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Revisiting Rational Numbers and the Decimal Expansion

Theorem : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form , where p and q are coprime, and the prime factorization of q is in the form 2n5m , where n, m are non-negative integers.


Theorem : Let be a rational number, such that the prime factorization of is of the form , where are non-negative integers . Then has a decimal expansion which terminates.

Theorem :  Let be a rational number, such that the prime factorization of is not of the form , where are non-negative integers . Then has a decimal expansion which is
non-terminating repeating.

 

Example

Which of the following rational numbers have the terminating decimal representation?
(i)    (ii) (iii)
(iv) (v) (vi)
[Making use of the result that a rational number where and have no common factors will have a terminating representation if and only if the prime factors of are 2's or 5's or both.]

Solution

(i) The prime factor of 5 is 5. And hence the denominator can be written in the form of 20 x 51 (i.e.) 1 x 5 = 5

Hence has a terminating decimal representation.
(ii) 20 = 4 x 5 = 22 x 5.
The prime factors of 20 are both 2's and 5's. Hence has a terminating decimal.
(iii) The prime factor of 13 is 13. The denominator cannot be expressed in the form of 2n x 2m Hence has non-terminating decimal.
(iv) 40 = 23 x 5.
The prime factors of 40 are both 2's and 5's. Hence has a terminating decimal.
(v) 125 = 53
The prime factor of 125 is 5's. Hence has a terminating decimal.
(vi) The prime factor of 7 is 7. Hence has a non-terminating decimal representation.

 

Example

Express as a decimal fraction.

Solution

0.109375

    
600
    576
       240
      192
          480
          448
          320
          320
              0
Therefore = 0.109375

 

Example

Represent 0.57  in the form of .

Solution
= 0.57                               ……………(i)
100= 57.57                        ……………(ii)

(ii) – (i)
99 = 57
Therefore = .




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