In summary, depending on the intended interpretation, the number 1.00‾1.001 can be either a mixed repeating decimal (rational) or an irrational number based on the nature and length of the repeating block.
While the overline suggests a repeating decimal, a closer look reveals that the repeating block contains two different digits ("00" and "1"). This makes it a different type of number.
Here are two possibilities, depending on the intended interpretation:
1. Mixed repeating decimal: If the number is meant to represent an infinitely repeating block of "001," then it can be classified as a mixed repeating decimal. This combines a non-repeating decimal part (1.00) with a repeating block that contains more than one digit (001). Mixed repeating decimals are still considered rational numbers as they can be expressed as simplified fractions.
2. Irrational number: If the number is meant to represent an infinitely alternating sequence of "00" and "1," then it is likely an irrational number. Irrational numbers cannot be expressed as simplified fractions and have decimal expansions that continue infinitely without repeating or settling into a predictable pattern. This interpretation aligns with the alternating sequence of digits, although additional context or information about the specific number and its origin might be needed for a definitive conclusion.