The cycle of 0.3 can be regarded as a cycle with countless 3' s after the decimal point. No matter how many threes there are, we can't turn it into four. Similarly, if we divide 1 into three points, we can't use four to represent the last extra one, so we have to write a cycle of 0.3.

Looking at the answers of the brothers downstairs (sumeragi693), I suddenly feel that this is a question about infinite expressions. Because we can't use a finite decimal to represent 1/3, the cycle of 0.3 was born to represent the vacancy of fractional representation. Imagine, if it is 1/2, infinity multiplied by ten or infinity, how will these disputes arise? Because we can't explain how many digits there are in the cycle of 0.3 at all, because 0.3 is less than 1/3, so we use 0.33, because 0.33 is less than 1/3, so we use 0.333 ... Because we can't really use what we know to represent the size of the fraction, there is a cycle of 0.3.

Imagine that 0.3 is less than 1/3, 0.33 is less than 1/3, 0.333 is less than 1/3, 0.3333...(n threes) ... 333 is less than 1/3, so if we add a 4. ...

But in fact, the cycle of 0.3 only represents countless cycles of 3 and 3 0.3 added after the decimal point, that is, every digit after the decimal point has the addition of 3, all of which are equal to 9 but can never be carried to 0, so the three cycles of 0.3 add up to 0.9.

In addition, 1/3 means dividing the unit 1 into three parts and getting one part, then 1/3×3 is naturally 1.