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

Abstract

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.

Perceiving the “smallest” or “largest” multi-digit number: A novel numeric-scale end effect

“…That is, responses were faster when comparing compatible pairs (i.e., the larger/longer number contained the larger left digit; e.g., 4,000,000,000 vs. 2,000,000) than for incompatible pairs (i.e., the larger/longer number contained the smaller left digit; e.g., 2,000,000,000 vs. 4,000,000). Similar findings were obtained by García-Orza et al (2022) when testing comparisons of 3-digit versus 4-digit numbers that varied in their digit values and lengths (e.g., 2384 vs. 107; 2675 vs. 398). Together, the findings of both studies indicate that participants processed number lengths, as well as left-digits, when comparing number magnitudes.…”

“…Namely, end effects in multi-digit number comparisons emerged only when the compared numbers differed in their lengths. However, we believe that differences in the numbers' lengths alone do not tell the whole story (see also, García-Orza et al, 2022;Lozin & Pinhas, 2022) because similar effect sizes were obtained for comparisons to an end-value that differed from the non-end-values by two or more scales; that is, the end-value differed in its length. Such a pattern is not expected if differences in number length alone guide the comparative decision-making.…”

“…This place-value structure links number values with certain positions in a string of digits and associates the overall number length with its magnitude; the more digits there are, the larger the value of the number. Thus, the number's length serves as a primary cue for a crude evaluation of its value (e.g., Dixon, 1978;García-Orza et al, 2022;Hinrichs et al, 1982;Huber, et al, 2014;Lozin & Pinhas, 2022;Varma & Karl, 2013). Some models postulate that a number's length and other syntactic and lexical aspects are encoded during early visual analysis of multi-digit number stimuli.…”

“…It should be considered that magnitude comparison in such a context, namely, comparing a "longer" number to a "shorter" one, could rely (at least in part) on the numbers' lengths (e.g., Dixon, 1978;García-Orza et al, 2022;Hinrichs et al, 1982;Huber, et al, 2014;Lozin & Pinhas, 2022;Varma & Karl, 2013). If end-values are detected based on numerical syntax alone (i.e., identified as such based on a rough visual analysis of the numbers' lengths) then end effects may occur without fully realizing the exact value each of the compared numbers represents.…”

Adults systematically underestimate decimals and whole number exposure induces further magnitude-based underestimation.

“…Notably, adding zero to previously incongruent stimuli removes this interference effect (e.g., 0.90 vs. 0.23 resulted in faster response time), suggesting that activating the appropriate magnitude (0.90 = >90) resolves the interference (Coulanges et al, 2021; Varma & Karl, 2013). Further evidence that string length is not the only factor in mismatched length comparison comes from recent research in multidigit comparison which finds that both the string length and the left-digit magnitude simultaneously impact large whole number processing (García-Orza et al, 2023; Lozin & Pinhas, 2022). For example, Lozin & Pinhas (2022) found that adults take longer when comparing pairs of numbers whose left digits are incongruent with the longer string length (e.g., 2,000 vs. 500) than those whose left digits are congruent with the longer string length (e.g., 5,000 vs. 200).…”