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Overview Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Appendix Online Material

Dead Reckoning: Calculating Without Instruments

Link to book on author has gone as far as Doerfler in covering methods of mental calculation beyond simple arithmetic. Even if you have no interest in competing with computers you'll learn a great deal about number theory and the art of efficient computer programming.

Martin Gardner (renowned author of the Mathematical Games column in Scientific American for 25 years, as well as many books on mathematics)

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This section provides readers with resource material related to my book, Dead Reckoning: Calculating Without Instruments, published in 1993 by Gulf Publishing Company.

This book describes techniques of computation and approximation that can be used to rapidly and mentally calculate mathematical quantities. It's an advanced book on mental calculation, aimed at those of us who, through an interest in mathematics, have acquired a general appreciation for numbers (i.e., a number sense). Selected methods of past and present lightning calculators are combined with number theory, early computer algorithms, and relatively modern techniques of numerical analysis to provide a unique set of tools for rapidly and mentally finding products, quotients, roots of non-perfect powers, trigonometric values, and logarithmic and exponential values. The idea is to stretch our capabilities for personal satisfaction and recreation, while experiencing a the rich field of elementary mathematics that mankind has had thousands of years to develop, but which is disappearing from our lives and educational systems at an alarming rate.

Click on chapter titles below to access webpages providing background, book errata, additional examples and exercises, and the cited reference materials relating to that topic.

Chapter Links:

Chapter 1: A Time for Reckoning

This chapter provides an introduction to the motivation and purpose of the book, as well as an overview of its contents.

Chapter 2: Primitives

This chapter describes techniques related to multiplication, division and repeating decimals, greatest common divisor (GCD), error-checking and divisibility tests, and factoring numbers.

Chapter 3: Roots

This chapter provides a brief description of methods for finding the roots of perfect powers, then delves into methods for mentally approximating square roots, a general method for mentally extracting square roots to any number of digits that is more powerful than the traditional method, an iterative technique for reciprocal square roots, and extensions of these methods to cube and higher-order roots.

Chapter 4: Logarithms and Their Inverses

After a brief review of logarithms, this chapter discusses general approximation techniques for common and natural logarithms that are suitable for mental calculation, then details more accurate relations involving neighboring and intermediate values, offers methods for the more difficult problem of finding inverse logarithms, and concludes with an example of using logarithms and inverses to find higher-order roots of numbers.

Chapter 5: Trigonometric Functions and Their Inverses

This chapter provides convenient 3- to 4-digit approximations to the sine, cosine, tangent, arcsine, arccosine and arctangent functions in units of degrees or radians. Graphs provide error curves for each method as a means of evaluating them.

Chapter 6: Concluding Remarks

The main text of the book ends with a list of the best of the cited references in the text, an encouragement to search out these references (which are now gathered on this site), and a plea for the reader to experiment with number relationships as a means of exploring new algorithms and approximations that may be useful to us all.

Appendix: Finding Rational Approximations to Precomputed Constants

Often we encounter multi-digit numbers, such as scaling factors, that are difficult to use directly as a divisor or multiplier because of the number of digits involved. This appendix describes the general procedure (using continued fractions) for generating whole-number fractions that approximate multi-digit numbers to a required accuracy and in practice involve multiplication and division by reasonably small whole numbers.

Online: Web Material on Additional Topics

This area contains materials relating to advanced mental calculation that either do not appear in the book or are covered in greater detail here.  Work by others is also hosted here with their permission--please contact me if you have an idea or write-up for this section.



Acknowledgements of Contributors to this Book Section:

Contributors to individual chapter pages are given on those pages. However, the following individuals offered advice that was incorporated in creating the structure and content of this section. Feel free to contact me with any more suggestions.

John McIntosh has provided extensive feedback on the book. His website contains many very interesting and original essays on similar areas of mathematics, of which the top mathematics level is here.  He has also written an essay summarizing things he found particularly interesting in the book, with additional interesting commentary of his own--it can be found here.

Grant Nixon published a very nice review, and followed up with very interesting correspondence on the book.

Thomas Wall and John O., in their reviews of the book, commented on the need for additional material, such as exercises and errata, that would benefit the reader.  This led in part to the creation of this webpage.

Works Citing This Book:

Link to Chapter

Ron Hale-Evans, Mind Performance Hacks: Tips & Tools for Overclocking Your Brain  - Referenced

Link to book at Amazon

David Pimm, Symbols and Meanings in School Mathematics , Routledge, London, 1995  - Book excerpts quoted in discussions of modern mathematics education

Math Power
Dell, Dick, ed., The World Book of Math Power, Vol. 2, World Book, Chicago, 1996 - Suggested book "intended for those in high school, college, or professional occupations who are intrigued by mathematics" in Sources of Mathematical Materials.
Link to Entry
Wikipedia (German Edition), Kopfrechnen (“ Mental Arithmetic ”), Web Translation: “Genuine Mental Arithmetic: Techniques are offered only rarely for the general mental arithmetic. This area normally covers all functions, which more on the average train pocket calculators control must. One of the few good works in this area is the book Dead Reckoning - Calculating Without Instruments of Ronald W. Doerfler. Unfortunately no German translation of this book exists.”
Harold Thimbleby, “Calculators are Needlessly Bad,” International Journal of Human-Computers Studies (2000)  - Referenced
School of Mathematics and Statistics, University of St Andrews, Scotland, History of Mathematics Site, “Memory, mental arithmetic and mathematics.”  - Referenced
“Percent Change: Big Lies We Are Taught in Government Schools,”  - Referenced
"Dead Reckoning," an essay by John McIntosh on his website summarizing things he found particularly interesting in the book, with additional interesting commentary of his own.
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