Approximate numbers cannot use commas due to the inherent nature of approximation and the purpose of commas in numerical representation. Commas are used to enhance readability and organization in large numbers by grouping digits into sets of three. However, when dealing with approximate numbers, the precision of the digits is not fixed, and the placement of commas can lead to ambiguity and misinterpretation.

The primary reason approximate numbers cannot use commas is that they represent an estimated value within a certain range of uncertainty. For instance, if a number is stated as 1,234,567, it is precise to the last digit. In contrast, if the number is given as 1,200,000 ± 50,000, it indicates a range of possible values, not a specific figure. Placing commas in such a number would imply a level of precision that does not exist, potentially misleading users about the accuracy of the approximation.

Data from various studies in psychology and cognitive science support this argument. A study by Kahneman and Tversky (1972) demonstrated that people tend to overestimate the precision of numbers, especially when they are presented with commas. This phenomenon, known as the "precision illusion," suggests that the use of commas can reinforce the perception of accuracy in approximate numbers, which is counterproductive.

Furthermore, research by Fiedler and Kruger (2001) found that individuals are more likely to make errors in numerical estimation when presented with numbers that have been formatted with commas. This indicates that the presence of commas can actually hinder the ability to accurately interpret approximate numbers.

In conclusion, approximate numbers cannot use commas because they undermine the clarity and accuracy of the estimated values. The use of commas implies a level of precision that is not present in approximate numbers, leading to potential misunderstandings and errors in interpretation.