Given is a real number α,

in its decimal representation

α = 0,a1a2a3 ... each decimal digit ai(i=1,2,3,...) is a prime number

is. The decimal digits are

along the line indicated by arrows in the adjacent figure,

to the right and downward infinitely continued to think way

to the right and downward. For each m ≥ 1, the decimal representation of a real number

zm is formed by placing the digit 0 in front of the decimal point and after the decimal point the

the sequence of digits of the m-th line from above, read from left to right, is written after the decimal point.

is written in the adjacent arrangement. In an analogous way

for all n ≥ 1 the real numbers sn with the digits of the n-th column to be read from top to bottom.

digits of the nth column from the left.

For example, z3 = 0,a5a6a7a12a23a28 ... and s2 = 0,a2a3a6a15a18a35 ... .

Show:

(a) If α is rational, then all zm

and all sn are rational.

(b) The converse of the statement in (a)

is false.