Gradience as a cognitive principle for evaluating numerical notations

Significance

This study presents a cognitive principle, gradience, for comparing the communicative efficiency of numerical notations, along with two indices for measuring gradience, jitter and inversion. In general, larger numbers require more signs than smaller ones, but this is inconsistent across the world’s 100+ numerical notations used over the past five millennia. Gradience helps analyze the cultural evolution of numerical systems and test the assumption that the present ubiquity of decimal, positional notation is optimal.

Abstract

More than 100 historically, archaeologically and ethnographically attested numerical notations have been used over the past 5,000 y; however, because most of those systems are no longer used, experimental investigation is challenging. Prior research often assumes, rather than demonstrates, the inferiority of notations like Roman numerals. Gradience is a principle whereby the length of numeral phrases correlates with the magnitude of the numbers being represented. In general, but inconsistently, larger numbers require more signs, just as larger sets of abstract dot patterns occupy more space. This study compares the gradience of 13 numerical notations varying in phylogenetic family, linguistic family, and structural properties, using two indices, inversion (how often N + 1 requires fewer signs than N) and jitter (the mean length difference between successive numbers). Place value systems are highly gradient; i.e., their length indexes numerical magnitude more accurately than other systems. The relationship between the structural properties of notations and the two indices of jitter and inversion is complex. Next, a cultural-evolutionary analysis compares the older additive Roman numerals to the eventually predominant subtractive variant (e.g., XIX = 19). Subtractive Roman numerals are more concise and have a lower jitter than additive ones, but at the cost of a significantly higher inversion ratio. This analysis permits the evaluation of cultural-evolutionary hypotheses grounded in the representational properties of attested notations, even when cognitive tests are not feasible. Several avenues for future experimental investigation using attested or constructed notations of varying properties are proposed.

Over the past 5,500 y, well over 100 structurally distinct numerical notation systems, and hundreds more variations of these, have been used cross-culturally (1). Numerical notations are translinguistic, visual, highly structured systems for representing numbers, and thus stand in contrast to other systems like number words (whether spoken or written) as well as numerical gesture, calculating devices, and other modalities. Numerical notations are often described as being “written,” and normally occur alongside written language, but are not necessarily fully integrated with writing systems. They have their own properties separate from the number word systems of the world’s 7,000+ languages, most of which were, at least historically, not written.

Any successful numerical notation must, at minimum, allow the quick and accurate assessment of quantity to be read. So, for instance, consider the following problem: which of the following two integers is larger: 2494923932921 or 24949618? In place value or positional numeral systems in which each power of ten has a digit from 0 to 9, the expression with more signs will always be greater. Of these two expressions, the first has 13 digits and the second has eight. Even though they both start with the same five digits, it is relatively easy to assess that the former is (much) larger than the latter. The reader need not compare digit by digit–and in fact, can get a rough sense of the difference in magnitude between two expressions just from digit length. Now consider a second problem: which of the following two integers is larger: MMDCCLXXII or MMM? The shift from Western numerals^* to Roman numerals immediately poses a problem, because the former expression (2772) is smaller than the latter (3000) even though the latter uses fewer signs (10 vs. three). Put formally, the conciseness of a Roman numeral phrase is a poor predictor of the numerical magnitude of the expression.

The assessment of numerosity has been a central question in numerical cognition for many decades (2) and in particular since the identification of the approximate number system (ANS) (3, 4). The ANS is a cognitive system in humans and many other species, allowing the rapid estimation–but not exact counting–of sets of perceived objects in the environment. Up to around four objects, subitizing allows the rapid perception of exact quantity, but beyond that point, rapid assessments are estimations, prone to error (5). These errors follow the distance effect, a form of Weber’s law, a psychophysical regularity whereby the perceptibility of a difference between two stimuli is proportional to the ratio of their magnitude (3, 6). That is, it is easier to distinguish 10 from 20 objects than 110 vs. 120, but roughly the same difficulty as assessing the difference between 100 and 200 objects. One of the major empirical demonstrations of the properties of the ANS derives from its application to patterns of dots or other abstract objects as a means of estimating cardinalities. But the problems above draw on the same principle, even as they involve comparison of expressions in numerical notation. The ANS is usable to assess the approximate cardinality of sets of objects, and numeral signs or digits qualify as those objects. Thus, a reader can assess the rough value of at least some numerical notations without even having to read the digits themselves, because the relative lengths of the numerical expressions roughly indexes the difference in magnitude between them.

Over the past 30 y, there has been ample experimental research on reading numerals using place value decimal (Indo-Arabic/Western) notation, the family of common systems used by almost all literate people worldwide today (7–11). However, this represents only a tiny fraction of the total variability among all numerical notations used by cognitively and evolutionarily modern humans. There is far less empirical research to date on the cognitive properties of the vast majority of numerical notations (but see, e.g., refs. (12–14). The work of Krajcsi and Szabo (14) is an interesting counterpoint to the general assumption that place value systems are representationally superior. They develop artificial base-4 notations, some with place value and others without (i.e., positional vs. additive systems) and show, contrary to expectations, that for unfamiliar notations, at least, positional systems are more complex for users.

Of course, historical or archaic systems present a methodological challenge to experimental research, in that these notations have no regular, “native-like” users, and their unfamiliarity to untrained participants makes comparing them to familiar systems difficult if not impossible. But because most number-users historically were not familiar with decimal, positional numerals like those studied in modern experiments, our knowledge of the cognitive properties of the full range of numerical notations remains poor. This is a “WEIRD societies” problem (15), but even more importantly, it is a modern problem–we cannot simply fix it by sampling more broadly among living people. But this does not foreclose all research. An important recent study (16) investigates communication of numbers drawing on the historical and cross-cultural literature on numerical representation, in which participants were asked to communicate about visual arrays of sets of objects presented as stimuli without using numerical words or familiar numerical notations. Expectedly, many respondents drew figures employing one-to-one correspondence, but many also used properties of visual arrays, such as organization of objects into rows and columns, to structure their ad hoc representations. By doing so, they suggest some key ways by which humans exploit properties of number in the environment when creating novel strategies for numerical communication. Such promising avenues of investigation are beginning to open up, using theory and data from cognitive science, anthropology, archaeology, and the philosophy of mathematics.

The present study is an empirical but nonexperimental examination of the properties of numerical notations using cross-cultural historical and anthropological data (1, 17). An assessment is offered of the range of properties previously identified as relevant to the informational efficiency of different notations, all of which are shown to be insufficient, on their own, to fully assess the merits of various notations. A principle, gradience, is identified, and two indices for its assessment are outlined. These principles are applied comparatively, first, to a cross-cultural sample of notations across different phylogenetic traditions, and then to two historically interrelated variants of the Roman numerals as a cultural-evolutionary case study. Suggestions are then offered for future experimental research that might use gradience to assess its effects on numerical cognition directly.

Evaluating Informational Efficiency of Numerical Notations.

All numerical notations ever to have existed fall into one of five basic types (1, 17); see also ref. 18 for a modified typology. Each system uses an intraexponential principle that determines how signs are used within each power of the base, as well as an interexponential principle that determines how the different values for each power are structured into an entire numeral phrase. Intraexponentially, cumulative systems use multiple repeated signs within a power, ciphered systems use at most one sign per power, and multiplicative signs use a multiplier-sign and a power multiplicand, whose product is taken. Interexponentially, additive systems simply add the values of each power together, while positional systems use the principle of place value to imply multiplication by different powers of the base, based on the position of the sign(s) in the phrase. Table 1 illustrates this typology along with examples of the inter- and intraexponential principles underlying the expressions for 228 in five different systems.

Table 1.

Typology of numerical notations (cf. refs. 1, 17)

One feature consistently identified as relevant to numerical cognition is conciseness (19, 20). The conciseness of a particular numerical expression can be defined as the number of signs it uses. These signs could be morphemes (neunhundertachtundneunzig has six morphemes) or graphemes (CMXCVIII or 998). Beyond simply assessing the conciseness of any one expression, we could ask about the conciseness of an entire system. Systemic conciseness is an analytical construct, defined here as: For a set of integers from 1 to X in some notation, the mean conciseness of all its numerical expressions. Most numerical notations are not infinite in extent and because most written numbers are small, it usually suffices to use relatively small values of X to assess systemic conciseness. Several studies examine the relative conciseness of the lexical numeral systems of Polynesia, using the term “compactness,” demonstrating that some of their features often thought to be weaknesses actually convey advantages in information transfer (21, 22).

It is often claimed that the Roman numerals are less concise than the various Indo-Arabic systems (i.e., any decimal positional system of the family including the Western numerals, but also Devanagari, Thai, Nepali, etc.). At a systemic level, that is accurate; on average, Roman numerals are less concise than their Indo-Arabic counterparts. However, this is not true for all numeral phrases; for instance, MMM = 3,000, as in the second example above. Additionally, the expressions where the Roman numerals are more concise than Western numerals are round numbers like 3,000 (MMM) or 50 (L). Because round numbers (along with small numbers) are used more commonly in language than nonround numbers, as shown in a variety of corpus-based research (23, 24), this mitigates this disadvantage.

Moreover, nonpositional systems like the ciphered-additive Greek alphabetic numerals are actually more concise than Western numerals, because they have no sign for zero. So, for instance, 300 is the single Greek sign τ. Following the typology above, ciphered systems are indeed more concise on average than cumulative ones, but positional systems are in fact less concise than an otherwise similar additive one. Conciseness is thus a poor candidate for explaining why decimal, positional numerical notations are preferred cross-culturally or tend to prevail over time.

A second candidate for assessing the informational efficiency of a numerical notation system might be its sign inventory; that is, the total number of numerical signs needed to create numerical expressions in some system. However, this is not a productive approach. The Roman numerals use only seven basic signs (IVXLCDM) whereas the various Indo-Arabic systems use ten digits from 0 to 9, but there has been no suggestion that this matters in terms of representational efficiency or processing time. Some systems (like the Babylonian sexagesimal numerals shown above in Table 1) use only two signs (for 1 and 10), along with place value, to express an infinite range of expressions, while the various alphanumeric systems in which letters are assigned numerical values piggyback on the corresponding writing systems in their alphabetic order, and thus have dozens of signs. It also raises the question of what counts as a “numerical sign”–for instance, is the decimal point a sign? Is the interdigital comma used after every three digits a sign?

A third criterion might be extensibility. Some systems, like the Indo-Arabic family of decimal, ciphered-positional systems, are truly infinite in extent, while others (mainly nonpositional systems) reach some maximum, beyond which new signs must be invented. Without denying that extensibility is useful, there is limited historical evidence that this mattered in the way that modern critics have often imagined, for instance, affording numerical cognition of infinity as a feature motivating its adoption (25). Extensibility is relevant to a very small number of users of any notation. There really is no difficulty for the users of a system to invent new signs as needed, including (as in the much-derided Roman numerals) using multiplicative structures such as placing a bar (vinculum) over an expression to multiply its value by 1,000. And in any case, most numbers expressed in any notation are relatively small numbers, because in almost any context or number system, lower numbers (as well as some round numbers like 10, in decimal systems) are more frequent than higher and nonround ones. Finally, for numbers beyond one million, it is common to use a hybrid of words and notations (226 million rather than 226,000,000), rendering the question of extensibility moot (26).

Finally, general claims are often advanced about the computational efficiency of different numerical notations—that is, how effectively they can be used for arithmetic. Such propositions are difficult to test directly, because most experimental participants would be so familiar with arithmetic in a system of the Indo-Arabic family that comparison with, say, Roman numerals would not be possible. There have been multiple efforts to show that the Roman numerals could have been used for arithmetic without too much difficulty (12, 27–29). Some historians of accounting have noted that one of the factors behind the adoption of Western numerals in Europe may have been their greater utility than Roman numerals for double-entry bookkeeping (30).

The challenge with all these demonstrations is the countervailing historical fact that we know how Greeks and Romans did arithmetic, and it was not through written notation, but rather, primarily through highly efficient cognitive artifacts like the abacus (31, 32). In fact, most numerical notations known historically, as best as can be assessed, were never used directly for computation. Even when, as in Western numerals, pen-and-paper arithmetic is normative, most instances of numerals one encounters in daily life are not arithmetical, but simply representational. Tasks like bookkeeping do require some notation, but it is not at all clear whether conciseness, sign inventory, or extensibility truly mattered in a way that supported a long-term cultural-evolutionary preference for Western numerals; the Roman numerals were just not discussed or evaluated in those terms at the time (33). Numerical notation is thus principally a representation and communication tool, not a computational one, and assessing it for a function it was not designed or used for is inappropriate (34, pp. 55–90).

Gradience, Jitter, and Inversion.

This paper outlines a principle, gradience, for assessing informational efficiency in numerical notations, treating them as representations. Gradience is the degree to which the conciseness of a series of numerical representations is well ordered, or put another way, the degree to which the number of graphemes in numeral phrases for integers indexes their numerical value. Gradience piggybacks on the cognitive capacity to rapidly assess the numerical magnitude of two groups of unlike size, either through subitizing (for quantities up to four signs) or approximation. In a perfectly gradient numerical notation, the conciseness of the numeral phrase for a natural number N is always equal to or less than that for its successor N + 1; in other words, it is a monotonically nondecreasing function that either stays constant or increases. Gradience does not apply to fractions, decimals, or similar numerical expressions.

Gradience can be conceptualized as a curious kind of notational iconicity; in a system that is highly gradient, there is a sustained correlation between visual size (number of signs per expression) and numerical value. Iconicity, the resemblance of signs to their referents, has been an active subject of research in notations, both from scholars of writing (35) and cognitive scientists (36). Even in a highly gradient numerical notation where each expression is no shorter than the one before it, gradience is imperfectly iconic. A tally-based notation, in contrast, in which one mark corresponds to one object, is nearly perfectly iconic (setting aside spacing and orientation), as has been often noted (37, p. 9). Wescott (38, p. 420) goes so far as to develop this into an evolutionary theory in which iconicity gives way to symbolism. But tallying is very different in function than numerical notation and is not merely a precursor to symbolic notation; it is a highly effective way to represent an open-ended, ongoing count rather than a completed expression, which is why it is not merely a historical curiosity but an active modern practice (39).

One way to reconceptualize gradience is as a kind of logarithmic iconicity–conciseness approximates magnitude only relative to different powers of the system’s base. For instance, the conciseness of any Western numeral expression is the whole-number component of the logarithm in base 10 of that number, plus one, so that all numerical expressions starting with 10² but less than 10³ (100-999) are of conciseness 3. For numerical notations following other principles, however, that relationship is far more complex, but all systems provide at least some iconic correspondence with numerical magnitude. The imprecision of gradience’s iconicity is not necessarily a disadvantage. Chrisomalis (34, pp. 33–45) discusses the phenomenon of conspicuous computation, the public display of numerical expressions whose visual magnitude is the point–not simply a big magnitude like six quadrillion but a large number like 6,000,000,000,000,000. In public texts like the National Debt Clock on display in Manhattan, the goal is not to be numerically precise, but rather, to overawe the viewer with the sheer, literal magnitude of the expression for rhetorical and cognitive effect. This form of iconicity may be similar to the metaphorical relationship between size and quantity observed in language, e.g., the almost-invisible metaphor of quantities being “larger” or “smaller” (in English) than others (40). These associations are evident in the growing literature on multimodal numerical communication, where, for instance, manual gestures for larger quantities involve more space between the fingers (36, 41). Gradience is not simply iconicity nor simply metaphor, however, but a system of (often irregular) mappings between conciseness and magnitude that vary cross-culturally.

To assess gradience across an entire numeral system requires some further definition of terms. Jitter, for some set of numbers {1, 2, 3 … N}, is the mean of the absolute differences in conciseness between all pairs of successive numbers. A system of pure tally-marks, made one after the other, has a jitter of 1, because every expression is one longer than the one before it, while a hypothetical system in which every number used a single (different) sign would have a jitter of 0. Inversion, for some set of numbers {1,2,3 … N}, is the proportion of numbers such that a numeral phrase N + 1 is shorter than that of its predecessor N. So, for instance, across the most commonly used Roman numerals from 1 to 10 {I, II, III, IV, V, VI, VII, VIII, IX, X}, the system’s jitter is 1.11 because the absolute difference in conciseness is 10 across nine intervals (eight of the intervals are ±1, excepting VIII to IX which is −2). Similarly, 4 of 10 (IV, V, IX, X) signs are more concise than their predecessor, so its inversion index is 0.4. A decimal, ciphered-positional system like Western numerals has a jitter close to 0, because phrases increase in size only at powers of the base (9 to 10, 99 to 100, etc.), and an inversion of 0 because no expression is shorter than that for its predecessor. The difference between the two indices is that inversion counts both large and small decreases in conciseness equally, while ignoring increases, whereas jitter accounts for the degree of change and includes the possibility of large increases (more than one sign), not merely decreases, between two numbers.

In this study, thirteen numerical notations were selected from across the comparative range of known historical systems used over the past 5,000 y, as described in Table 2. These were selected from a previous published reference of numerical notations, divided into several distinct phylogenies or lineages (1, 17). Except for the two variants of the Roman numerals, the systems chosen were relatively historically unrelated to one another phylogenetically. Their user base eventually included a wide range of speakers of various languages, and their originators came from speakers of at least seven language families.

Table 2.

Phylogeny, language family, and duration of use of 13 numerical notations

System	Phylogeny	Language Family	First*	Last	Duration
Aramaic	Levantine	Afro-Asiatic	−750	−250	500
Aztec	Mesoamerican	Uto-Aztecan	1200	1600	400
Chinese	East Asian	Sino-Tibetan	−250	Pres.	2,275+
Egyptian	Hieroglyphic	Afro-Asiatic	−3250	400	3,650
Greek alphabetic	Alphabetic	Indo-European	−575	Pres.	2,600+
Western	South Asian	Indo-European	1000	Pres.	1,025+
Inka	Inka	Quechuan	1300	1600	300
Oberi Okaime	African	Constructed	1931	1975	44
Old Persian	Mesopotamian	Indo-European	−500	−350	150
Roman-additive	Italic	Indo-European	−500	Pres.	2,575+
Roman-subtractive	Italic	Indo-European	−100	Pres.	2,125+
Sinhalese	South Asian	Indo-European	500	Pres.	1,525+
Sumerian	Mesopotamian	Sumerian	−2900	−1500	1,400

^*Negative dates are BCE; positive dates are CE.

The sample includes numerical notations of all the various attested bases among the world’s notations (10, 20, and 60), includes both systems having and lacking subbases, and includes representatives of each of the five basic types of numerical notation. Three systems (Aramaic, Sinhalese, and Old Persian) use one structure for numbers up to 100, and then are hybrid systems above that point, using multiplicative-additive structuring for numbers greater than 100. Table 3 highlights these structural features along with examples of two numbers, 228 and 409, expressed in each notation. Table 4 describes the mean conciseness, jitter, and inversion of all expressions in each system up to and including 1,000.

Table 3.

Structural features and example expressions for 228 and 409 across the sample of numerical notations

Systems whose type is marked with an asterisk obey the given principle for numbers <100, and then are multiplicative-additive beyond 100.

Table 4.

Conciseness, jitter, inversion, and comparison to Indo-Arabic numeration

System	Base	Subbase	Mean Conciseness	Jitter	Inversion	<IA	≦IA
Aramaic	10	20	12.40	1.797	100	4	13
Aztec	20		18.81	1.887	50	5	13
Chinese	10		4.49	0.215	90	11	118
Egyptian	10		13.50	1.798	100	6	15
Greek alphabetic	10		2.70	0.198	90	181	1,000
Western	10		2.89	0.003	0	NA	NA
Inka	10		13.50	1.798	100	6	15
Oberi Okaime	20		2.58	0.002	0	310	1,000
Old Persian	10		14.40	1.797	100	3	9
Roman-additive	10	5	7.50	1.598	200	15	49
Roman-subtractive	10	5	6.00	1.198	400	17	69
Sinhalese	10		3.60	0.197	89	19	271
Sumerian	60	10	11.21	1.788	100	7	21

Results: Cross-Cultural Comparison

Fig. 1 visualizes the conciseness of all expressions from 1 to 1,000 across four systems: the Indo-Arabic systems like Western numerals, which have very little jitter and no inversion; Chinese numerals, with low-moderate jitter and moderate inversion; the Roman numerals (with subtractive notation), with moderate jitter and very high inversion, and the Sumerian numerals, with very high jitter and moderate inversion. Magnitudes where each system is less concise than the corresponding Indo-Arabic notation are visible where these systems dip below the baseline set by the Indo-Arabic line.

Fig. 1.

Conciseness of expressions for 1 to 1,000 in four numerical notations (black: Indo-Arabic; blue: Roman-additive; red: Chinese classical; orange: Sumerian).

Cumulative systems are on average less concise than multiplicative and ciphered ones. However, additive and positional systems are roughly equally concise, other factors being equal. These findings echo those of (1, pp. 390–392). Indo-Arabic systems like our own are quite concise, but are less concise either than a ciphered-additive system (because it lacks a zero) or a ciphered-positional system with a higher base (because it takes longer to reach higher powers that require more signs).

The eight cumulative systems in the sample have relatively high jitter (mean = 1.708) and inversion (mean = 157.142) compared to the five noncumulative systems (mean jitter = 0.123; mean inversion = 53.8). This is because moving from a large quantity of a lower power-sign (nine, in a decimal system) to a single, higher power-sign causes such disjunctures. Ciphered-positional systems have almost zero jitter (because they increase in length only at powers of the base) and zero inversion. Ciphered-additive and multiplicative-additive systems fall between these two extremes, having low but nonzero jitter and a moderate degree of inversion. Positionality plays almost no role in any of this assessment. For instance, the Egyptian cumulative-additive and Inka cumulative-positional systems have identical conciseness (13.501), jitter (1.798), and inversion (100), because both are base-10 and cumulative.

The use of a base other than 10 does matter, but not in a consistent direction. A cumulative-additive base-20 system like the Aztec numerals is less concise than a base-10 one, because signs can be repeated up to 19 times. Conversely, a ciphered-positional base-20 system like Oberi Okaime is more concise than a base-10 one, because numbers from 11-19 and 100-399 use one fewer sign than their decimal counterparts. The Oberi Okaime system is included in this study despite its relative obscurity because it is one of the only ciphered-positional base-20 systems in existence. Sign inventory must surely play some role here–one could in principle develop a system with 100 signs for 0 to 99, or 1,000 signs for 0 to 999, but the difficulty of memorizing such a large body of signs would be exceptional. But there is nothing magical about base-10 other than the “handy” fact that we have two hands with five fingers each. Adding a subbase (like the Roman subbase of 5 in an otherwise base-10 system) improves conciseness (compare VIII to IX) and jitter, but has a much higher inversion because the points of disjuncture where many signs are replaced by one are more numerous.

Results: Diachronic Transformation

Cultural-evolutionary approaches to cognitive technologies allow the analysis of patterns of change in terms of recurrent processes, recognizing that the histories of such phenomena are multilinear. We can examine evolutionary change in numerical notations in two key ways. First, we can examine diachronic transformation within a particular system or phylogeny–that is, the changes in a system over time–in terms of the factors outlined above. Second, we can analyze replacement–instances where a numerical notation is replaced by another (regardless of its phylogenetic relationship between them). To analyze the first case, a system whose historical transformation is well attested, the Roman numerals, is instructive. The Roman numerals survive today as a popular auxiliary notation–for instance, used for paginating the front matter of books, or for enumerating prestige concepts such as popes and Super Bowls. From their origins in ancient Italy to the present, they have an uninterrupted history of 2,500 y of consistent use, and counting. Nonetheless, they have been the subject of much derision both in popular and academic work for their supposed inefficiency (34, pp. 55–90).

One of the features of the Roman numerals often remarked upon is its use of subtractive notation in phrases that use multiples of four and nine of the base (e.g., 14 or 900). The Roman numerical notation began in the early republican period (around 500 BCE) as a cumulative-additive base-10 system with a subbase of 5. Starting around 100 BCE, however, subtractive numeral phrases were introduced. In subtractive numeral phrases, rather than repeating a sign four times (like XIIII or DCCCC), a sign is preceded by the sign immediately lower than it in value to indicate subtraction, as in XIV, 10+ (5–1), or CM (1,000–100). Subtraction was rare in the late republican period, but became more common during the Roman Empire. Subtractive notation never fully replaced additive notation–it is quite typical to find IIII instead of IV on modern Roman numeral clocks (42). Nevertheless, it is by far the most common form of Roman numerals used today.

The obvious advantage of subtractive notation is the conciseness of numeral phrases. As shown above, across all numbers up to 1,000, the average additive Roman numeral is 7.5 signs while the average subtractive one is 6.0 signs long. While neither system uses repetition beyond four of the same sign, subtraction reduces that further, to three repeated signs at most. However, the disadvantage of subtraction is that reordering signs out of their ordinary high-to-low power order is informationally more complex and requires reading order outside the normal left to right (for the Roman script). One of the key advantages of strict power-ordering is that regardless of conciseness, the first sign is the one with the highest value, allowing more rapid estimation of the value of a phrase (43, p. 274). It is easy to see that DCCCC (900) is larger than DCCCI (801)–both have five signs, but the final sign is C rather than I, which is not as clear when comparing CMIII (903) to DCCCI (801). Additionally, the Latin language has very limited subtraction (only for 18, duodeviginti, and 19, undeviginti) so subtraction creates a disjuncture between language and notation which may produce errors in transcoding–similarly to how German speakers have slower response times when doing arithmetic that involves tens and ones, which in German number words are in unit-first order (10, 44). There is an ample tradition of research demonstrating a variety of cognitive effects involved in multidigit numerical processing (45). A recent study in this tradition that has implications for gradience is Feder et al. (46), who artificially alter multidigit expressions so that the stimuli have one digit physically larger than others (e.g., 3₄ vs. ₃4). They demonstrate that in Western numeral expressions, place-value effects on numerical cognition are nearly inseparably intertwined with the effect of the physical size of digits. While gradience is about the size of whole expressions, rather than individual digits, this is a rich area for future research employing a range of notations of different structural principles.

Using the principle of gradience, the two measures of jitter and inversion demonstrate a complex informational balance across additive and subtractive Roman numerals. Fig. 2 compares the conciseness of each numeral in these two systems for the numerals 1 to 50–the same principle would be evident across the full sample from 1 to 1,000 but would be less visually clear. Across all subtractive Roman numerals from 1 to 1,000, 400 are inverted (less concise than their predecessor) whereas only 200 have this property in the additive system. This ratio is straightforwardly explained: additive III – IIII – V has only one inversion (from 4 to 5) whereas subtractive III – IV – V has two inversions. However, using jitter instead of inversion as a measure of gradience, the subtractive system is “smoother,” in that while it has more inversions, they are less dramatic, with a lower mean change in the conciseness of consecutive numeral phrases (1.20) than the additive system (1.60). The subtractive system inverts length more often, but lacks the extreme jumps as happen in the additive system where XXXXVIIII for 49 is followed by L for 50. It is thus somewhat smoother than the additive system, which is characterized by regular large jumps down in conciseness.

Fig. 2.

Conciseness of numeral phrases from 1 to 50 in the purely additive Roman numerals (purple) and the subtractive Roman numerals (green).

While there is no rigid requirement that efficiency always increases in language or notations over time, there are good reasons to expect that, all other things being equal, there are sometimes cultural-evolutionary pressures toward brevity in numeral systems (47). For the Romans there must have been some advantage to making this change. Other cultural-evolutionary changes in numerical notations may have similar explanations, although of course social, historical, and other factors surely also play key roles. Ideally, the study of systems in the abstract would be accompanied by future diachronic analyses of the frequency of these notations using paleographic and epigraphic data from one or more premodern corpora that exhibit numerical change. This research would parallel that of Berg and Neubauer (48) who analyze the morphological transition from unit-and-ten (four-and-twenty) to ten-before-unit (twenty-four) in a corpus of early modern English lexical numerals. The choice of the Roman numerals for this particular comparison is not the only one that could have been made; it has the benefit of familiarity for a broad audience, but represents a single type of structural change, among many that could be examined. Further study examining the gradience of other long-lived families of notations that underwent structural evolution, including both place value and nonplace value systems, is desirable; the East Asian and Mesopotamian phylogenies offer highly promising cases for future examination.

Discussion

The previously identified repertoire of factors on which the informational efficiency of numerical notations has been evaluated is insufficient. Indo-Arabic systems like the common Western system used ubiquitously today are less concise than systems like the Greek alphabetic numerals, have a larger sign inventory than many ancient systems, and are not unique in their infinite extensibility. Gradience (as assessed through both jitter and inversion) varies across numerical notations of different structure, in ways that do not directly follow from these better-known properties like conciseness. However, ciphered-positional systems with a zero (regardless of base) are consistently gradient—their conciseness perfectly predicts their magnitude, with no decreases, and consistent, regular increases at each power of the base. Gradience therefore a more plausible candidate as a cognitive factor underlying the adoption of such systems, to accompany the range of social and historical factors known to be at play (34). Because gradience indexes phrase length to numerical magnitude, it can be thought of as a kind of iconicity facilitating numerical communication and cognition (49).

There are multiple future directions for research into gradience. We do not yet know how important the fact of having monotonically nondecreasing conciseness is, relative to other factors like conciseness alone, either at the level of individual reading and processing of numbers, or at a diachronic scale, predicting the cultural evolution of numerical notations. No user of numerical notation, as far as can be told from existing historical data, has ever specifically noted this feature when comparing notations, so we must be careful in inferring that it was relevant to users’ understandings. Experimental studies using unfamiliar or invented notations having various properties would allow a fuller analysis of the role of these different factors in accurately learning and rapidly responding to stimuli. Similarly, because there is an obvious parallel drawn between the capacity to distinguish 2494923932921 and 24949618 and the ability to estimate the relative magnitude of dot patterns using the ANS, gradience is an evident area for future investigation into how approximate numeration affects numerical cognition of written numerical expressions. Other features, like the use of interdigital commas (e.g. 46000 vs. 46,000) provide additional cues to size by chunking numerals into groups of signs below the subitizing limit of four.

There is also potential for examining corpus data of numerical representations using gradience as an analytic tool. As previously discussed, there is already strong evidence that smaller magnitude and round numbers are more frequent in natural-language corpora, at least in modern English (26). The data from comparative numerical notations here show that these same magnitudes, regardless of notation, use fewer graphemes than higher and nonround ones. By comparing the modern corpus research to studies of ancient corpora (e.g., Babylonian cuneiform, Maya inscriptions, etc.) we could assess the cross-cultural validity of these findings. In turn, this research could be compared to the body of Zipfian analyses of graphic notations such as heraldic signs (50) and writing systems (51). These studies look at the complexity of individual signs, while gradience assesses the complexity of numeral phrases in terms of their length.

Because numerical notation is a cognitive technology (52, 53) it is likely that it has some identifiable properties relevant to information processing. As in the case of measurement systems, though, there is a great risk in assuming that a particular system is efficient for some purpose, and then assuming that that is why it was adopted, transmitted culturally, or replaced other options (54). Body-based measurement may seem old-fashioned or archaic but has clearly identifiable motivations underlying its continued retention or use. Similarly, looking at conciseness, or utility for arithmetic, underlying the decline of the Roman numerals, is not sufficient. Gradience is not intended to be a panacea to the question of why particular notations succeed or not. It does, however, identify a feature that can be added to the explanatory repertoire underlying not only the modern prevalence of Western numerals, but various episodes of cultural evolution in the history of this important phenomenon in numerical cognition.

Materials and Methods

Thirteen of the approximately 110 well-described numerical notations used between 3500 BCE and the present day (1) were selected. Sampling was nonrandom in order to ensure inclusion of systems from a range of geographical regions/continents, language families, phylogenetic families of numerical notations, and structural features within the typology described above in Fig. 1. With the exception of the Oberi Okaime system (a 20th century CE invention in West Africa), all of the systems selected were in active, regular use for 100 or more years. Both additive and subtractive variants of the Roman numerals were included to permit diachronic analysis of the structural effects of the change from the (older) additive to the (newer) subtractive system.

Each selected system was assessed across the range of integral expressions from 1 to 1,000, for the conciseness of each numeral phrase, by a simple count of the number of characters in each expression. These data are included in the Dataset S1. The features of each system were assessed (Table 4). First, the mean conciseness of all the standard expressions from 1 to 1,000 was taken. 1,000 was chosen as the limit because numbers higher than 1,000 are quite rare in both premodern and contemporary corpora (55). Second, the jitter of each system was calculated as the mean difference between each numerical expression and its successor, across the same range. Third, the inversion of each system was calculated as the count of expressions for 1 to 1,000 in the system where the conciseness of N + 1 is less than the conciseness of N. Finally, to address the theory that Indo-Arabic systems are superior because of their conciseness, the count was calculated of expressions from 1 to 1,000 that are more concise, or equally or more concise, than a system of the Indo-Arabic family, such as the common Western numerals.

Supplementary Material

Dataset S01 (XLSX)

Data, Materials, and Software Availability

Previously published data were used for this work (1). Other data are included in the article and/or SI Appendix.