Method 1: We know that 1/3 equals 0.33333…2/3 equals 0.66666…, so 1/3+2/3 must equal 0.3333…+0.6666…

Adding the two sides, the result is 1 = 0.999.

Method 2: Given a set of intervals, exactly one point on the number axis is contained in all these intervals; 0.999 ... corresponds to interval sets [0, 1], [0.9, 1], [0.99, 1], [0.999, 1] ... and the only intersection of all these intervals is/kloc-.

Method 3: All rational numbers less than 0.999 ... are less than 1, and it can be proved that all rational numbers less than 1 are always different from 0.999 ... after decimal point (therefore less than 0.999 ...), which indicates that 0.999 ... and 1 are exactly the same set.

analyse

In order to confirm whether a number is a cyclic number, it is necessary to ensure that the number is multiplied by several consecutive numbers to appear cyclic. So 076923 will not be considered as a cyclic number, even if the number after each cycle is a multiple of it.

For example, the following numbers are cyclic numbers.

1, single digit, such as 5.

2. Number of unit repetitions, such as 555.

3. The cycle number is repeated, such as 142857.

If leading 0 is not allowed, 142857 will be the only decimal cycle number. If leading zeros are allowed, the first few periods are:

142857 (6 digits).

0588235294117647 (bit16).

052631578947368421(1bit 8).