Prove 1: Because 0. 1( 1 cycle) = 1/9, 0.2(2 cycles) = 2/9; 0.3(2 cycles) = 3/9; 0.4(4 cycles) = 4/9 ... so 0.9(9 cycles) =9/9= 1.

Method 2:

0.9(9 cycles) =0.3(3 cycles) ×3= 1/3×3= 1.

Method 3: Fill in the blanks

Try to fill in the blanks: 1-( )=0.9(9 cycles). Analysis will find that; No matter which number in parentheses is greater than 0, the difference will be less than 0.9(9 cycles). Because it is an infinite decimal, it can only be filled with 0, so 0.999999 ... =1-0 =1.

The second question:

0.36 (cycle of 36) =36/99

0.123 (cycle of123) = 123/999

0.23 (cycle of 3) =23/99

The third question: through exploration, I can get such enlightenment:

When a pure cyclic decimal is converted into a fraction, the numerator of the fraction is a number composed of the number of cyclic segments; The denominator consists of several numbers 9, and the number of 9 is equal to the number of digits in the loop segment. For example, the number of 0.36 (a cycle of 36) is 36, so as a numerator, the denominator consists of two 9s.