The Book of Numerals

Of all the sketchbooks a perhaps-former artist could ever keep, the Book of Numerals has enjoyed the deepest impact in my life. Perhaps this is because many of the ideas borne within it, some useful, many not, were assembled and represented in the Book of Numerals.

New! In celebration of the decadal anniversary of the Book of Numerals, I have begun digitally recreating some of the charts and graphs through Wolfram Mathematica. The remastered content will appear at the foot of the pages to which they pertain. Since the new charts are fully algorithmic, they are quite flexible and will be applied to the future “Tour des Bases” project. These edits will take place over the coming weeks; in the menu below the remastered content page designation appears in green

In 1981, at age 10, the Apple ][+ introduced me to base 16, hexadecimal. As a middle school boy I was fascinated by the fact that we could write a number in base 16, have it look completely different, with letters serving as numerals, and still mean the same quantity as in decimal. As social conditions declined between me and my classmates, I began to encrypt my thoughts. Along with the learning of Greek and Russian alphabets in the seventh and eighth grades, I began to use alternative number bases to mask the numbers I wrote. Duodecimal proved better than hexadecimal for computation because common fractions terminated. By age 15, I had developed the tayya date system, a duodecimal count of days since 1 July 1970, using it in class notes and in sketchbooks.

Throughout my teen years I remained fascinated with number bases among other things, studying ever larger bases. Numerals for transdecimal bases were required; letters may be interesting as numerals but certainly not satisfactory. In the early 80s I borrowed the first two letters of a constructed alphabet for digits ten and eleven. The digit ten actually resembled a written "11" so I changed it to a 2 rotated 180°. At first, the digit twelve resembled a C, with the digits thirteen through fifteen being Hindu Arabic numberforms adapted to resemble the hexadecimal transdecimals. Later the digit twelve resembled Greek lowercase gamma written on the line mdash; a 6 with the involuted loop allowed to emerge and extend on the left equal to the riser on the right. This served as the nucleus of what would be the sixty-numeral argam arimaxa in 1992. After university, I used duodecimal to represent measurements in feet and inches, facilitating the arithmetic associated with dimension strings. This evolved into a method still used to represent dimensions in a three-digit mixed-radix called the taqga reneya (TAWNG-ga sheh-NAY-uh) one of the means by which I still produce construction visualization. Many of my childhood predilections, toward constructed languages, astronomy, melted away shortly after marriage.

The argam were retained and were reformed between November 2004 and mid 2006. This very different sketchbook records my discovery and innovations regarding number bases, spurred by a forum quest to build a case for duodecimal vs. decimal. The book is a travelogue not through a physical land or my own coming of age, but through a discovery in number. Elementary number theory continues to be one of my passions.

Material that appeared in this sketchbook and unattached scratch paper lead to the writing and production of a dozen major articles. Thoughts on the relationships of digits and bases, and their impact on multiplication tables appear at “Analysis of Multiplication Tables”, 1½ Mb PDF d872a. These relationships are the subject of “Digit Base Relationship”, exploring elementary number-theoretical relationships of digits to bases 2-120, 2 Mb PDF d87a9. I expanded this study to incorporate the effects of such relationships in the multiplication table, divisibility tests, and digital representation of fractions in the mid-2011 presentation “Digit Base Relationships (A Guide toward Measuring Radix Utility?)”, 11 Mb PDF d87b5. The neutral relationships are described and illustrated in “Neutral Digits”, 1.15 Mb PDF d8895, and material in the article “Exploring Number Bases as Tools”, published in the March 2012 edition of ACM-Inroads, a magazine of computer science education. The material has folded into the 2011 paper on merits of duodecimal, “Dozenal FAQs”, 5.6Mb PDF d8907. I've written many posts at DozensOnline over the years. Chief among these are < Dare I admit good things about Decimal? > dated 11 April 2011 (tayya 8752) available here→ , and < Number Base Theory 101 >, a thread initiated 10 October 2011 (tayya 8884) summarizing ideas here→ .

This work eventually led to authoring integer sequences beginning in summer 2014, as well as a continued interest in the elementary number-theoretical entities explored in its pages.

The Book of Numerals was handwritten in an acid-free leather-bound book acquired on an August 2005 visit to Gubbio, Umbria, Italy. The sketchbook was recently taken on the Grand Tour of 2019. Dr. Neil Sloane had written a sequence (Recamán’s sequence, A5132) and illustration in it, so as to explain illustration work I was executing at the time. This developed into a video according to Neil’s specification. Others who've seen the Book of Numerals include British author Alex Bellos (2008) and certain members of the construction industry. The work is dedicated to my wife.


a00 Title Page The title page and dedication, in line with previous sketchbooks	a01 Introduction Page 1 History of the argam numerals, including a cyclic diagram of the number twelve and argam for SHCNs.	a02 Introduction Pages 2-3 Continuation of the written introduction, along with the argam reforms of 1992 (arimaxal) and of 2006 (xarikavincal).	a03 Introduction Page 4 Conclusion of the introduction, giving the rationale behind the fascination with numbers.

a06 Leverage vs Resistance This page explores the balance between divisors (further, regular numbers) and totatives, complete with cyclic diagrams.	a07 Study of the Ratio of Divisor and Totient Functions for SHCNs Comparison of divisor and totient counting functions for highly composite numbers	a08 Study of Totient Ratios and the Diminishing Return of Ever Larger Primorials Exploration illustrating how primorials minimize the totient ratio	a11 “Infinite Multiplication Table” Table of products used to produce transdecimal argam numerals for composite digits.

a12 Study of Totatives and Prime Argam The sexagesimal totatives are used to strain out primes less than 540, and the chemical elements help symbolize and name them.	a13 Study of Nomenclature Geometric progressions, or powers of primes help develop argam numerals for composite numbers.	a16 2008-Canonical Ismarragam Studying the multiples of the numbers 5-17 to produce argam and names for transdecimal numerals.	a19 “The Optimized Sequence” Numerals and names for superior highly composite numbers. These SHCNs are expressed in bases 10, 12, 60, 120, 360, and 2520.

a20 Studies of Divisors and Totatives and Cubes of Prime Powers Divisors of the cube of prime powers that are number bases expressed in the number base.	a26 Omega Divisibility Tests for Bases 2 ≤ r ≤ 93 Identification of omega divisors for bases less than or equal to 93.	a31 Sexagesimal Fractions Multiples of reciprocals of select integers less than or equal to 45, expressed in sexagesimal argam notation.	a34 Powers of Primes in Select Bases Powers of primes less than or equal to 7 expressed in select bases, calculated on flights by hand.

a35 Prime Power Residue Systems (mod r) Studying power residues of 2, 3, and 5 in bases 6, 8, 10, 12, 14, 16, 20, 36, 60, 120 and 360.	a36 Powers of Common Number Bases in Duodecimal, Hexadecimal, Sexagesimal, and Base 120 Powers of 12, 16, 20, 60, etc. in bases 12, 16, 60, and 120.	a37 “The Factorial Page” Factorials in duodecimal, sexagesimal, base 360, and base 2520.	a41 “The Irrational Page” and Reciprocals of Primes 2 ≤ p ≤ 37 Irrationals and reciprocals of small primes in bases {6, 8, 10, 12, 14, 16, 18, 20, 36, 60, 120, 360, 2520}.

a51 Sexagesimal Cyclical Diagrams and Totatives Regular polygons produced by joining vertices of an 60-gon that are multiples of integers that divide 60 evenly.	a52 Summary of the Complementary Divisor Multiplication Method The abbreviated multiplication table for sexagesimal and three cases of applying it.	a53 “The Prime Page” Primes less than 2160 sieved using the sexagesimal totatives. Ulam's spiral in sexagesimal. Sexagesimal totatives.

b00 Reference Appendix Reference tables in the back cover used to standardize argam transdecimal numerals.	b01 “The Prime Reference Page” and the “The Totative Page” Totatives of select numbers and their additive complements.	b02 Abbreviated Multiplication Table for Select Bases between 60-120 Tables for reference when using the complementary divisor method.	b03 Abbreviated Multiplication Table for Select Bases between 10-56 Tables for reference when using the complementary divisor method.

b09 Kinoctoval Geometric Progressions and SHCN Totatives Examination of digits coprime to their bases and powers of small primes in base 360	b14 Multiplicative Matrices Multidimensional multiplication fields based on prime powers arranged on each axis.

This page last modified Sunday 16 October 2022.

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The Book of Numerals