. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not

If they are rational, and of the form  p/q what can you say about the prime factors of q

(i) 43.123456789

(ii) 0.120120012000120000

(iii) 43.123456789

Best Answer

Answer: (i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form  and q is of the form 2m×5n

That is, the prime factor of q will be 2 or 5 or both.

(ii) 0.120120012000120000

The decimal expansion is neither terminating nor recurring.

Therefore, the given number is an irrational number.

(iii) 43.123456789

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q and q is not of the form 2m×5n that is, the prime factors of q will also have a factor other than 2 or 5.


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