• 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 ** 10 ^{l} ** for some positive integer

**1/(10 ^{i} -1)=10^{-i} +
10^{-2i} + .... + 10^{-ki} + ...... **

**135 / 22 = 6075 / (10*(10 ^{2} -1)) = 6075 * (
10^{-3} + 10^{-5} + 10^{-7} +...... )**

**135/22 = 6.1 + 36*10 ^{-3} + 36*10^{-5}
+ ...... = 6.1363636....[36]... **

Now the positive real numbers are obtained by considering all possible sequences of digits. The set of real numbers is denoted by

**2 ^{1/2} = 1.414213562.....
** or

• 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 **2 ^{1/2}
** , is also an algebraic number, being solution of the equation