Any real number whose decimal representation terminates or repeats a certain pattern indefinitely is a rational number.
If #0.33333# is intended as just that, i.e. the terminating decimal #0.33333000...#, then we can just multiply and divide by a power of #10# to find its fractional representation.
#0.33333 = (0.33333xx10^5)/10^5 = 33333/100000#
If it is intended as #0.33333... = 0.bar(3)#, that is, an unending string of #3"'s"#, then we can divide the repeating portion by #10^k-1#, where #k# is the number of digits the repeating portion. As #3# is what is repeating, and has a single digit, we have
#0.33333... = 3/(10^1-1) = 3/9 = 1/3#
In either case, we can represent the given number as a ratio of two integers, meaning it is a rational number.
