How do systems of measurement influence our conceptualization of relative magnitudes? This study investigates the cognitive interplay between measurement precision and magnitude categorization. By employing morphed shapes organized by an arbitrary dimension, we examine whether exposure to high- vs low-precision numerical systems affects conceptual judgments and well-known phenomena such as semantic distance and semantic congruity effects as found for familiar dimensions. Participants trained on novel scales revealed differences in their sensitivity that depended on the precision of the trained measurement system, consistent with high-precision systems leading to relatively expanded dimensional encodings compared to low-precision systems. Our findings also shed light on other topics such as the interplay of perception and language in learning novel dimensions and the association of directionality with a mental number line.