### Real Numbers Class 10th Mathematics CBSE Solution

##### Question 1.Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: Answer:(i) Factorize the denominator we get,3125 = 5×5×5×5×5 = 55The denominator is of the form 5mHence, the decimal expansion of is terminating.(ii) Factorize the denominator we get,8 = 2 × 2 ×2 = 23The denominator is of the form 2mHence, the decimal expansion of is terminating.(iii) Factorize the denominator we get,455 = 5×7×13Since, the denominator is not in the form of 2m × 5n, and it also contains 7 and 13 as its factors,Its decimal expansion will be non-terminating repeating.(iv) Factorize the denominator we get,1600 = 26×52The denominator is in the form 2m × 5nHence, the decimal expansion of is terminating.(v) Factorize the denominator we get,343 = 73Since the denominator is not in the form of 2m × 5n, it has 7 as its factors.So, the decimal expansion of non-terminating repeating.(vi) The denominator is in the form 2m×5nHence, the decimal expansion of is terminating.(vii) Since, the denominator is not in the form of 2m × 5n, as it has 7 in denominator.So, the decimal expansion of is non-terminating repeating.(viii) The denominator is in the form 5nHence, the decimal expansion of is terminating.(ix) Factorize the denominator we get,10 = 2×5The denominator is in the form 2m×5nHence, the decimal expansion of is terminating.(x) Factorize the denominator we get,30 = 2×3×5Since the denominator is not in the form of 2m × 5n, as it has 3 in denominator.So, the decimal expansion of non-terminating repeating.Question 2.Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.Answer:(i)  (ii)  (iv)  (vi)  (viii)  (ix)  Question 3.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 what can you say about the prime factors of q?(i) 43.123456789(ii) 0.120120012000120000...(iii) Answer:(i) 43.123456789Since this number has a terminating decimal expansion, it is a rational number of the form and q is of the form 2m×5nThat 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) Since the decimal expansion is non-terminating but recurring, the given number is a rational number of the form 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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