# MyReckonings.com

## Dead Reckoning: Calculating Without Instruments

Chapter 3: Roots

Background and Topics

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.

Notes and Errata (If you have any more to contribute, please email me! This is a compendium of feedback)  A printer-friendly summary table for all chapters is found here.

 Color Code Type Meaning Note A clarification or elaboration of the text. Typo A simple mistake that does not affect the method presented. Error An error that may affect a method or the reader interpretation of it.

Additional Materials Related to Topics in This Chapter

Square Root With Low Number Groups:  A paper demonstrating the general square root algorithm is given here for N=51, but unlike the book all the values of b are kept less than or equal to 50, which is recommended in the chapter in order to ease the multiplications and to limit the ripple effects on neighboring number groups.

Another Square Root Example:  A paper providing another, more detailed example of the general square root algorithm is given here.

Yet Another Square Root Example:  A paper providing yet another detailed example of the general square root algorithm is given here.

An Alternate Derivation of the General Square Root Algorithm:  The derivation given in the book of the general square root algorithm is not very intuitive and is rather complicated.  Here is an alternate derivation that provides a better visual feel for why the algorithm works as it does.  The extension of this algorithm to cube roots is also discussed.

Cited Reference Materials

A.C. Aitken's fascinating talk given in 1954, The Art of Mental Calculation; With Demonstrations, which is discussed in the first part of this chapter, can be found here.  I remember waiting for weeks in the late 1980's to locate the journal and get a copy of it (a University of Minnesota librarian graciously photocopied the article and mailed it to me).

Lehmer's 1923 article in which he describes a variation of the general square root algorithm for use with mechanical calculators can be found here.  This is not an easy read, and it took me awhile to verify that it is indeed related to the method presented in the book.

In the near future, this section will contain additional relevant sections of some of the references cited at the end of this chapter.