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A Reviewon the Theoryofcontinued Fractions

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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 4 Ser. III (Jul.–Aug. 2020), PP 21-32 www.iosrjournals.org A Reviewon the TheoryofContinued Fractions M.Narayan Murty1 andPadala Navina2 1(Retired Reader in Physics, Flat No.503, SR Elegance Apartment, Gazetted officers’ colony, Isukhathota, Visakhapatnam-530022, Andhra Pradesh, India.) 2(Mathematics Teacher, Sister Nivedita School, Ameerpet, Hyderabad – 500016, Telangana, India) Abstract: In this article, the theory of continued fractions is presented. There aretwo types of continued fraction, one is the finite continued fraction and the other is the infinite continued fraction. A rational number can be expressed as a finite continued fraction. The value of an infinite continued fraction is an irrational number. The ratio of two successive Fibonacci numbers, which is a rational number, can be written as a simple finite continued fraction. The golden ratio can be expressed as an infinite continued fraction. The concept of golden ratio finds application in architecture. Using the convergents of finite continued fraction, the relation between Fibonacci numbers can be calculated and Linear Diophantine equations will be solved. Keywords:Continued fraction, Convergent, Fibonacci numbers, Golden ratio. ----------------------------------------------------------------------------------------------------------------------------- ---------- Date of Submission: 17-07-2020 Date of Acceptance: 01-08-2020 ----------------------------------------------------------------------------------------------------------------------------- ---------- I. Introduction The credit for introducing continued fraction goes toFibonacci. The nickname of famous Italian mathematician Leonardo Pisano(1170-1250) is Fibonacci. In his book LiberAbaci, while dealing with the resolution of fractions 1 1 1 into unit fractions, Fibonacci introduced a kind of “continued function”. For example, he used the symbol 3 4 5 as an abbreviation for 1 1+5 1+ 1 1 1 4 = + + (1) 3 3 3.4 3.4.5 Continued fraction is two types, (a) Finite continued fraction and (b) Infinite continued fraction.A fraction of the form given below is known as finite continued fraction. 1 푎0 + 1 (2) 푎1+ 1 푎2+ 1 푎3+ ⋱ 1 1 푎 + 푛−1 푎푛 Where, the numbers 푎1, 푎2, … , 푎푛 are the partial denominators of the finite continued fraction and they all are real numbers. The number 푎0 may be zero or positive or negative. This fraction is denoted by the symbol 푎0; 푎1, 푎2, … . , 푎푛 and it is called simple if all of the 푎 are integers. The value of any finite simple continued fraction will always be a rational number. 1 If 푎 > 1 in the finite continued fraction (2), then 푎 = 푎 − 1 + 1 = 푎 − 1 + , where 푎 − 1 is a 푛 푛 푛 푛 1 푛 positive integer. Hence, every rational number has two representations 푎0; 푎1, 푎2, … . , 푎푛 and 푎0; 푎1, 푎2, … . , 푎푛−1, 1 as a simple finite continued fraction if 푎푛 > 1. If 푠 = 푎0; 푎1, 푎2, … , 푎푛 , where 푠 > 1, then 1 = 0; 푎 , 푎 , 푎 , … , 푎 (3) 푠 0 1 2 푛 Although due credit is given to Fibonacci for introducing continued fractions, most authorities agree that the theory of continued fractions began with Rafael Bombelli, the great algebraist of Italy. In his book L’Algebra Opera (1572), Bombelliattempted to find the value of square roots of integers by means of infinite continued fractions.He expressed 13 as an infinite continued fraction. 4 13 = 3 + 4 (4) 6+ 4 6+6+ ⋱ In general, an infinite continued fraction is written as 푏1 푎0 + 푏2 (5) 푎1+ 푏 푎 + 3 2 푎3+ ⋱ DOI: 10.9790/5728-1604032132 www.iosrjournals.org 21 | Page A Reviewon The Theory of Continued Fractions Where 푎0, 푎1, 푎2, …. and 푏1, 푏2, 푏3, …. are real numbers. The expansion of an infinite continued fraction continues for ever. The infinite continued fraction in which there is 1 in all the numerators is called simple infinite continued fraction. Putting 푏1 = 푏2 = 푏3 = ⋯ = 1 in (5), the simple infinite continued fraction is written as 1 푎0 + 1 (6) 푎1+ 1 푎2+ 1 푎 + 3 푎4+ ⋱ The notation 푎0; 푎1, 푎2, … indicates a simple infinite continued fraction. The value of any infinite continued fraction is an irrational number. II. Finite Continued Fractions Theorem2.1: Any rational number can be written as a simple finite continued fraction. Proof: Let 푎 푏, where 푏 > 0, is an arbitrary rational number. Now, let us write the following equations. 푎 = 푏푎0 + 푟1 0 < 푟1 < 푏 푏 = 푟1푎1 + 푟2 0 < 푟2 < 푟1 푟1 = 푟2푎2 + 푟3 0 < 푟3 < 푟2 (7) ....................... ................... 푟푛−2 = 푟푛−1푎푛−1 + 푟푛 0 < 푟푛 < 푟푛−1 푟푛−1 = 푟푛 푎푛 + 0 The above equations are rewritten as follows. 푎 푟 1 = 푎 + 1 = 푎 + 푏 0 푏 0 푏 푟1 푏 푟2 1 = 푎1 + = 푎1 + 푟1 푟1 푟1 푟2 푟1 푟3 1 = 푎2 + = 푎2 + 푟2 (8) 푟2 푟2 푟3 .................................. 푟푛−1 = 푎푛 푟푛 Eliminating 푏 푟1 in the above first equation using the second equation, we get 푎 1 = 푎 + (9) 푏 0 1 푎1+푟1 푟2 Eliminating 푟1 푟2 in (9) using third equation of (8), we obtain 푎 1 = 푎 + 푏 0 1 푎1 + 푟2 푎2+ 푟3 Continuing in this way we get the following expression. 푎 1 = 푎 + 푏 0 1 푎1 + 1 푎2+ 1 푎3+ ⋱ 1 1 푎 + 푛−1 푎푛 Thus, the rational number 푎 푏is expressed as a simple finite continued fraction. Hence, the Theorem2.1 is 71 proved. As an example, let us apply this Theorem 2.1 to the rational number . 55 71 16 71 = 1 × 55 + 16 = 1 + 55 55 55 7 55 = 3 × 16 + 7 = 3 + 16 16 16 2 16 = 2 × 7 + 2 = 2 + 7 7 7 1 7 = 2 × 3 + 1 = 3 + 2 2 From the above equations, we obtain 71 1 1 1 = 1 + = 1 + = 1 + 55 55 7 1 3 + 3 + 2 16 16 2+ 7 71 1 ⇒ = 1 + 1 (10) 55 3+ 1 2+ 1 3+2 DOI: 10.9790/5728-1604032132 www.iosrjournals.org 22 | Page A Reviewon The Theory of Continued Fractions This is the simple finite continued fraction of the rational number 71/55. Since, in general, the finite continued fraction (2) is denoted by the symbol 푎0; 푎1, 푎2, … . , 푎푛 , the above continued fraction is denoted by the symbol 1; 3,2,3,2 . As, 2 =1 + 1/1, this continued fraction can also be denoted by the symbol 1; 3,2,3,1,1 . That is, 71 = 1; 3,2,3,2 = 1; 3,2,3,1,1 55 Now, using (10), let us represent 55/71 as continued fraction. 55 1 1 = 71 = 1 (11) 71 1+ 55 1 3+ 1 2+ 1 3+2 Hence, the rational number 55/71 is represented as [0;1,3,2,3,2] or [0;1,3,2,3,1,1].The examples (10) & (11) 19 172 710 15 prove the statement (3).Similarly, the rational numbers , , − and − are denoted as given below. 51 51 457 23 19 = 0; 2,1,2,6 = 0; 2,1,2,5,1 51 172 = 3; 2,1,2,6 = 3; 2,1,2,5,1 51 710 − = −2; 2,4,6,8 = −2; 2,4,6,7,1 457 15 − = −1; 2,1,6,1 23 The sequence of numbers introduced by Italian mathematician Leonardo Pisano is known as Fibonacci sequence, given by 1,1,2,3,5,8,13,21,34,55,89,144,233, ......... (12) Each term in the sequence after the first two is the sum of the immediately preceding two terms. The 푛푡ℎ term, 푡ℎ denoted by 퐹푛 , is called 푛 Fibonacci number. Note that 퐹3 = 퐹2 + 퐹1 = 1 + 1 = 2 and 퐹6 = 퐹5 + 퐹4 = 5 + 3 = 8 In general, we can write 퐹푛 = 퐹푛−1 + 퐹푛−2, (푛 ≥ 3)(13) Let us write the following equations for Fibonacci numbers using (13). 퐹푛+1 퐹푛−1 퐹푛+1 = 1 × 퐹푛 + 퐹푛−1 = 1 + 퐹푛 퐹푛 퐹푛 퐹푛−2 퐹푛 = 1 × 퐹푛−1 + 퐹푛−2 = 1 + 퐹푛−1 퐹푛−1 ................................ ......................... (14) 퐹4 퐹2 퐹4 = 1 × 퐹3 + 퐹2 = 1 + 퐹3 퐹3 퐹3 퐹3 = 2 × 퐹2 + 0 = 2 퐹2 Using the above equations, we obtain 퐹 1 1 푛+1 = 1 + = 1 + 퐹푛 1 퐹푛 1 + 퐹푛−1 퐹푛−1 퐹푛−2 Continuing in this manner, one will get 퐹푛 +1 1 1 = 1 + 퐹푛 = 1 + 1 (15) 퐹푛 1+ 퐹푛−1 퐹푛−1 퐹푛−2 ⋱ 1 1+2 퐹 ⇒ 푛 +1 = 1; 1,1, … ,1,2 = 1; 1,1, … ,1,1,1 (16) 퐹푛 The above expression (16) shows that the ratio of two successive Fibonacci numbers 퐹푛+1 퐹푛 , which is a rational number, can be written as a simple finite continued fraction. III. Convergentsof Finite Continued Fraction 푡ℎ Definition: The continued fraction made from 푎0; 푎1, 푎2, … . , 푎푛 by cutting off the expansion after the 푘 푡ℎ partial denominator 푎푘 is called the 푘 convergent of the given continued fraction and denoted by 퐶푘 . 퐶푘 = 푎0; 푎1, 푎2, … , 푎푘 (17) 퐶0 = 푎0(18) The convergent 퐶0 is called the zeroth convergent. Going back to the example (10), let us write the successive 71 convergents of = 1; 3,2,3,2 . 55 DOI: 10.9790/5728-1604032132 www.iosrjournals.org 23 | Page A Reviewon The Theory of Continued Fractions 퐶0 = 1 1 4 퐶 = 1; 3 = 1 + = = 1.3333. .. 1 3 3 1 9 퐶2 = 1; 3,2 = 1 + 1 = = 1.2857 …(19) 3+ 7 2 1 31 퐶 = 1; 3,2,3 = 1 + = = 1.2916. .. 3 1 24 3 + 1 2+ 3 71 퐶 = 1; 3,2,3,2 = = 1.2909 … 4 55 The above values of convergents show that, except for the last convergent 퐶4, these are alternately less than or greater than 71/55, each convergent being closer in value to 71/55 than the previous one.
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  • Proportions: a Ratio Is the Quotient of Two Numbers. for Example, 2 3 Is A

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    Proportions: 2 A ratio is the quotient of two numbers. For example, is a ratio of 2 and 3. 3 An equality of two ratios is a proportion. For example, 3 15 = is a proportion. 7 45 If two sets of numbers (none of which is 0) are in a proportion, then we say that the numbers are proportional. a c Example, if = , then a and b are proportional to c and d. b d a c Remember from algebra that, if = , then ad = bc. b d The following proportions are equivalent: a c = b d a b = c d b d = a c c d = a b Theorem: Three or more parallel lines divide trasversals into proportion. l a c m b d n In the picture above, lines l, m, and n are parallel to each other. This means that they divide the two transversals into proportion. In other words, a c = b d a b = c d b d = a c c d = a b Theorem: In any triangle, a line that is paralle to one of the sides divides the other two sides in a proportion. E a c F G b d A B In picture above, FG is parallel to AB, so it divides the other two sides of 4AEB, EA and EB into proportions. In particular, a c = b d b d = a c a b = c d c d = a b Theorem: The bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent side.