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From natural integers
to real numbers

 

• Natural integers

The simplest numbers are the so called natural integers:

0 , 1 , 2 , 3 , ...
The set of natural integers is denoted by N . Among these numbers there is an important subclass whose elements are called prime numbers. The prime numbers are defined as the natural integers greater than one which are only divisible by one and themselves. The set of prime numbers, denoted by P , is infinite and its first elements are :
2 , 3 , 5 , 7 , 11 , 13 , ...
Observe that 2 is the only even prime number.
• Rational integers

By considering the opposite of the natural integers, we obtain the set of rational integers, denoted by Z , whose elements are :

... , -3 , -2 , -1 , 0 , 1 , 2 , 3 , ...
• Rational numbers

By taking the inverse of all non-zero rational integers and their multiples, we obtain the set of rational numbers, denoted by Q . For example -135/11 is a rational number. Rational numbers are commonly written in decimal expansion. For example

1/2 = 0.5                                 1462/50 = 29.24                      230/3 =76.666666...
and
-135/11 = -12.27272727....
  We recall the meaning of this writting
                           1/2 = 5/10                                  1462/50 = 29 + 2/10 + 4/100                     
and

230/3 = 76 + 6/10 + 6/100 + 6/1000 + 6/10000 +....

or else

230/3 = 7*101 + 6*100 + 6*10-1 + 6*10-2 .....

 So every positive rational number can be written as

x=an10n + an-110n-1 + an-210n-2 +......

  Here the digits ai are natural integers between 0 and 9. Notice that the multiplicative symbol is traditionally omited between ai and 10i . Besides, clearly the sequence of the digits may be finite or infinite.
If n > -1 , then the begining of the expansion u = an10n + an-110n-1+.... + a0 is a natural integer called the integral part of x, while the remaining part v = a-110-1 + .... is the fractional part of x .

 

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