1 / ( a 1 + 1 / ( a 2 + . . . + 1 /a n )) we have (1) α = a + 1 / ( a 1 + 1 / ( a 2 + . . . + 1 / ( a k- 1 + 1 /r k ))) = ”[ a ; a 1 , a 2 , . . . , a k- 1 , r k ]” for any positive k ≤ n . Quotation signs appear because we consider the expressions of this kind only with integer entries but the quantity r k may be a non-integer. It is not difficult to expand any rational number α into a continued fraction. Indeed, let a = [ α ] be the greatest integer not exceeding α . Thus the difference δ = α- a < 1 and, of course, δ ≥ . If δ = 0 then we are done. Otherwise put r 1 = 1 /δ , find a 1 = [ r 1 ] and non-negative δ = α 1- a 1 < 1 . Continue the procedure until you obtain δ = 0 ....
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- Spring '09
- pH, Rational number, Irrational number, Continued fraction, Qk Qk, continued fractions