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Inside Your Calculator: From Simple Programs to Significant Insights



Inside Your Calculator: From Simple Programs to Significant Insights

Gerald R. Rising

ISBN: 978-0-470-12459-8 June 2007 288 Pages


Uncover the mysteries that lie within your calculator

This remarkable book explores the simple internal calculator processes—algorithms and programs—that tell us, for example, that the cosine of 56? is 0.5591929035. Using carefully constructed diagrams and figures, the author effectively demonstrates how calculator keys compute powers, roots, logarithms, and trigonometry functions, while also providing insights into simple programming, the conversion between decimal and binary numeration, and perhaps most importantly, the structure of our numeration systems. Many people believe that the processes that drive calculators demand advanced mathematical concepts; however, this book proves that a minimal understanding of algebra and geometry is all that is needed to follow the step-by-step explanations of how scientific calculators work.

Inside Your Calculator: From Simple Programs to Significant Insights is a complete and multifaceted exercise in critical thinking. This book features:

  • A detailed explanation of how to use a graphics calculator and program basic functions

  • A discussion of the history of mathematics when appropriate, which provides a foundation for further learning

  • Fundamental mathematical lessons and interesting applications of pre-calculus mathematics

  • A thorough review of the fundamentals of programming, algebra, and geometry needed to gain insight into why the algorithms work and how the results are meaningful in our lives

While the simultaneous use of a calculator is not needed to gain insight into how the algorithms work, those who do have a programmable graphics calculator can experiment with the programs presented in the book. These programs may be used on TI-84 and TI-83 calculators, and additional information for other Texas Instruments calculators as well as the Casio FX series is available on the book's related web site.

As a result of over fifty years of award-winning teaching experience in both high school and college classrooms, Dr. Rising anticipates and answers potential questions from readers, and he successfully brings this subject alive in an illuminating and entertaining way. This book is therefore not only ideal for undergraduate mathematics majors as either a primary or supplemental text, but it also appeals to anyone with an interest in mathematics and its ideas.

View Dr. Rising's book presentation:

Preface xi


1 Introduction 3


2 Numbers, Algorithms, and Programs 19

3 Integer Powers 40

4 Square Root 51

5 Rational Powers 73

6 Logarithms 84

7 Archimedes’ Calculation of π 106

8 Calculating Trigonometric Functions 117

9 CORDIC Calculation of Cosine 138


10 Graphing 163


A A Primer on Programming 179

B Interpolation 188

C Pre–Electronic Calculation Tools 191

D Fermat’s Last Theorem 204

E An Extension and an Application of Integer Division 206

F Binary Arithmetic 211

G Binary Subtraction 222

H The Rapid Convergence of Newton’s Method 228

I How Newton’s Method Applies to the Square Root Algorithm and the Rth Root of N 230

J The Ancient Greeks Approximate √2 233

K Continued Fraction Approximations 238

L Multiplying Numbers with Many Digits 243

M Finding Equation Roots by Binary Search 249

N Derivation of the Logarithm Change of Base Formula 256

O The Ratio of Decimal to Binary Digits 257

P Constructing a Log Table 260

Q Relations between Sides of Inscribed and Circumscribed Polygons 265

R Change in Form of a Polygon Formula 273

S An Area Approach to Archimedes’ Problem 276

FURTHER READING: A Personal Selection 281


"This book would appeal to many readers who have a minimum background in mathematics and/or programming." (CHOICE, January 2008)

"People who want to know how things work will enjoy this book…" (Journal of Recreational Mathematics)

"I recommend this book for anyone who has…an interest in understanding how calculators work or in programming calculators." (Mathematics Teacher, February 2008)

"A book on algorithms for scientific pocket calculators that is written with much dedication." (Zentralblatt MATH, 2008)

Related Web Site
Watch Dr. Rising’s presentation on Inside Your Calculator
  • Written by an expert in mathematical education, this well-organized book reflects the author's forty-plus years of teaching and writing experience.
  • This book constitutes a complete and multi-faceted exercise in critical thinking. 
  • The author focuses on explanations that are based on only high school basic algebra and geometry (both of which are reviewed in context) in an effort to appeal to a broad range of readers who are attracted to mathematics and its ideas.
  • Well-conceived, illuminating, and entertaining diagrams and other figures are found throughout the book.
  • Teachers, at both the high school and college level, will especially enjoy this book since it provides insight to answer student questions such as "Why is this important?"
  • The history of mathematics is discussed when appropriate and provides a foundation for further learning.