Place-value and physical size converge in automatic processing of multi-digit numbers.

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.

No power: exponential expressions are not processed automatically as such.

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.

Numerical comparisons of exponential expressions: The saliency of the base component.

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.