Math 453 Definitions and Theorems 6 6 1 4 18 2011 A J Hildebrand Continued fractions Definitions and notations Definition 6 1 Continued fractions A finite or infinite expression of the form 6 1 1 a0 1 a2 a1 where the ai are real numbers with a1 a2 0 is called a continued fraction c f The numbers ai are called the partial quotients of the c f The continued fraction 6 1 is called simple if the partial quotients ai are all integers It is called finite if it terminates i e if it is of the form 6 2 1 a0 1 a1 a2 1 an and infinite otherwise Notation Bracket notation for continued fractions The continued fractions 6 1 and 6 2 are denoted by a0 a1 a2 and a0 a1 a2 an respectively In particular a0 a0 a0 a1 a0 1 a1 a0 a1 a2 a0 1 1 a1 a2 Remarks i Note that the first term a0 is allowed to be negative or 0 but all subsequent terms ai must be positive This requirement ensures that there are no zero denominators and that any finite c f 6 2 and all of its convergents are well defined ii In the sequel we will focus on the case of simple c f s i e c f s where all partial quotients are integers 6 2 Convergence of infinite continued fractions Definition 6 2 Convergents The convergents of a finite or infinite c f a0 a1 a2 are defined as C0 a0 C1 a0 a1 C2 a0 a1 a2 If the c f is simple its convergents Ci represent rational numbers denoted by Ci pi qi where pi qi is in reduced form Definition 6 3 Convergence of infinite continued fractions An infinite c f a0 a1 a2 is called convergent if its sequence of convergents Ci a0 a1 ai converges in the usual sense i e if the limit lim Ci lim a0 a1 ai i i 21 Math 453 Definitions and Theorems 4 18 2011 A J Hildebrand exists and is a real number In this case we say that the continued fraction a0 a1 a2 represents the number or is a continued fraction expansion of and we write a0 a1 a2 Theorem 6 4 Convergence of infinite simple c f s Any infinite simple c f a0 a1 is convergent and thus represents some real number 6 3 Properties of Convergents Proposition 6 5 Formulas for pi and qi Let a0 a1 be a simple c f with convergents Ci a0 a1 ai pqii i Recursion formula The numbers pi and qi are given by the recurrence pi ai pi 1 pi 2 qi ai qi 1 qi 2 for i 1 2 along with the initial conditions p0 a0 p 1 1 q0 1 q 1 0 ii Matrix representation For i 0 1 2 a0 1 a1 1 a i 1 0 1 0 1 1 pi 0 qi pi 1 qi 1 Theorem 6 6 Properties of convergents The convergents Ci pi qi of an infinite simple continued fraction a0 a1 a2 satisfy i pi qi 1 for i 0 1 i e the fractions pi qi are reduced ii q1 q2 i e for i 1 the denominators qi are strictly increasing iii C0 C2 C4 C5 C3 C1 That is the even indexed convergents form an increasing sequence while the odd indexed convergents form a decreasing sequence with the value of the c f sandwiched between both sequences iv Ci 1 Ci v 1 i for i 0 1 2 qi qi 1 pi 1 for i 0 1 2 qi qi qi 1 vi Best approximation property For any rational number a b with a Z b N and 1 b qi a pi qi b with equality if and only if a b pi qi That is the convergent pi qi is the best possible approximation to among all rational numbers with the same or smaller denominator 22 Math 453 Definitions and Theorems 6 4 4 18 2011 A J Hildebrand Expansions of real numbers into continued fractions Proposition 6 7 Continued fraction algorithm Given a real number define successively real numbers 0 1 and integers a0 a1 by 0 a0 0 1 0 0 1 2 1 1 1 a1 1 a2 2 where x denotes the integer part of x i e the floor function Stop the algorithm if n is an integer and thus an n otherwise continue indefinitely Then a0 a1 is a simple c f that represents the number Moreover for any i 0 we have i ai ai 1 a0 a1 ai 1 i Theorem 6 8 Continued fraction expansion of rational numbers Any finite simple c f represents a rational number Conversely any rational number can be expressed as a simple finite c f a0 a1 an Moreover under the requirement that an 1 this representation is unique Thus there is a one to one correspondence between rational numbers and finite simple c f s with last partial quotient greater than 1 Theorem 6 9 Continued fraction expansion of irrational numbers Any infinite simple c f represents an irrational number Conversely any irrational number can be expressed as a simple infinite c f a0 a1 a2 and this representation is unique Thus there is a one to one correspondence between irrational numbers and infinite simple c f s Theorem 6 10 Continued fraction expansion of quadratic irrationals The c f expansion of a quadratic irrational i e a solution of a quadratic equation with integer coefficients is eventually periodic i e of the form a0 aN aN 1 aN p where the bar indicates the periodic part Conversely any infinite simple c f that is eventually periodic represents a quadratic irrational Thus there is a one to one correspondence between quadratic irrationals and infinite eventually periodic simple c f s 23