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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

Online: Web Material on Additional Topics



Background and Topics

This area contains materials relating to advanced mental calculation that either do not appear in the Dead Reckoning 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!


No man but a blockhead ever wrote, except for money.  -- Samuel Johnson


Bogdanov-Belsky, 1895

Online Material

Links to Papers Found on Webpages for Individual Chapters

My online papers on mental calculation that relate to topics covered in the book are found on the webpages for the corresponding chapters.  For convenience, though, here is a set of links to those papers:

Notes and Errata for Entire Book gathered from all chapters
NEW! Developing a "Number Sense" Through Mental Multiplication from Chapter 2
Yet Another Square Root Example from Chapter 3
An Alternate Derivation of the General Square Root Algorithm from Chapter 3
The Practical Use of the Bemer Method for Exponentials from Chapter 4
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions from Chapter 4 and Chapter 5


Other Papers and Software

Mental Square Root Strategies: Joe Maruszewski has done further investigation of the general square root algorithm described in the book and in papers here. To prevent the intermediate remainders from increasing in magnitude in successive iterations, these earlier works suggested adjusting the quotient so that the remainders lie within +\-50 when using 2-digit groups. To my surprise, Joe has shown in tests he has run that simply keeping the remainder positive, even using single-digit groups, is extremely effective in keeping the remainders from spiraling upward. In fact, when calculating single-digit a_i values such that the remainder is positive in each step, you can find the the square root of 7 to 26 digits before you get a two-digit value of a_i that requires backing up to adjust the previous value. Joe demonstrates this in examples using 1-digit, 2-digit and 3-digit division for several roots, and discusses three different strategies for finding values of a_i that yield positive remainders, in a paper found here.

"Lightning Calculators" Series: I have written up a three-part series on Lightning Calculators on my blog Dead Reckonings: Lost Art in the Mathematical Sciences that discusses historical and modern calculators, methods of mental calculation, and the media aspect of it all. The first part is located here. There are links to the other parts at the ends of the posts.

A General Method for Extracting Roots using (Folded) Continued Fractions: Manny Sardina has produced very nice approximation algorithms for integer and fractional roots of numbers in original research based on continued fraction representations. The derivations and methods are presented very clearly, describing in the process what continued fractions are and how they arise. The results are formulated so that they can be exploited for mental calculation. For example, the paper shows how continued fractions can be expanded to a sum of reciprocals that can be approximated mentally. Simple patterns in continued fractions for the square root, cube root and the general case of a fractional root m/n are also derived, again to facilitate a mental solution. In other sections Manny discusses manipulating the given number and selecting an initial estimate of the root to yield a calculated root of greater accuracy. This paper is a really nice piece of mathematics, just what people interested in this site would like, and I'm very happy to present it here.

Two Tough Puzzles: Prior to the publication of my book, the editor asked me if I could create two puzzles based on the book for possible use as competitions in Tough Puzzles magazine and Mensa Magazine. The recommendation by the Tough Puzzles editor was for a challenging puzzle solvable without guessing and having a multiple-answer format to allow grading of solutions. I found it very difficult to formulate such a puzzle based on a book on mental calculation, particularly because of the possibility of using a computer to do exhaustive searches for numbers matching any given criteria. However, I did come up with two puzzles (to be solved with pencil and paper) that make such a computer search impractical, at least compared to the effort in reasoning out the problem. To my knowledge these were never published, so here they are if you are interested:

The Sensational Mentalist (and its solution)
Edward's Challenge (and its solution)

Mentally Expressing a Number as a Sum of Four Squares:  The book does not treat some of the less useful types of mental calculation, including day-date calculations and so forth.  One of these topics is reducing a given integer to a sum of four squares, of which every integer can be expressed in at least one way.  A while back I responded to a query on the Yahoo Mental Calculation site about how to handle difficult cases of this problem in the 4-digit range.  I have since polished it into a paper that is available here.

Calculating Compound Interest:  Compound interest is perhaps the one topic specifically excluded from the book that may be of general use to certain people.  George Parker Bidder, a famous lightning calculator, described his method for this in a talk titled "On Mental Calculation" given in 1856, which can be found here.  The mathematical basis of Bidder's method (as well as the text of this part of the talk) is found as Chapter 17 of Steven B. Smith's book, The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present," New York, Columbia University Press, 1983.

(Newly Updated 12/25/08) Mnemonics for Squares and Cubes of Two-Digit Numbers:  As a tool for mental calculation, I have never been particularly fond of mnenomics, that is, methods for memorizing items by word association or letter patterns.  Perhaps I would be if I were a performer called upon regularly to recall numbers or items, but my interests lie more along the lines of calculations one might encounter in the course of daily life.  Since these can be varied and infrequent, I have a bent toward general rather than particular methods.  That said, this chapter from a 1910 book presents a refreshing and fun way to quickly call to mind the cubes of two-digit numbers (it apparently dates back a few hundred years).   I have modernized and extended their scheme to provide the squares as well as the cubes of two-digit numbers, as these are so important for mental calculation, and I wrote it all up in a paper to be found here. The 12/25/08 update is due to an error in the mnemonic for 19^3 discovered by Al Stanger.

(Actually, I also like the memory technique for Morse Code given here.)

Software Program for Finding Convenient Fractions Approximating Decimal Numbers:  I wrote this useful, interactive program to generate rational approximations to entered decimal numbers (e.g., pi=3.14159265... can be approximated by 22/7 to 3 digits, 355/113 to 7 digits, etc).  These can be used as convenient fractions for calculations involving these constants.  Often these are conversion factors--for converting radians to degrees, the value 180/pi=57.2958... can be represented by 401/7=57.2857 to nearly 4 digits.  The efficient algorithm used here is from the Appendix of the book.   Please consult the README file for simple instructions on its use.  The zip file includes the Tcl/TK program for Macs and Unix machines, as well as a stand-alone binary executable file for Windows.

Click the image to the right for a full-size screenshot.  Note that the 355/113 approximation for pi lies below the visible area of the scrollbox, as the results are sorted by decreasing error. Click for full-size screenshot

External Links of Exceptional Note

I usually keep external, related links in the sidebar area, but  I have to give a special thanks to Jimmy Ruska for creating a really great, free Speed Math Trainer program found here.

John McIntosh has created an alternate approach for calculating exponentials (the McIntosh-Doerfler algorithm) that is based on the Bemer method but significantly minimizes the memorization required--log 2, log 3 and a conversion factor.  His essay, titled Exponentials, is referenced within my papers here that deal with calculating this function, but it also deserves a separate listing.  It can be found here.

Acknowledgements of Contributors to this Page

Much thanks goes to Oleg Stepanov, who has made available historical articles on mental calculation and lightning calculators, a good number of them in English.

Manny Sardina has made available on this site his new paper on methods of approximating integer and fractional roots based on continued fraction representations reformulated for fast mental calculation, as described above.

Joe Maruszewski has contributed his paper on strategies for mental square roots that can significantly simplify the effort involved in the general method for square roots given in my book.

Al Stanger discovered an error in the paper on mnemonics for squares and cubes that has now been fixed. Thanks, Al!

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