Rationale of the Argam numeral set

This page endeavors to explain the rationale behind the argam numerals. It is not complete but is detailed enough perhaps to give an idea of the ingredients in the set, some of its history and phases of development, and some insight via material at De Vlieger's website regarding work conducted on and with the argam. Some of the methods may prove useful in producing your own multiplicatively-defined numeral set. First we need to explain a couple concepts.

Let’s define a few terms so that we have a common frame of reference. Let the positive integer

x

{\displaystyle x}

 be an arbitrary quantity. Let the integer

r
≥
2

{\displaystyle r \geq 2}

 be a **number base,** and the integer

0
<
n
≤
r

{\displaystyle 0<n\leq r}

 be a **digit,** with the digit “

0

{\displaystyle 0}

” signifying

x
≡
0

(mod
r
)

{\displaystyle x\equiv0\ \text{(mod }r)}

. We will ignore the special case where

# x
0

{\displaystyle x = 0}

 and the digit “

0

{\displaystyle 0}

” signifies actual zero. A **numeral** is a graphic symbol (sometimes called a glyph, symbol, digit, figure, cipher, or number) intended to signify one of the digits

n

{\displaystyle n}

 of base

r

{\displaystyle r}

. A **transdecimal** numeral is a symbol for a digit greater than 9, i.e., a digit larger than the range of numerals used in decimal notation. The **Hindu-Arabic numerals** are those decimal numerals in the range {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. The Hindu-Arabic numerals are internationally used and recognized, as is decimal numeration.

Ralph Beard described two principal approaches toward creating duodecimal numeral proposals in the *Duodecimal Bulletin’s* *“The Opposed Principles”.* The “least change” or conservative approach more or less retains the decimal Hindu-Arabic numerals, and appends transdecimal numerals to the set. The “separate identity” or innovative approach replaces most or all of the familiar decimal numerals with new symbols along with the addition of new numerals for digits greater than nine. The argam numeral set is a strict “least change” system. The Hindu-Arabic numerals set the stage for subsequent transdecimal numerals. This leverages the familar numberforms to produce novel transdecimal numerals that “fit” with the existing numerals in the decimal range. Innovation applies only where the need occurs, minimizing the need to memorize new numerals. Stylistically, at least in the sexagesimal range, the argam resemble the Hindu-Arabic numerals in their form.

## A Systemic versus Piecewise approach

A **piecewise definition** is one that suits a single instance, e.g., suppose we are considering a numeral for digit-ten. We may ignore any systemic resolution for a series of related numerals and simply produce a solution just for digit-ten. Piecewise definition is the “hand-to-hand combat” of defining elements in a system. The Hindu-Arabic numerals may be considered a mixed bag of some numerals with a more or less obvious rationale and some with no currently definitive rationale; we can consider all of the numerals to be arbitrary and the set a sort of black box that we can use as the basis for new numerals if a “least-change” approach is desired. Once we have a range of piecewise-defined numerals, i.e., a range of numerals designed to suit individual integers, we can take two systematic approaches to define a great range of numerals. **Systemic approaches** will involve producing **compound numerals,** i.e., numerals that are combinations of smaller syntactic symbols, similar to Mandarin radicals and base characters, vehicles of significance that are welded together to make a single-piece symbol with a new meaning. We’ll call the smaller syntactic symbols **figures.** A compound numeral is not the decimal-coded sexagesimal representation we observe in the digital expression of time of day, e.g., 4:12 pm. This representation may be seen as a limited-range mixed radix rather than a compound numeral as we have defined it. Looking at the “12”, we see a sexagesimal numeral expressed in decimal form, with “1” in the decade-place and “2” in the unit-place. The decade-place can be seen as a senary (base-6) system alternating with a decimal unit-place. This would indeed be a mixed radix system if decimal-coded sexagesimal were used to express all magnitudes of time; instead only minutes and seconds are expressed in decimal-coded sexagesimal. Suppose we wrote the decade-place in a half-height figure above a half-height unit place, perhaps similar to the vertical form of the fraction

1
2

{\displaystyle \frac{1}{2}}

, minus the solidus (

1

2

{\displaystyle \begin{matrix} 1 \\ 2 \end{matrix}}

). We would then have a simulated compound numeral De Vlieger calls a “stacked numeral”; I devised a set of stacked numerals called “decker” in 2009 at the DSA website. Here the decade-figure rides atop the unit-figure to produce the stacked numeral. If the figures were joined to form a single piece, we would have a compound numeral. Jean-Michel Smith’s sexagesimal numeral system is a one of compound numerals.

## Definition of new numerals

There are three basic rules for the definition of a new numeral.

### Creative definition

We can devise arbitrary forms to convey the notion of a digital quantity. The form we choose to use to signify an integer need not have an apparent reason for its shape or configuration to validly convey the meaning. Such a form bears the meaning merely by having a mnemonic association with the integer. Looking at the Hindu Arabic numerals, we know the numeral “8” means the cube of two because we’ve memorized it. In fact we know the meaning of all the numerals because they were presented as signifying a particular integer, without needing to know the rationale behind the design of the numeral. The word “arbitrary” imparts a negative connotation in our society, as we are often attempting to extract a rationale from the world around us, and we are wary of ideas that appear baseless. For this reason we'll use the term “creative”, as an apparently arbitrary approach is quite valid and sufficient in defining a new numeral.

### Additive definition

Suppose we have a compact range of small numerals such as the existing decimal Hindu-Arabic system. We can call this compact range a **numeral base.** One method of creating numerals is to use the numeral base to define subsequent contiguous ranges of numerals by adding a group-figure to a unit-figure. Let's call a power of *r* a rank, with the unit-figure representing the zeroth rank, the group-figure the first rank (as it is *r*¹), the square-figure the second rank (as it is *r*²), etc. Decimally we have unit-figures, decade-figures, century-figures, etc.; we might use other decimal terminology for the general names of the figures. Suppose we use a small circle (°) below each of the Hindu Arabic numerals to define the transdecimal digits ten through nineteen. In this case, the group-figure is a decade-figure, as it is signifying a group of ten. Thus, the modified “7°” would signify digit-seventeen. Effectively, we are using a *number base,* in this case, base ten to define the transdecimal numerals in this range. We can decode the new numeral by observing the presence of the figure (°), and the unit-figure “7”. The group-figure imparts the addition of ten to the unit-figure, which communicates *n* ≡ 7 (mod 10). Alternatively, using the integer *k,* the transdecimal numeral “7°” reads as *n* = *kn* + 7. We could use Hindu Arabic numerals as decade-figures, as we described the decimal-coded sexagesimal stacked numerals above. In this case, we could write a vertically-stacked 17 to signify digit-seventeen. Using two ranks of figures, we have sufficient compound numerals to represent all digits 0 ≤ *n* < 99. If we were to use three ranks of figures, we could represent all digits 0 ≤ *n* < 999. If we use *e* ranks of figures, we can represent all digits 0 ≤ *n* < (*re* − 1). Since we use addition systemically to define numerals, I call this approach “additive”. The benefits of using an additive approach include a compact (thus finite) range of small numerals and the complete coverage of a range of digits related to the quantity of ranks of figures we decide to use. Obviously there is some practical limit to the number of ranks we can incorporate; it would be difficult to envision using more than several ranks. We might define a larger numeral base and thus increase the coverage of our ranks. Though conceptually, the set of additively-defined numerals can be  infinite, we are practically limited by human cognition and the physical ability to produce a legible numeral. The major drawback of additive definition regards the numeral base. We are imparting a phantom number base via the numeral base on any integer outside the numeral base. This might be acceptable for target digits that are regular to the numeral base, i.e., decimal-coded digit-forty shares the decimal prime divisors. In this case however, the decimal representation of forty incorporates “four-ness” and “ten-ness”, but we could define forty using “five-ness” and “eight-ness”, “evenness” and “twenty-ness”, etc. The situation is more awkward when the target digit is coprime to the numeral base and *k*. The example, digit-seventeen, has nothing in common to the unit-figure 7 and the numeral base 10 = 2 · 5. Seventeen often has nothing in common with the other numerals (mod 10), i.e., {27, 37, 47, 57, 67, 77, …}, though these numerals will resemble the others in the set. When we use a numeral base, we are defining numerals larger than the numeral base in terms of the numeral base, resulting in a sort of indirect definition, a definition based on extrinsic properties.

### Multiplicative definition

Suppose we have a range of defined numerals for elemental integers (i.e., the primes). The primes constitute an infinite number sequence; right from the start we know we can never define numerals for an infinite set; we'll set that drawback aside to examine multiplicative definition. We can then define composite numerals using figures that signify prime powers. A numeral so-defined would be encoded according to intrinsic properties, that is, the prime decomposition of the digit. We would avoid projecting a number base via a numeral base, instead we are multiplying the figures in the numeral to arrive at the digit’s value. The key to producing such a system is generating a sufficiently-large pool of small prime numerals and names for the numerals, called **elemental numerals.** Let’s use the prime counting function Π(*δ*), with the integer *δ* signifying the number of the prime that is largest in the set; then Π(*δ*) represents the largest prime we’ve symbolized via an elemental numeral and name. Once we had a sufficient range of elemental numerals, we could produce a large range of composites. The range would be continuous up to the prime succeeding the largest-defined prime Π(*δ* + 1). We could produce all composite numerals using the defined elemental numerals, even those very much larger than Π(*δ*). For example, suppose we had two figures, an “E” called “tri” meaning 3 and a “T” called “sef” meaning seven. We could produce a stacked numeral “ET” “trisef” to signify the product of (3 · 7). Perhaps this stacked numeral would weather to become “&”, the name weathering to “tress”. Then we would have a numeral for digit-twenty-one defined entirely by intrinsic properties of the integer twenty-one. If we permit the numerals to weather, we could then use the weathered numeral as a compact figure to produce larger composites. Suppose we wanted to produce numerals for digit-sixty-three and digit-one-hundred-five. We could produce stacked numerals based on the weathered numeral “&”, “tress”. So we could somehow combine “E” with “&” to arrive at a stacked numeral “E&” “tritress”, or we may dissociate “&” and use a squared form of three to produce a combination more like (9 · 7), calling it something more like “sefnove”, or weathered, “senove”. Suppose we represent fivefoldness using a cedilla (¸) called “kin”. Then our numeral-one-hundred-five would be “&” with a cedilla “¸” at the bottom, called “trikinsef”, or we might combine a weathered name to arrive at a briefer name like “kintress”; the name might be allowed to weather and likewise become more concise, like “kress” for example. Because numerals are defined solely though multiplication, I call this method “multiplicative”. The upside of a multiplicative definition method is that numerals so-produced are defined by intrinsic properties. This liberate the digits from a projected number base. We can use these base-less digits to represent any number base without comtamination from a numeral base. This leads to a purer representation of a transdecimal number base, with digits that are defined by their number-theoretical intrinsic properties. The digits themselves are purer symbols for that integer as a number base. The downside includes the following. We are limited by our ability to produce elemental numerals. This drawback is mitigated somewhat by the fact that additively defined numerals have a practical limit based on the magnitude of the numeral base and the practical quantity of ranks used to define the numeral. The figure-combinations of a multiplicatively-defined numeral set can be quite complicated. Some methodology would be necessary to encode highly composite numerals, i.e., digit-2520 = (2³ · 3² · 5 · 7). We can release some of the pressure of incorporating four or more figures by allowing some frequently-encountered numerals (i.e., more highly composite numerals involving small primes) to weather; we might use (5 · 7 · 8 · 9), even  (40 · 63) or something similar to approach a numeral for 2520. Finally, for highly composite numerals, there are many ways to encode a numeral multiplicatively; we can devise multiple numerals for the same digit. We thus would need some methodology to control the possibilities so that a standard might emerge. Alternatively, we can use a preferred definition for certain numerals and let this definition propagate. When we define a multiplicative numeral set using a set of elemental numerals and figures, the resultant numerals are defined by intrinsic properties.

The benefit of a piecewise assembly approach is that we can address an individual numeral problem without reference to a systemic concept, and arrive at a solution that is often simpler (i.e., that it is not a compound solution and that it is less complicated) than a systemic solution. We are prone to considering the digit’s intrinsic properties, as we are focused on the digit and not necessarily referring to an overarching scheme. I refer to a set produced piecewise as having “accreted” like the formation of a planet because it is an empirical process. One may define a numberform and play it in a context and find it unsatisfactory. Subsequent numberforms may influence change on numerals that have already been defined. This is akin to the bombardment of a young planet by planetesimals that at once bring new mass to the planet, yet modify its surface features. Because the piecewise definition of a numeral set requires time and is empirical, the resultant set tends to have a greater degree of differentiation. Numerals in the set will tend to be more easily distinguished from other numerals, improving legibility of numbers written with the set. If the numeral set grows from the smaller to larger numerals, the tendency will be to assign the smaller and thereby more common numerals less complicated numberforms, while the greater numerals take on more involved designs. Thus the set scales well, with common numerals having simpler forms. The drawbacks include the following. The  numeral set resulting from piecewise accretion may not possess the logical unity that a systemic scheme would automatically have. Numerals for digits with similar intrinsic properties may not resemble one another, especially if a strictly creative approach is used to the exclusion of a scintilla of logic. Finally, a piecewise-assembled system will require memorization efforts upfront in order for one to use the set. The learning curve may be steeper because of this. The benefit of a systemic assembly approach is that we can concoct a strong rule for the generation of a class of numerals, thereby producing solutions for very many of them. We can compose numerals we haven’t really used before simply by knowing the rules for composition. Likewise we can decipher individual numerals we encounter without having ever seen them. In this way the systemic approach is akin to writing sentences; there are syntactic composition rules given a set of elements. If our rules are well-aligned, then we will be able to define numerals based on a digit’s intrinsic properties. Drawbacks include the following. A systemic assembly may generate simple solutions (i.e., those that are not complex or those made of a couple or several parts) rather efficiently. When the numeral requires a large number of treatments, a concise and compact solution may not be available, and the numeral may become difficult to write or produce. Similarly, a fully rule-based solution may create complicated numerals for small and often encountered numerals. If the rules are strictly followed, the solutions across the entire range of numerals may have too much unity and be difficult to distinguish, harming legibility. Also, consequences of strict rules must be investigated thoroughly, because one may end up with contradictory graphic results or conflicts in execution. A solution that might tend to allay some of the drawbacks of both extremes is to employ a middle-ground application. We might allow numerals to be conceived in a loose logic of guidelines, play them in context and allow them to “weather” with use. This perhaps supplies a new syntactic element, helping to abbreviate the figures for more complex but often-encountered concepts. Such weathering, though severing some of the logically-clear attachment to a systemic rule, tends to simplify the numeral and improve its legibility. In letting both approaches play, we sacrifice some strictness of logic and thereby some ease of systemic learning to attain a more legible set of numerals that are simpler according to their commonness.

## References

1. ↑ http://www.dozenal.org/drupal/sites\_bck/default/files/db4a205\_0.pdf
2. ↑ 2.0 2.1 http://www.dozenal.org/drupal/sites\_bck/default/files/DSA-SymbologySynopsis\_0.pdf. See Note E for 2.0, and see Note S for 2.1.

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