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Exponential expressions represent series that grow at a fast pace such as carbon pollution and the spread of disease. Despite their importance, people tend to struggle with these expressions. In two experiments, participants chose the larger of two exponential expressions as quickly and accurately as possible. We manipulated the distance between the base/power components and their compatibility. In base-power compatible pairs, both the base and power of one expression were larger than the other (e.g., 23 vs. 34), while in base-power incompatible pairs, the base of one expression was larger than the base in the other expression but the relation between the power components of the two expressions was reversed (e.g., 32 vs. 24). Moreover, while in the first experiment the larger power always led to the larger result, in the second experiment we introduced base-result congruent pairs as well. Namely, the larger base led to the larger result. Our results showed a base-power compatibility effect, which was also larger for larger power distances (Experiments 1-2). Furthermore, participants processed the base-result congruent pairs faster and more accurately than the power-result congruent pairs (Experiment 2). These findings suggest that while both the base and power components are processed when comparing exponential expressions, the base is more salient. This exemplifies an incorrect processing of the syntax of exponential expressions, where the power typically has a larger mathematical contribution to the result of the expression.

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Previous research has shown that multi-digit number processing is modulated by both place-value and physical size of the digits. By pitting place-value against physical size, the present study examined whether one of the attributes had a greater impact on the automatic processing of multi-digit numbers. In three experiments, participants were presented with two-digit number pairs that appeared in frames. They were instructed to select the larger frame while ignoring the numbers within the frames. Importantly, we manipulated the physical size of the digits (i.e., both decade/unit digits were physically larger) within the frames, the unit-decade compatibility (i.e., the relationship between the numerical values of both decade and unit digits was consistent or inconsistent), and the congruity between the numerical values of the decade digits and the frames' physical size (i.e., decade-value-frame-size congruity). In Experiment 1, where all pairs were unit-decade compatible, a decade-value-frame-size congruity effect emerged for pairs with physically larger decade, but not unit, digits. However, when adding unit-decade incompatible pairs (Experiments 2-3), in unit-decade compatible pairs, there was a decade-value-frame-size congruity effect regardless of the digits' physical size. In contrast, in unit-decade incompatible pairs, there was no decade-value-frame-size congruity effect, even when the physically larger digit (i.e., unit) contradicted the place-value information, presumably due to the cancellation of the opposing influences of the digits' physical sizes their place-values. Overall, these findings suggest that place-value and physical size are intertwined in the Hindu-Arabic numerical system and are processed as one.

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Our mental representation of the infinite has received little research attention in cognitive psychology. In countably infinite sets, the infinity symbol (∞) is presumed to be perceived as larger than any finite natural number. The present study sought to explore if the infinity symbol is processed as "larger than" natural numbers, and, if so, whether it is associated with the special status of "the largest." In a series of four experiments (N = 40, 20, 20, and 40, respectively), participants performed numerical and physical comparisons of the infinity symbol against single- and multidigit numbers. Overall, numerical comparisons yielded slower responses for comparisons between infinity and a number than for comparisons between two numbers. Furthermore, distance-like effects were obtained for comparisons to infinity, suggesting the infinity symbol was treated as larger than all numbers presented. Importantly, however, physical comparisons revealed a normal size congruity effect for comparisons of infinity and single digits, but a reversed effect for comparisons of infinity and multidigit numbers, suggesting that the infinity symbol was automatically processed as smaller than multidigit numbers. These novel findings reveal limitations in abstracting the meanings of infinity from its symbol, indicating that the infinity symbol is not perceived as "the largest" and can be misconceived as a "number" mapped onto the numerical magnitude system. More generally, the results seem to reflect a crude, automatic evaluation of numerical magnitude based on the physical magnitude of the stimuli, namely, their overall length and the number of symbols of which they are comprised. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

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BACKGROUND: A hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers. METHOD: ERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task. RESULTS: Only children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks. CONCLUSION: Results demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.

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Post-Traumatic Stress Disorder (PTSD) research indicates that hyper-reactivity to trauma-related stimuli reflects reduced prefrontal cortex (PFC) modulation of amygdala reactivity. However, other studies indicate a dissociative "shutdown" reaction to overwhelming aversive stimuli, possibly reflecting PFC over-modulation. To explore this, we used an Event-Related Potential (ERP) oddball paradigm to study P3 responses in the presence of the following: 1. Trauma-unrelated morbid distractors (e.g., "injured bear") related to the Rorschach inkblot test, and 2. Negative distractors (e.g., "significant failure"), among participants with high post-traumatic stress symptoms (PTS; n = 20), low PTS (n = 17), and controls (n = 15). Distractors were presented at 20% frequency amongst the more frequent (60%) neutral standard stimuli (e.g., "desk lamp") and the equally frequent (20%) neutral trauma-unrelated target stimulus ("golden fish"). P3 amplitudes were high in the presence of morbid distractors and low in the presence of negative distractors only amongst the control group. Possible mechanisms underlying the lack of P3 amplitude modulation after trauma are discussed.

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Little is known about the mental representation of large multidigit numbers that are usually beyond our personal experience. The present study explored the processing mechanisms of these numbers in a series of experiments, using the numerical comparison task. Experiment 1 included within and between-scale comparisons of multidigit numbers varying in their left digits (e.g., 8,000,000), with one group comparing small numbers (tens, hundreds, and thousands) and the other large ones (millions, billions, and trillions). In Experiment 2, comparisons of small (tens, hundreds) and large (millions, billions) multidigit numbers that varied in their left and right digits (e.g., 8,000,003) were presented in separate blocks. Experiment 3 presented small and large multidigit numbers (from tens to trillions) that varied in their left digits in the same block. We found novel compatibility effects between the left digit and scale components, as well as between the left digit, right digit, and scale components, and extended the previously reported unit-decade compatibility effect to larger scales. We also obtained global and scale distance effects for all scales in most conditions. Both compatibility and distance effects showed context dependency in large, but not small, multidigit numbers. Overall, these results demonstrate that small and large multidigit numbers are processed differently. We discuss these differences and propose a processing model that accounts for them. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

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Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for "different" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1-9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

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Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In this study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as "zero," but rather as "nothing," when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

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We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set - the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

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Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers, specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.

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The question whether human beings process empty sets as zero has received little research attention. In this study, we used the distance and end effects as indicators for treating empty sets as a numerical entity that represents an absence of quantity. In a series of experiments, participants performed a magnitude comparison task. They were presented with empty sets and other numerosities from 1 to 9, presented as dot arrays. We manipulated task instructions relevant to the target (i.e., "choose the target that contains more/less dots" in Experiment 1) or the given numerical range mentioned in the instructions (i.e., 0-9 or 1-9 in Experiment 2) to create conditions in which an empty set would be perceived as the smallest value of the experimental numerical range. The results revealed distance effects for comparisons to empty sets, irrespective of task instructions. In Experiment 3, we manipulated the response mode. Two groups of participants responded to target location, one group with a key-press and the other vocally, while the third group responded vocally to target color. The results revealed distance effects for comparisons to empty sets only when responding to target location, regardless of the response mode, indicating that spatial features should be primed in order to perceive an empty set as a numerical entity. These findings show that perceiving an empty set as nothing or as zero depends on the context in which it is presented.

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BACKGROUND: Event-Related Potential (ERP) studies of PTSD have reported enhanced P3 amplitudes in response to trauma-related stimuli that are less likely to habituate over time. METHODS: In the present study, we compared ERPs to the first and last half of an auditory novelty oddball task using neutral (trauma-unrelated) stimuli. Participants were 59 young students who were: trauma-exposed with "Probable PTSD", trauma-exposed without PTSD, or non-traumatized controls. RESULTS: Reduced P3 amplitudes were observed for the last half of the trials for the entire sample, but this habituation was less profound for both trauma-exposed groups, demonstrating reduced habituation over time. Arousal symptom severity and trauma history negatively correlated with P3 amplitude habituation across the entire sample. Reduced N1 amplitudes for the last half of the trials were found in both trauma-exposed groups, but not among controls. CONCLUSIONS: Our findings suggest that trauma-exposed individuals exhibit information processing alterations in response to neutral environmental stimuli that may be related to a general pattern of heightened activity of the Salience Network. Implications for the neurobiological model of PTSD and PTSD psychotherapy are discussed.

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The brain activity associated with processing numerical end values has received limited research attention. The present study explored the neural correlates associated with processing semantic end values under conditions of automatic number processing. Event-related potentials (ERPs) were recorded while participants performed the numerical Stroop task, in which they were asked to compare the physical size of pairs of numbers, while ignoring their numerical values. The smallest end value in the set, which is a task irrelevant factor, was manipulated between participant groups. We focused on the processing of the lower end values of 0 and 1 because these numbers were found to be automatically tagged as the "smallest." Behavioral results showed that the size congruity effect was modulated by the presence of the smallest end value in the pair. ERP data revealed a spatially extended centro-parieto-occipital P3 that was enhanced for congruent versus incongruent trials. Importantly, over centro-parietal sites, the P3 congruity effect (congruent minus incongruent) was larger for pairs containing the smallest end value than for pairs containing non-smallest values. These differences in the congruency effect were localized to the precuneus. The presence of an end value within the pair also modulated P3 latency. Our results provide the first neural evidence for the encoding of numerical end values. They further demonstrate that the use of end values as anchors is a primary aspect of processing symbolic numerical information.

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Number processing evokes spatial biases, both when dealing with single digits and in more complex mental calculations. Here we investigated whether these two biases have a common origin, by examining their flexibility. Participants pointed to the locations of arithmetic results on a visually presented line with an inverted, right-to-left number arrangement. We found directionally opposite spatial biases for mental arithmetic and for a parity task administered both before and after the arithmetic task. We discuss implications of this dissociation in our results for the task-dependent cognitive representation of numbers.

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Little is known about the neural underpinnings of number word comprehension in young children. Here we investigated the neural processing of these words during the crucial developmental window in which children learn their meanings and asked whether such processing relies on the Approximate Number System. ERPs were recorded as 3- to 5-year-old children heard the words one, two, three, or six while looking at pictures of 1, 2, 3, or 6 objects. The auditory number word was incongruent with the number of visual objects on half the trials and congruent on the other half. Children's number word comprehension predicted their ERP incongruency effects. Specifically, children with the least number word knowledge did not show any ERP incongruency effects, whereas those with intermediate and high number word knowledge showed an enhanced, negative polarity incongruency response (N(inc)) over centroparietal sites from 200 to 500 msec after the number word onset. This negativity was followed by an enhanced, positive polarity incongruency effect (P(inc)) that emerged bilaterally over parietal sites at about 700 msec. Moreover, children with the most number word knowledge showed ratio dependence in the P(inc) (larger for greater compared with smaller numerical mismatches), a hallmark of the Approximate Number System. Importantly, a similar modulation of the P(inc) from 700 to 800 msec was found in children with intermediate number word knowledge. These results provide the first neural correlates of spoken number word comprehension in preschoolers and are consistent with the view that children map number words onto approximate number representations before they fully master the verbal count list.

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Mental arithmetic shows systematic spatial biases. The association between numbers and space is well documented, but it is unknown whether arithmetic operation signs also have spatial associations and whether or not they contribute to spatial biases found in arithmetic. Adult participants classified plus and minus signs with left and right button presses under two counterbalanced response rules. Results from two experiments showed that spatially congruent responses (i.e., right-side responses for the plus sign and left-side responses for the minus sign) were responded to faster than spatially incongruent ones (i.e., left-side responses for the plus sign and right-side responses for the minus sign). We also report correlations between this novel operation sign spatial association (OSSA) effect and other spatial biases in number processing. In a control experiment with no explicit processing requirements for the operation signs there were no sign-related spatial biases. Overall, the results suggest that (a) arithmetic operation signs can evoke spatial associations (OSSA), (b) experience with arithmetic operations probably underlies the OSSA, and (c) the OSSA only partially contributes to spatial biases in arithmetic.

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The representation of numbers is commonly viewed as an ordered continuum of magnitudes, referred to as the mental number line. Previous work has repeatedly shown that number representations evoked by a given task can be easily altered, yielding an ongoing discussion about the basic properties of the mental number line and how malleable they are. Here we studied whether the resolution of the mental number line is fixed or depends on the relative magnitudes that are being processed. In 2 experiments, participants compared the same number pairs under 2 conditions that differed in terms of the overall range of numbers present. We report a novel number range effect, such that comparisons of the same number pairs were responded to faster under the smaller versus larger number range. This finding is consistent with the idea that the resolution of the mental number line can be adjusted, as if a unit difference is perceived as larger in smaller ranges.

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The representation of 0 in healthy adults was studied with the physical comparison task. Automatic processing of numbers, as indicated by the size congruity effect, was used for detecting the basic numerical representations stored in long-term memory. The size congruity effect usually increases with numerical distance between the physically compared numbers. This increase was attenuated for comparisons to 0 or 1 (but not to 2) when they were perceived as the smallest number in the set. Furthermore, the size congruity effect was enlarged in these cases. These results indicate an end effect in automatic processing of numbers and suggest that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line. The implications of these findings are discussed with regard to models of number representation.

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Research in cognitive science has documented numerous phenomena that are approximated by linear relationships. In the domain of numerical cognition, the use of linear regression for estimating linear effects (e.g., distance and SNARC effects) became common following Fias, Brysbaert, Geypens, and d'Ydewalle's (1996) study on the SNARC effect. While their work has become the model for analyzing linear effects in the field, it requires statistical analysis of individual participants and does not provide measures of the proportions of variability accounted for (cf. Lorch & Myers, 1990). In the present methodological note, using both the distance and SNARC effects as examples, we demonstrate how linear effects can be estimated in a simple way within the framework of repeated measures analysis of variance. This method allows for estimating effect sizes in terms of both slope and proportions of variability accounted for. Finally, we show that our method can easily be extended to estimate linear interaction effects, not just linear effects calculated as main effects.

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To address the ongoing debate about the origins of the size effect (faster comparison time for smaller than larger numbers, given a fixed intrapair distance), an indication of automatic number processing was searched for. Participants performed a quantity comparison task in which they had to decide which of two sketched cups contained more liquid, while ignoring the number superimposed on each cup. In the congruent condition, the larger number appeared on the cup containing more liquid, while in the incongruent condition the larger number appeared on the cup containing less liquid. The size effect was found in a numerical comparison task, while in the quantity comparison task the size congruity effect decreased as the magnitude of the irrelevant numbers increased. Together, these results suggest that the size effect reflects a basic feature of the mental number line.

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