Why is 3 repeating is a rational number?

Answer 1

The fraction is rational, the repeating decimal is not.  A fraction is a fraction and a repeating decimal is a repeating decimal. They are different representaions of a quantity. One is finite the other is not. It is unreasonable to say a repeating decimal is a fraction and therefore the repeating decimal is rational. The repeating decimal is what it is, albeit derived from a fration, it is not a fration. You can hold 1/3 of a Pizza in your hands but you cannot hold .333333... of a pizza in your hands.  It is literally irrational for you to say that you can.  Theoretically, if you accept that you can always add another decimal place to any decimal number, and that inner space is infinite, without limits, then my point is valid.  .333333... is moving infinitly closer to 1/3 without ever getting there, and the more 3's are piled on, the closer it gets, but is forever approaching 1/3, the same as the universe of decimals never arrives at zero. A pizza the size of .333333... is a work in progress, never terminating at any size at all, but continually sizing itself towards 1/3 forever - quite irrational, possessing a character equal to the famously irrational pi. 1/3 of a pizza is precise, easily sliced, finite, rational, and I'll have mine with sliced tomatoes and a caffeine-free, diet root beer.

Answer 2

0.3333333333... is a rational number because it can be expressed as the ratio of two integers-- 1/3.

This is the definition of a rational number.