by Michael Thomas De Vlieger, 14 August 2014, St. Louis, Missouri.
This is a fair question. I don't have a math education; my education is in architecture, and I've been better-known as a creative professional, skilled in hand-drawings and digital imagery of construction sites. How does this translate into an interest in mathematics?
When I was 10 in December 1980, my father bought me an Apple II+ (64k ram instead of only 48!) I'd learned to read at age 3½ and my text of choice was astronomy books. Sr. Gerald made me very popular in first grade when she asked me to read to the class to demonstrate kids should be able to read. I read from a college textbook on astronomy. I wanted to be an astronomer since age 5, having read Carl Sagan's Other Worlds and scintillated by Cosmos: A Personal Journey. My father had to turn me around, hence the computer. No games; the "fun" would have to come from programming it. Thus I began to develop a penchant for coding that has stuck with me, through college and into my professional career. I learned javascript and HTML, some PHP, and eventually Wolfram language.
Around the same time, in May of 5th grade when the classroom would be in a swelter, there was an enrichment exercise in number bases; I'd finished the normal math work and didn't have to redo. Something is fascinating about writing numbers in another way, practicing the rules of arithmetic to match, and having the result mean exactly what the decimal result would be, just written differently. I would feel the same way about languages in 8th grade, teaching myself Russian so I could pass notes the teacher couldn't read. The Applesoft manual described hexadecimal; I was in love. Soon everything had to be hexadecimal. This ran all through sixth grade until I would learn about base 12.
In base 12, "the fractions come out", they'd say in my blue collar hometown. They were right. In hex, ¼ = .4 and 1/8 = .2 came out, meaning they terminated after a brief run of digits. But in duodecimal, the five most common fractions come out 1/2 = .6, 1/3 = 0.4, 2/3 = .8, 1/4 = .3, and 3/4 = .9. The multiplication table was as compact as the one I learned, and was very simple to learn. I was in love again at age 12, but this love would last despite my testing her.
At age 15 and throughout my teen years I would develop a duodecimal calendar that I still use today, akin to Julian dates, and study higher bases, inventing "argam" transdecimal numerals. (The original name was "argam Arimaxa", named after my girlfriend). These extended to base 20 in 1986, then to 30 and 60 in the late 80s. In 1992 I had 100 numerals and programmed my HP-28s to do arithmetic in any base desired, presenting the results in argam. Shortly after graduation from university, I was on jobsites and would measure existing conditions, taking the measurements down in dozenal. This allowed me to do arithmetic as easily in the field as if the US Customary measures were metric. Because I could check and total dimension strings in the field, I was more efficient at work and that contributed to a beneficial opinion of my work by managers.
When I started my company at age 34, in 2004, I'd fly around the country now and then and have some time to kill. I began firming up 120 argam numerals and studied sexagesimal arithmetic. In 2006 I joined the Dozenal Society of America, and for the 2007 general meeting, produced a 100+ page manual on sexagesimal arithmetic. This is when my interest in elementary number theory took hold.
In order to multiply in sexagesimal, memorization of the 1860 unique products in the table is out of the question (there are just 55 unique facts in decimal; 78 if we count the 12x table many have memorized). Instead, having read that the Babylonians used "reciprocals" (complementary divisors) to arrive at results, I tried the same. Though 60 has a dozen divisors, four dozen of the numerals are nondivisors; they come up more often than divisors in arithmetic. I found that the worst numbers to work with were the relatively prime "totatives". There were numbers "in the middle", that were non-totative non-divisors. Some of these were nearly as easy to work with as divisors; others were tough and involved totatives.
After the 2007 "Reciprocal Divisor Method" (RDM, 6 Mb PDF) was presented, I began to delve deeper into what made these numbers tick. Can an RDM be useful for even larger bases-maybe 120? The language, "nondivisor nontotatives" was clumsy. There seemed to be two types of nondivisor nontotatives, one friendlier to use than the other. I studied very large highly composite bases and found they were swamped with totatives.
In 2008 I bought Wolfram Mathematica 6.0 and subscribed to the service so got updates. The program was alien and at first all I did was convert numbers to bases, basically an expensive calculator. In 2010 I began to educate myself in number theory, reading Oystein Ore's Number Theory and Its History, Underwood Dudley's Elementary Number Theory, and other works. I learned about the Fundamental Theorem of Arithmetic, Euler's work, modular math, etc. This helped the reframing of the "nondivisor nontotative" concept over the next year into my first set of proofs.
In 2011 I wrote a paper called "Neutral Digits", attempting to explain what I'd called "nondivisor nontotatives", using the Greek word for "neither" = neutral to refer to these numbers. The proofs served to underscore that there are only two types, which I called semidivisors and semitotatives, how to construct the types for any base, the circumstances that lead to their appearance as digits of those bases. This lead to "digit maps" and eventually the Tour des Bases at the DozensOnline forum, which assessed and explored many bases in terms of the arithmetic qualities of the digits. I presented Digit-Base Relationships in 2011 at the DSA annual meeting. I wrote an article for the ACM Inroads magazine, published in March 2012 that explained much of what appears in the forum, based on Digit-Base Relationships. Meanwhile I discerned an education in math at U of I through NetMath, but determined that it was too focused on math I wasn't interested in learning, and none of the math that excited me. I figured it would kill my interest in the subject.
In November 2013, I wrote Wolfram code that produced the digit maps I'd become known for on the forum. These color-coded the digits of bases according to their direct and indirect arithmetic relationship to the base. Now I could generate maps for very large bases nearly instantly and effortlessly. For bases larger than 1000, I could produce "digit spectra", bar graphs that showed the number and proportion of each type of digit. In spring 2014, my capabilities in Mathematica continued to increase, charting the "proper regular" numbers of a base (a paper I have yet to write).
On 11 June 2014 I submitted two sequences to the Online Encyclopedia of Integer Sequences. The first, A243822, counted the number m of semidivisors of n, while the second A243823, the number of semitotatives. These were published nearly on my 44th birthday. I've submitted other sequences and found great joy in coding Mathematica functions that would help define, generate data, or extend other sequences. This helped me learn how to Reap-Sow, Catch-Throw, encapsulate loops in modules, and generate self-referential functions.
I wrote a proof regarding anti-divisors as they relate to neutral arithmetic relationships on 12 August 2014. The jury is out on whether this is useful to anyone, but it was thrilling to prove things about these numbers. I realize I am at a disadvantage, not having the education, skill, and talent of the professional mathematician. I do share the joy in writing functions that work, in studying what makes numbers tick. I think I get as much joy out of this as I do in drawing a successful portrait or building models of construction sites for a living. I hope you do to!
(Updated 14 August 2014)