chapter 3 - Bentham Science

64 Applications of Spreadsheets in Education The Amazing Power of a Simple Tool, 2011, 64–83
Spreadsheet Conditional Formatting Illuminates
Investigations into Modular Arithmetic
David Miller 1 and Stephen Sugden 2,∗
1 Department of Mathematics
West Virginia University, Morgantown, West Virginia, USA
2 Faculty of Business
Bond University, Queensland, Australia
CHAPTER 4
∗ Address correspondence to: Dr. Stephen Sugden, Faculty of Business, Bond University, QLD 4229, Aus-
tralia; Tel: (+61) 7-5595-3325; E-mail: ssugden@bond.edu.au
Abstract: Modular arithmetic has often been regarded as something of a mathematical
curiosity, at least by those unfamiliar with its importance to both abstract algebra
and number theory, and with its numerous applications. However, with the ubiquity
of fast digital computers, and the need for reliable digital security systems such as
RSA, this important branch of mathematics is now considered essential knowledge
for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular.
This chapter describes how the modern graphical spreadsheet may be used to clearly
illustrate the basics of modular arithmetic, and to solve certain classes of problems.
Students may then gain structural insight and the foundations laid for applications to
such areas as hashing, random number generation, and public-key cryptography.
Keywords: modular arithmetic, conditional formatting.
4.1 Introduction
Sometimes described as “clock-arithmetic,” at least at high-school level, modular arithmetic was
sometimes presented there as bit of a curiosity, with little practical application. Perhaps with the
advent and dominance of digital timepieces, the “clock-arithmetic” terminology has faded, however
this field of mathematical study is now more important than ever. Pioneered by Euler, and greatly
advanced by Gauss in his Disquisitiones Arithmeticae in 1801 [1], modular arithmetic was indeed
regarded as a curiosity at that time. Given that integer arithmetic on a digital computer is inherently
modular (see Section 4.2), and that computers permeate our daily lives, it seems somewhat strange
that our pre-tertiary students learn essentially nothing of the very important topic of modular arithmetic.
More than this, a working knowledge of modular arithmetic is essential for the mathematical
study of modern cryptographic systems such as RSA. We believe that the time for this material to
Mark Lau and Stephen Sugden (Eds)
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