People can identify the number of objects in sets of four or fewer items with near-perfect accuracy but exhibit linearly increasing error for larger sets. Some researchers have taken this discontinuity as evidence of two distinct representational systems. Here, we present a mathematical derivation showing that this behaviour is an optimal representation of cardinalities under a limited informational capacity, indicating that this behaviour can emerge from a single system. Our derivation predicts how the amount of information accessible to viewers should influence the perception of quantity for both large and small sets. In a series of four preregistered experiments (N = 100 each), we varied the amount of information accessible to participants in number estimation. We find tight alignment between the model and human performance for both small and large quantities, implicating efficient representation as the common origin behind key phenomena of human and animal numerical cognition.

人们可以以近乎完美的准确度识别四个或更少项目的集合中的对象数量，但对于较大的集合会表现出线性增加的误差。一些研究人员将这种不连续性视为两种不同表征系统的证据。在这里，我们提出了一个数学推导，表明这种行为是有限信息容量下基数的最佳表示，表明这种行为可以从单个系统中出现。我们的推导预测了观众可访问的信息量应如何影响大型和小型集的数量感知。在一系列四个预先注册的实验中（N = 每个 100），我们改变了参与者在数量估计中可获得的信息量。我们发现模型与人类在小批量和大批量的表现之间紧密结合，暗示有效表示是人类和动物数值认知关键现象背后的共同起源。