2020 IB Extended Essays

3.2.1 Continued Fraction Notation Consider the continued fraction expansion of √ 7 and √ 13 , for example: √ 7 = 2 + 1 1 �√ 7 − 2 � = 2 + 1 √ 7 + 2 3 = 2 + 1 1 + 1 3 √ 7 − 1 = 2 + 1 1 + 1 √ 7 + 1 2 = 2 + 1 1 + 1 1 + 1 2 √ 7 − 1 = 2 + 1 1 + 1 1 + 1 √ 7 + 1 3

= 2 + 1 1 + 1 1 + 1 1 + 1 3 √ 7 − 2

= 2 + 1 1 + 1 1 + 1 1 + 1

= 2 + 1 1 + 1

1 + 1 1 + 1 4 + 1 1 √ 7 − 2

√ 7 + 2

1 1 + 1

= 2 +

= ⋯

1 + 1 1 + 1 4 + 1 1 + 1 √ 7 + 1 2

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