On Mental Calculation of Repeating Decimals, Finding Fibonacci Numbers and a Connection to Pascal’s Triangle

Marjorie Bicknell-Johnson

A chapter in Applications of Fibonacci Numbers, 1990, pp 191-195 from Springer

Abstract: Abstract There is a well-known expression for 1/89, $$\begin{gathered} {1 \mathord{\left/ {\vphantom {1 {89}}} \right. \kern-\nulldelimiterspace} {89}} = .0112358 \hfill \\ 13 \hfill \\ 21 \hfill \\ 34 \hfill \\ 55 \hfill \\ + ... \hfill \\ - - - - - - - - - - \hfill \\ = .01123595505... \hfill \\ \end{gathered} $$ where successive terms of the Fibonacci sequence F n+1 = F n + F n−1, F0 = 0, F1 = 1, appear and are summed. Of course, the decimal expansion of 1/89 can be found by long division and will have a period length of 44 digits, but there is a short-cut for obtaining those digits which leads to other entertainments.

Keywords: Mental Calculation; Fibonacci Number; Fibonacci Sequence; Successive Term; Decimal Expansion (search for similar items in EconPapers)
Date: 1990
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-1910-5_21

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DOI: 10.1007/978-94-009-1910-5_21

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