Are all terminated in repeating decimals rational numbers and why?

Answer 1

Such numbers can be written in the form of a ratio of two integers. In the first case as a ratio of an integer and a power of ten, and in the second case as a ratio of an integer and one less than a power of ten. Since the denominators in both cases are non-zero integers, these are rational numbers.

Examples:

0.356 = 356/1000

0.356356356... (repeating) = 356/999

In the first case the number of zeros after 1 is the number of digits after the decimal point. In the second case, the number of 9s is one fewer than the length of the repeating string.