Continued Fractions Algorithm (cont.) i Xi a; = = int(x;) €¡ = X; — a; 0 €¡-1 3.1415 3 0.1415 1 7.067137809 7 0.067137809 2 14.89473684 14 0.894736842 3 1.117647059 1 0.117647059 4 8.5 8 0.5000000002 5 1.99999999926 1 0.99999999926 6 1.000000001 1 7.44052E-10 Hence, the simple continued fraction representation of x = 3.1415 is [3, 7, 14, 1, 8, 1, 1]. Continued Fractions Algorithm • So far we discussed CF representations of rational numbers. Is there a way to find the CF representation of irrational numbers? The CF representation of an irrational number will have to be an infinite expression. • We will demonstrate an algorithm that is similar to the Euclidean Algorithm, but this algorithm also works for any real number™ Let x Є R. Step 0: x0 = x Step 1: Let ao = int(xo) Step 2: 0x0 - ao Note that each error term is the (m+1)th complete quotient: 1 =am + am+1 So a+ is then m If € 0, then let x₁ = 1 and go back to Step 1 to compute am+1 If o 0, then stop

Continued Fractions Algorithm (cont.) i Xi a; = = int(x;) €¡ = X; — a; 0 €¡-1 3.1415 3 0.1415 1 7.067137809 7 0.067137809 2 14.89473684 14 0.894736842 3 1.117647059 1 0.117647059 4 8.5 8 0.5000000002 5 1.99999999926 1 0.99999999926 6 1.000000001 1 7.44052E-10 Hence, the simple continued fraction representation of x = 3.1415 is [3, 7, 14, 1, 8, 1, 1]. Continued Fractions Algorithm • So far we discussed CF representations of rational numbers. Is there a way to find the CF representation of irrational numbers? The CF representation of an irrational number will have to be an infinite expression. • We will demonstrate an algorithm that is similar to the Euclidean Algorithm, but this algorithm also works for any real number™ Let x Є R. Step 0: x0 = x Step 1: Let ao = int(xo) Step 2: 0x0 - ao Note that each error term is the (m+1)th complete quotient: 1 =am + am+1 So a+ is then m If € 0, then let x₁ = 1 and go back to Step 1 to compute am+1 If o 0, then stop

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 13E

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