Question

Integers are always rational nimbers

Answer 1

In the context of combinatorial game theory, a nimber is a form of an equivalence class used to analyze games like Nim. A nimber can be thought of as an impartial game position and is typically represented as a non-negative integer (0, 1, 2, etc.) or a "position" in a game setup.

Every integer is indeed a rational nimber. The term "rational nimbers" refers to a broader class of numbers representing positions in certain games, where we can think of rational numbers as potential outcomes in complex game scenarios. However, it is important to note that in the traditional sense of nimbers within the framework of combinatorial game theory, the focus is typically on non-negative integers.

In standard game theory terms:

- The nimber 0 represents a losing position (no moves available).
- Nimbers 1, 2, 3, etc., represent winning positions, with each integer representing a unique game state.

Thus, while integers (whole numbers) are categorized as rational numbers, the framework you are asking about specifically refers to the idea of nimbers in games. Therefore, you can say:

- Yes, integers can be regarded as rational nimbers in this game theory context, but the classical nimber notation mainly concerns non-negative integers.

If you have a more specific question about nimbers or need examples, feel free to ask!