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• 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 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 10^l*( 10^m -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 + ......

We illustrate this with an example. For instance 22 divides 10*( 10² -1) and consequently

135 / 22 = 6075 / (10*(10² -1)) = 6075 * ( 10^-3 + 10^-5 + 10^-7 +...... )

which leads to

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

The period is indicated between square brakets.
Now the positive real numbers are obtained by considering all possible sequences of digits. The set of real numbers is denoted by R . The real numbers which are not rational are called irrational numbers. Thus for an irrational number taken at random, the sequence of digits is unpredictible. For example

2^1/2 = 1.414213562..... or pi = 3.1415926535....

The complexity of the sequence of the digits for such numbers is a mystery.

• 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 x² - 2 = 0 . On the other hand pi is a very famous example of a transcendental number.

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