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.. Classify the following numbers as rational or irrational:

(i) √23

(ii) √225

(iii) 0.3796

(iv) 7.478478

(v) 1.101001000100001…

Answer:

(i) √23

Since, 23 is not a perfect square,

√23 is an irrational number.

(ii) √225

√225 = \sqrt{(15)^{2}}=15

Since, 225 is a perfect square,

√225 is a rational number.

(iii) 0.3796

0.3796 = 3796/1000

Since the decimal expansion is terminating decimal.

0.3796 is a rational number.

(iv) 7.478478

Let x = 7.478478

Since there are three repeating digits after the decimal point,

Multiplying by 1000 on both sides, we get

1000x = 7478.478478…

Now, subtract both the values,

999x = 7478 – 7

999x = 7471

x = 7471/999

7.478478 = 7471/999

Hence, it is neither terminating nor non-terminating or non-repeating decimal.

7.478478 is an irrational number.

(v) 1.101001000100001…

Since the number of zero’s between two consecutive ones is increasing. So it is non

terminating or non-repeating decimals.

1.101001000100001… is an irrational number.

Let x = 345.0456456

Multiplying by 10 on both sides, we get

10x = 3450.456456

Since there is three repeating digit after the decimal point,

Multiplying by 1000 on both sides, we get

1000x = 3450456.456456…

Now, subtract both the values,

10000x – 10x = 3450456 – 345

9990x = 3450111

x = 3450111/9990

Since, it is non-terminating repeating decimal.

is a rational number.

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