Real Numbers
It is a well known fact that the sequence of the digits for a rational number is either finite or ultimately periodic ( i.e. periodic after a finite number of terms). Clearly, if a positive integer n divides 10l for some positive integer l then the decimal expansion for n'/n is finite. In the general case, the periodicity will result from the two following facts. First, every positive integer n divides 10l*( 10m -1) for some positive integers l and m . Secondly, for each positive integer i the following expansion holds
1/(10i -1)=10-i + 10-2i + .... + 10-ki + ......
135 / 22 = 6075 / (10*(102 -1)) = 6075 * ( 10-3 + 10-5 + 10-7 +...... )
135/22 = 6.1 + 36*10-3 + 36*10-5 + ...... = 6.1363636....[36]...
21/2 = 1.414213562..... or pi = 3.1415926535....
Algebraic and Transcendental Real Numbers
Among the real numbers there is another important classification between the algebraic numbers and the remaining numbers which are called transcendental. The algebraic numbers are those which are solutions of algebraic equations with integer coefficients. The rational numbers are clearly algebraic as solutions of equations of first degree. So for example 135/11 satisfies the equation 11x-135 = 0 . The square root of two, denoted by 21/2 , is also an algebraic number, being solution of the equation x2 - 2 = 0 . On the other hand pi is a very famous example of a transcendental number.