The numbers that can be represented by 9 dots are 1 through 9. This is because each dot can be used to represent a single unit, and with 9 dots available, one can create any number from 1 to 9. For example, a single dot represents 1, two dots can be arranged to represent 2, and so on, up to 9 dots for the number 9.

The concept of representing numbers with dots is a simple and ancient method of counting and arithmetic. It is often used in educational settings to teach children the basics of numbers and arithmetic operations. The use of 9 dots is particularly useful because it allows for the representation of all single-digit numbers without the need for additional symbols or notation.

This method is based on the principle of combinatorics, which is the study of counting and arranging objects. In the case of 9 dots, the number of unique arrangements that can be made is equal to the number of ways the dots can be arranged, which is 2^9 (since each dot can either be used or not used in any given arrangement). This results in 512 possible combinations, but only 9 of these combinations represent unique numbers (1 through 9).

The simplicity and versatility of this method make it an effective tool for understanding the fundamental concepts of numbers and arithmetic. It is also a testament to the power of basic geometric shapes and their ability to represent abstract concepts. The use of 9 dots to represent numbers is a clear example of how simple tools and ideas can be used to convey complex information.