With the cumulative model, we get an added benefit: it yields precise hypotheses about the mental representation of each number.
For instance, 24 is most likely represented as 4×6, but also as 2×12, 3×8, or the successor of 23—in that order.

Our model was actually missing a crucial ingredient: approximation.
Locutors experience a tradeoff between accuracy of the expressed quantity and length of the expression. Adding a parameter to account for this accurately models the whole curve.

Fitting our models to the data, we find that both models accurately captured the downward trend as well as some local peaks, yet it failed to do so for decade numbers. Why?

We developed two models to explain how often a number appears:
- Shortest-path: only the simplest construction determines frequency
- Cumulative: all valid constructions contribute to frequency, weighted by their complexity
Here's how:

Building on this, we tested a new cognitive model based on a Language of Thought (LoT) introduced by Dehaene et al., 2025.
The idea: number concepts are constructed in the mind using simple building blocks—1, +, and ×.

Using the Google N-grams database, we looked at the frequency of number words from 1 to 99 in 6 languages.
Across the board, we found two patterns:
🔻 Frequency drops with size
📈 Local spikes at certain numbers—especially round ones and those with many small divisors