Are there more real numbers than natural numbers between 0 and 1? • page 5/5 • Off-Topic Discussion • lichess.org

Yeah there are more real numbers in [0,1] (or any other interval) as explained numerous times

Yeah there are more real numbers in [0,1] (or any other interval) as explained numerous times
also between i (imaginair number) and 2i?

@Cedur216 said in #41: > Yeah there are more real numbers in [0,1] (or any other interval) as explained numerous times also between i (imaginair number) and 2i?

Cedur216

#43

Obviously enough yes (same sort of interval)

Assume that all real numbers between 0 and 1 can be listed. This mens we can create a list where each real number is paired with a natural number. Create a new real number between 0 and 1 that is not on the list. For each number in the list, change the n-th digit after the decimal point. And repea this process. The new number is different from every number on the list. It differs in at least one digit from each number. This contradicts the assumption that the list contains all real numbers between 0 and 1. Thus proming that there are a greater amount of real numbers that natural numbers betweeen 0 and 1.

@KevinWZG Assume that all real numbers between 0 and 1 can be listed. This mens we can create a list where each real number is paired with a natural number. Create a new real number between 0 and 1 that is not on the list. For each number in the list, change the n-th digit after the decimal point. And repea this process. The new number is different from every number on the list. It differs in at least one digit from each number. This contradicts the assumption that the list contains all real numbers between 0 and 1. Thus proming that there are a greater amount of real numbers that natural numbers betweeen 0 and 1.

#45