Short Article: Comparing Two-Digit Numbers: The Importance of being Presented Together

Length is not all that matters: testing the role of number identity and the ratio of fillers in comparisons of multi-digits with different digit length

Research in multi-digit number comparison usually considers stimuli with the same number of digits (e.g., 3452 vs. 7831). Surprisingly, there is almost no research on the comparison of numbers that differ in length (e.g., 995 vs. 1000), which demands a focus on the number of digits in each multi-digit, despite the fact that the role of number length has been explicitly acknowledged in componential models of multi-digit processing. Our study explores whether the comparison of pairs of natural numbers that differ in length is affected by the identity of the leftmost digit of each multi-digit, and asks what is the effect of having variable proportions of trials with pairs of numbers of the same-length in the task. Across three studies participants compared numbers in blocks with different proportions of same-length multi-digit pairs (Experiment 1 and 2: 25% vs. 50% vs. 75%; Experiment 3: 0% vs. 50%). Stimuli in the different-length condition were length-digit congruent (the number with more digits starting with a larger digit: 2384 vs. 107) or length-digit incongruent (the number with more digits starting with a smaller number: 2675 vs. 398). Response times were shorter in length-digit congruent pairs than in the incongruent pairs. Unexpectedly, this effect was only slightly modulated by the proportion of same-/different-length multi-digit pairs in the experimental set. Despite its perceptual saliency, length is not the only information considered when comparing different-length numbers. The leftmost-digit is also taken into account, with variable relevance here, depending on the characteristics of the stimuli set.

Modeling the left digit effect in adult number line estimation

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

Incongruity in fraction verification elicits N270 and P300 ERP effects

Understanding how the numerical magnitudes of fractions are accessed is a topic of major interest in numerical cognition and mathematics education. Only a few studies have investigated fraction processing using EEG methods. In the present study, 24 adult participants completed a fraction magnitude verification task while EEGs were recorded. Similar to other arithmetic verification tasks, behavioral results show increased response times to validate mismatching magnitudes compared to matching ones. ERP results show an early frontal N270 component to mismatching trials and a late parietal P300 component during matching trials. These ERP results highlight that participants treat matching fractions as targets and suggest that additional cognitive resources are needed to process mismatching targets. These results provide evidence that fractions processing shares a similar neurocognitive process as those observed during the processing of arithmetic operations and open the door to further explore fraction processing using ERP methods.

Attention allows the SNARC effect to operate on multiple number lines

To investigate whether participants can activate only one spatially oriented number line at a time or multiple number lines simultaneously, they were asked to solve a unit magnitude comparison task (unit smaller/larger than 5) and a parity judgment task (even/odd) on two-digit numbers. In both these primary tasks, decades were irrelevant. After some of the primary task trials (randomly), participants were asked to additionally solve a secondary task based on the previously presented number. In Experiment 1, they had to decide whether the two-digit number presented for the primary task was larger or smaller than 50. Thus, for the secondary task decades were relevant. In contrast, in Experiment 2, the secondary task was a color judgment task, which means decades were irrelevant. In Experiment 1, decades' and units' magnitudes influenced the spatial association of numbers separately. In contrast, in Experiment 2, only the units were spatially associated with magnitude. It was concluded that multiple number lines (one for units and one for decades) can be activated if attention is focused on multiple, separate magnitude attributes.

The processing of price comparison is not holistic but componential

In three experiments, we evaluate whether the processing of prices is holistic or componential. When participants receive two prices and they select the higher price, distance effects are found when the distances between the two prices are defined holistically but not when they are defined in terms of digits (Experiment 1). This result suggests that prices are processed holistically. However, we show that the holistic distance effect can be explained by the compatibility between the digits and the monetary category of prices (euro and cent). After controlling for the holistic distance, compatible trials (e.g., 8 euro-4 cent, 8 > 4, and euro > cent) are processed faster than incompatible trials (e.g., 8 cent-4 euro, 8 > 4 but cent < euro) with simultaneous and sequential presentation of prices (Experiment 2). Moreover, the compatibility effect is modulated by the ratio of intra monetary category comparisons (8 euro-6 euro) and inter monetary category comparisons (8 euro-4 cents) (Experiment 3). The existence of compatibility effects between the digits and the monetary category of prices suggests that cognitive processing of prices is not holistic but componential.

Spacing and Presentation Modes Affect the Unit-Decade Compatibility Effect During Number Comparison

Abstract. The unit-decade compatibility effect has challenged the model of holistic number magnitude processing, suggesting decomposed processing of multi-digit numbers. Recent evidence confirms that decomposed processing of decade and unit magnitudes occurs in parallel. However, the mode of presentation of multi-digit numbers may affect the processing mode (holistic vs. decomposed, parallel vs. sequential). We therefore investigated in two studies, whether presentation mode (vertical, horizontal, or consecutive) or the distance between two vertically presented numbers affects the unit-decade compatibility effect during number comparison. We found that the compatibility effect did not differ significantly between vertical and horizontal presentation, adding to previous results on perceptual generality, but was nonsignificant with consecutive presentation. However, the compatibility effect was significantly smaller, if numbers are spaced further apart. Thus, stimulus size and distance between numbers affect the processing of multi-digit numbers and should be reported in future studies.

Physical similarity or numerical representation counts in same-different, numerical comparison, physical comparison, and priming tasks?

Recent studies have highlighted the fact that some tasks used to study symbolic number representations are confounded by judgments about physical similarity. Here, we investigated whether the contribution of physical similarity and numerical representation differed in the often-used symbolic same-different, numerical comparison, physical comparison, and priming tasks. Experiment 1 showed that subjective physical similarity was the best predictor of participants' performance in the same-different task, regardless of simultaneous or sequential presentation. Furthermore, the contribution of subjective physical similarity was larger in a simultaneous presentation than in a sequential presentation. Experiment 2 showed that only numerical representation was involved in numerical comparison. Experiment 3 showed that both subjective physical similarity and numerical representation contributed to participants' physical comparison performance. Finally, only numerical representation contributed to participants' performance in a priming task as revealed by Experiment 4. Taken together, the contribution of physical similarity and numerical representation depends on task demands. Performance primarily seems to rely on numerical properties in tasks that require explicit quantitative comparison judgments (physical or numerical), while physical stimulus properties exert an effect in the same-different task.

On the limits of language influences on numerical cognition - no inversion effects in three-digit number magnitude processing in adults

The inversion of number words influences numerical cognition even in seemingly non-verbal tasks, such as Arabic number comparison. However, it is an open question whether inversion of decades and units also influences number processing beyond the two-digit number range. The current study addresses this question by investigating compatibility effects in both German- (a language with inverted) and English-speaking (a language with non-inverted number words) university students (mean age 22 years) in a three-digit number comparison task. We observed reliable hundred-decade as well as hundred-unit compatibility effects for three-digit number comparison. This indicates that, comparable two-digit numbers, three-digit numbers are processed in a parallel decomposed fashion. However, in contrast to previous results on two-digit numbers as well as on children’s processing of three-digit numbers, no reliable modulation of these compatibility effects through language was observed in adults. The present data indicate that inversion-related differences in multi-digit number processing are limited. They seem to be restricted to the number range involving those digits being inverted (i.e., tens and units in two-digit numbers) but do not generalize to neighboring digits. Possible reasons for this lack of generalization are discussed.

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction-decimal mixed pairs and decimal-fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.

Exploring the Boundaries of the Number–Temporal Order Association

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

Number Processing in Arabic and Hebrew Bilinguals

In the current study, a direct assessment of the effect of presentation language and format on the compatibility effect of two-digit numbers was made by contrasting performance of Arabic/Hebrew bilinguals in a digital (Hindi digits/Arabic digits) and verbal numerical comparison task (Arabic an inverted language: units-decades and Hebrew a non-inverted language: decades-units). Our data revealed in digital presentation format a regular compatibility effect in Hindi digits and Arabic digits characterized by lower reaction-time (RT) means for compatible number pairs than incompatible ones with no difference in the RT means of participants in the two languages, Arabic language–Hindi digits as a mother tongue and Hebrew language–Arabic digits as a second language. However, in verbal presentation format, different patterns of compatibility effect were found in Arabic and Hebrew verbal numbers. In Arabic number words, a regular compatibility effect was found, while in Hebrew number words, no compatibility effect was found. This reflects the influence and modulation of the lexical-syntactic structure of the language in two-digit numbers comparison. Evidently, these differences in the compatibility effect advocate and strengthen the claim that two-digit numbers comparison is influenced by the numbers presentation format. Different modes of presentation of two-digit numbers (digital vs. verbal) can lead to different number comparison styles. The parallel model accounts for the numerical comparison in digital presentation, while for the verbal numbers presentation, a revised sequential-syntactic model is preferable.

Flexible and unique representations of two-digit decimals

We examined the representation of two-digit decimals through studying distance and compatibility effects in magnitude comparison tasks in four experiments. Using number pairs with different leftmost digits, we found both the second digit distance effect and compatibility effect with two-digit integers but only the second digit distance effect with two-digit pure decimals. This suggests that both integers and pure decimals are processed in a compositional manner. In contrast, neither the second digit distance effect nor the compatibility effect was observed in two-digit mixed decimals, thereby showing no evidence for compositional processing of two-digit mixed decimals. However, when the relevance of the rightmost digit processing was increased by adding some decimals pairs with the same leftmost digits, both pure and mixed decimals produced the compatibility effect. Overall, results suggest that the processing of decimals is flexible and depends on the relevance of unique digit positions. This processing mode is different from integer analysis in that two-digit mixed decimals demonstrate parallel compositional processing only when the rightmost digit is relevant. Findings suggest that people probably do not represent decimals by simply ignoring the decimal point and converting them to natural numbers.

Language influences on numerical development-Inversion effects on multi-digit number processing

In early numerical development, children have to become familiar with the Arabic number system and its place-value structure. The present review summarizes and discusses evidence for language influences on the acquisition of the highly transparent structuring principles of digital-Arabic digits by means of its moderation through the transparency of the respective language's number word system. In particular, the so-called inversion property (i.e., 24 named as “four and twenty” instead of “twenty four”) was found to influence number processing in children not only in verbal but also in non-verbal numerical tasks. Additionally, there is first evidence suggesting that inversion-related difficulties may influence numerical processing longitudinally. Generally, language-specific influences in children's numerical development are most pronounced for multi-digit numbers. Yet, there is currently only one study on three-digit number processing for German-speaking children. A direct comparison of additional new data from Italian-speaking children further corroborates the assumption that language impacts on cognitive (number) processing as inversion-related interference was found most pronounced for German-speaking children. In sum, we conclude that numerical development may not be language-specific but seems to be moderated by language.

Are 1/2 and 0.5 represented in the same way?

Adults' processing of unit and decimal fractions was investigated using the numerical comparison task. When unit fractions were compared to integers, the pattern of distance effect found suggests that they were perceived to be on the same mental number line as integers; however, their representation was undifferentiated, as they were perceived to have the same magnitude. This was found both with simultaneous and with sequential presentation. When decimal fractions were compared to integers, the pattern of results suggests that they were also represented on the same mental number line with integers, but their representation was differentiated. Possible explanations for the different patterns found for unit and decimal fractions are discussed. Moreover, compatibility between the magnitude of the whole fraction and that of its components relative to the compared integer affected performance in the case of decimal fractions and unit fractions presented simultaneously, but not in the case of unit fractions presented sequentially. This suggests that sequential processing reduces the components representation of fractions and the whole number bias.

Magnitude representation in sequential comparison of two-digit numbers is not holistic either

There is accumulating evidence suggesting that two-digit number magnitude is represented in a decomposed fashion into tens and units rather than holistically as one integrated entity. However, recently, it has been claimed that this property does not hold for the case when two to-be-compared numbers are presented sequentially. In the present study, we pursued this issue in two experiments by evaluating perceptual as well as strategic aspects arising for sequential stimulus presentation in a magnitude comparison task. We observed reliable unit-decade compatibility effects indicating decomposed processing of tens and units in a magnitude comparison task with sequential presentation of the to-be-compared numbers. In particular, we found that both confounding low-level perceptual features and stimulus set characteristics determining cue validity of the units influenced the compatibility effect. Taken together, our results clearly indicate that decomposed representations of tens and units seem to be a general characteristic of multi-digit number magnitude processing, rather than an exception occurring under very specific conditions only. Implications of these results for the understanding of number magnitude representations are discussed.

"On the money" - monetary and numerical judgments of currency

Numerical and monetary judgments of currency were examined using two tasks-a monetary value task (which coin has a higher monetary value), and a numerical value task (which coin has a higher numerical value). In Experiment 1 participants were presented with pictures of coins of the Israeli currency. The Israeli currency is the shekel, which is composed of 100 agorot (equivalent to a dollar composed of 100 cents). Higher discriminability between shekels compared to agorot due to importance in everyday life was reflected in faster monetary comparisons of shekel pairs compared to agorot pairs. Automatic processing of numerical value was demonstrated for monetary judgments. When presented with pairs composed of one coin from each monetary category, responses were faster to pairs that were monetary-numerical congruent (e.g., 10 shekels vs. 5 agorot) compared to incongruent (e.g., 5 shekels vs. 10 agorot). Numerical value judgments were unaffected by such congruency. There was evidence for the automatic activation of physical size mainly in the numerical task. A similar picture was obtained in Experiment 2 where instead of pictures of coins we used verbal descriptions of currency, demonstrating the generality of our results.

Empirical and theoretical studies on number comparison: design parameters and research questions

For well over one-hundred years, several key factors have been well established in the study of number comparison, including mental number line, numerical distance effect, and effect of sensory representation on number processing. The purpose of this article is to put some of these studies together to discuss design parameters and research questions addressed in the mental number comparison studies. Most of the studies discuss sensory representation and abstract number representation as well as degree of their interaction. In order to give the different views on a particular research question, the author classified studies under the related research questions. For example, Stroop and size congruity effect studies are addressed under this title chronologically. It was very clear that the design parameters and research question might change the interpretation of a task. It may be time to shift attention from the question of the interaction degree of sensory representation and abstract representation to a larger scope. The larger scope would be to understand the differences and similarities between different groups using a universal approach.

Multi-digit number processing beyond the two-digit number range: a combination of sequential and parallel processes

Investigations of multi-digit number processing typically focus on two-digit numbers. Here, we aim to investigate the generality of results from two-digit numbers for four- and six-digit numbers. Previous studies on two-digit numbers mostly suggested a parallel processing of tens and units. In contrast, the few studies examining the processing of larger numbers suggest sequential processing of the individual constituting digits. In this study, we combined the methodological approaches of studies implying either parallel or sequential processing. Participants completed a number magnitude comparison task on two-, four-, and six-digit numbers including unit-decade compatible and incompatible differing digit pairs (e.g., 32_47, 3<4 and 2<7 vs. 37_52, 3<5 but 7>2, respectively) at all possible digit positions. Response latencies and fixation behavior indicated that sequential and parallel decomposition is not exclusive in multi-digit number processing. Instead, our results clearly suggested that sequential and parallel processing strategies seem to be combined when processing multi-digit numbers beyond the two-digit number range. To account for the results, we propose a chunking hypothesis claiming that multi-digit numbers are separated into chunks of shorter digit strings. While the different chunks are processed sequentially digits within these chunks are processed in parallel.

Working memory and two-digit number processing

The processing of two-digit numbers in comparison tasks involves the activation and manipulation of magnitude information to decide which number is larger. The present study explored the role of different working memory (WM) components and skills in the processing of two-digit numbers by examining the unit-decade compatibility effect with Arabic digits and number words. In the study, the unit-decade compatibility effect and different WM components were evaluated. The results indicated that the unit-decade compatibility effect was associated to specific WM skills depending on the number format (Arabic digits and number words). We discussed the implications of these results for the decomposed view of two-digit numbers.

Across-notation automatic processing of two-digit numbers

The existence of across-notation automatic numerical processing of two-digit (2D) numbers was explored using size comparisons tasks. Participants were Arabic speakers, who use two sets of numerical symbols—Arabic and Indian. They were presented with pairs of 2D numbers in the same or in mixed notations. Responses for a numerical comparison task were affected by decade difference and unit-decade compatibility and global distance in both conditions, extending previous findings with Arabic digits (Nuerk, Weger, & Willmes, 2001). Responses for a physical comparison task were affected by congruency with the numerical size, as indicated by the size congruency effect (SiCE). The SiCE was affected by unit-decade compatibility but not by global distance, thus suggesting that the units and decades digits of the 2D numbers, but not the whole number value were automatically translated into a common representation of magnitude. The presence of similar results for same- and mixed-notation pairs supports the idea of an abstract representation of magnitude.

The Processing of Two-Digit Numbers Depends on Task Instructions

Recently, a lot of research has focused on resolving whether two-digit numbers are processed holistically or compositionally. This has led to inconsistent results. In the present study we investigated effects of task instructions. Subjects performed magnitude or parity judgments on targets preceded by masked primes containing parts of the target at a task-congruent (3#_37) or task-incongruent (#3_37) position. Priming effects were influenced by the instructions: In the magnitude task, the priming effects were primarily mediated by the congruency of the decade digit, whereas in the parity task they were elicited by the congruency of the unit digit, which is in line with a flexible compositional processing style. These and previous findings show that two-digit numbers can be processed in a very flexible way, depending on the task context.

Extending the Mental Number Line

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

Cognitive control in number processing: evidence from the unit-decade compatibility effect

In this study we evaluated whether the reverse compatibility effect observed when participants compare two-digit Spanish number words might be modulated as a function of the percentage of filler trials (within-decade comparisons). The participants performed a comparison task with two-digit Spanish number words while the unit-decade compatibility in between decade comparison trials was manipulated. We biased the relevance of units by increasing the intra/inter decade comparison ratio (20%, 50%, and 70%). The compatibility effect shifted from reverse to regular depending on the unit relevance which indicates that the relative emphasis on the unit processing in two-digit number word can be modulated by the stimulus list. These results are discussed in terms of cognitive control in number processing.

Sequential Processing of Two-Digit Numbers

Two experiments using a number matching task (NMT) explored whether two-digit numbers are processed holistically or in a compositional fashion. In the NMT participants are required to decide whether one of the two numbers initially provided (cues) is presented some milliseconds later or not (probe). Probes which have some arithmetic relationship to the cues (e.g., cues: 2 3, probe: 6) are rejected more slowy than probes unrelated to their cues (e.g., cues: 2 3, probe: 7) – interference effect –, and this is considered as evidence of the automatic activation of that arithmetic relationship. Participants were presented with two-digit cues and probes which had an arithmetic progression relationship only detectable once the numbers were decomposed (Experiment 1: cues: 56 7, probe: 89; Experiment 2: cues: 45 67, probe: 89). Results showed longer response times in these conditions compared to unrelated conditions. Data support componential processing even when the numbers to be matched are presented serially.

Digit Repetition Effect in Two-Digit Number Comparison

Previous data from two-digit number naming show that when target and prime numbers share one digit at different positions (e.g., target 28 and prime 86 or 72) an inhibition effect may be observed (Ratinckx, Brysbaert, & Fias, 2005). Such an effect has been ascribed to the mechanism of morpho-phonological transcoding from Arabic to verbal format. We evaluate the alternative hypothesis of an inhibition effect arising during the Arabic form processing when two different syntactic values have to be assigned to the same single digit. In Experiments 1 and 3, a digit repetition effect was observed in number comparison tasks, even when phonological transcoding is blocked by an articulatory suppression task. Conversely, Experiment 2 showed that no digit repetition effect can be found with number comparison of verbal written numbers. Results are compatible with an Arabic-syntactic processing stage, where units and decades are decomposed and where each single digit is activated and recognized.

Holistic Representation of Negative Numbers is Formed When Needed for the Task

Past research suggested that negative numbers are represented in terms of their components—the polarity marker and the number (e.g., Fischer & Rottmann, 2005; Ganor-Stern & Tzelgov, 2008). The present study shows that a holistic representation is formed when needed for the task requirement. Specifically, performing the numerical comparison task on positive and negative numbers presented sequentially required participants to hold both the polarity and the number magnitude in memory. Such a condition resulted in a holistic representation of negative numbers, as indicated by the distance and semantic congruity effects. This holistic representation was added to the initial components representation, thus producing a hybrid holistic-components representation.

Two-digit number processing: holistic, decomposed or hybrid? A computational modelling approach

Currently, there are three competing theoretical accounts concerning the nature of two-digit number magnitude representation: a holistic, a strictly decomposed, and a hybrid model. Observation of the unit-decade compatibility effect (Nuerk et al. in Cognition 82:B25-B33, 2001) challenged the view of two-digit number magnitude to be represented as one integrated entity. However, at the moment there is no study distinguishing between the decomposed and the hybrid model. The present study addressed this issue using a computational modelling approach. Three network models complying with the constraints of all three theoretical models were programmed and trained on two-digit number comparison. Models were compared as to how well they accounted for empirical effects in the most parsimonious way. Generally, this evaluation indicated that the empirical data were simulated best by the strictly decomposed model. Implications of these results for our understanding of the nature of human number magnitude representation are discussed.