REAL NUMBERS (CLASS X - CBSE) Pre - exercise 1.4 examples

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Theorem 1.5 :  Let x be a rational number whose decimal expansion terminates.   
Then x can be expressed in the form  p  , 

                                                        q
where p and q are co-prime and the prime factorization  of q is of the form 2ⁿ5^m, where n and m are non-negative integers.

Theorem 1.6 :  Let x =  p   be a rational number such that the prime 

                                     q

factorization of  q is of the form 2ⁿ5^m, where n and m are non-negative  integers. Then x has a decimal expansion which terminates.

Theorem 1.7 :  x =  p   be a rational number such that the prime 

                               q

factorization of  q is not of the form 2ⁿ5^m, where n and m are non-negative  integers. Then x has a decimal expansion which is non - terminating repeating (recurring).

[NOTE: The decimal expansion of every rational number is either terminating or non - terminating repeating.]

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