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.

Cycle number attribute:

When multiplied by prime numbers that produce cyclic numbers, the result will be a series of 9s. For example, 142857× 7 = 99999.

If we divide it into several equal-length parts and add them together, the result will be a series of 9s. This is a special case of Midy theorem. Such as14+28+57 = 99142+857 = 9991428+5714+2857 = 9999. All cycle numbers are multiples of 9.