2) Finite Continued Fractions

Question 1

What is a finite continued fraction

Answer 1

Question 2

What is a simple finite continued fraction

Answer 2

A finite continued fraction is simple if all its elements are integers.

Question 3

How are the sequences (pk) and (qk) defined in terms of a sequence (xk) in continued fractions

Answer 3

Question 4

How can a continued fraction of the form [x0:x1,x2,…,xn,w] be expressed in terms of the sequences (pk) and (qk)

Answer 4

Question 5

How can a finite continued fraction [x0 :x1,x2,…,xn] be expressed in terms of the sequences (pn) and (qn)

Answer 5

Question 6

What key properties hold for the sequences (pk) and (qk) in continued fractions

Answer 6

Question 7

Describe the proof of the key properties of the sequences (pk) and (qk)

Answer 7

Question 8

For a finite simple continued fraction [x0:x1,…,xn] with associated sequences (pk) and (qk), what is gcd(pk ,qk ) for each k≥0

Answer 8

For each k ≥ 0, we have gcd(pk, qk) = 1

Question 9

Describe the proof that gcd (pk, qk) = 1for each k ≥0 in a finite simple continued fraction

Answer 9

Question 10

If the Continued Fraction Algorithm terminates what number does it represent

Answer 10

The Continued Fraction Algorithm terminates for every rational number