J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 17, 2011, Number 4, Pages 61–68

**Full paper (PDF, 119 Kb)**

## Details

### Authors and affiliations

J. V. Leyendekkers

*Faculty of Science, The University of Sydney
Sydney, NSW 2006, Australia
*

A. G. Shannon

*Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia
*

### Abstract

Classes of the modular ring Z_{4} were substituted into convergent infinite series for *π* and √2 to obtain *Q*, the ratio of the arc of a circle to the side of an inscribed square to yield *π* = 2√2 *Q*. The corresponding convergents of the continued fractions for π, √2 and *Q* were then considered, together with the class patterns of the modular rings {Z_{4}, Z_{5}, Z_{6}} and decimal patterns for *π*.

### Keywords

- Integer structure analysis
- Modular rings
- Prime numbers
- Fibonacci numbers
- Arctangents
- Infinite series
- Pell sequence
- Continued fractions
- Triangular numbers

### AMS Classification

- 11A41
- 11A55
- 11A07

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## Cite this paper

Leyendekkers, J., & Shannon, A. (2011). The structure of ‘Pi’. *Notes on Number Theory and Discrete Mathematics*, 17(4), 61-68.