**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)** **

The denominator is of the form 5*m*.

Hence, the decimal expansion ofis terminating.

(ii)** **

The denominator is of the form 2*m*.

Hence, the decimal expansion of is terminating.

(iii)

455 = 5 × 7 × 13

Since the denominator is not in the form 2*m* × 5*n*, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv)** **

1600 = 2^{6} × 5^{2}

The denominator is of the form 2*m* × 5*n*.

Hence, the decimal expansion of ** **is terminating.

(v)

Since the denominator is not in the form 2*m* × 5*n*, and it has 7 as its factor, the decimal expansion of is non-terminating repeating.

(vi)** **

The denominator is of the form 2*m* × 5*n*.

Hence, the decimal expansion of is terminating.

(vii)

Since the denominator is not of the form 2*m* × 5*n*, and it also has 7 as its factor, the decimal expansion of is non-terminating repeating.

(viii)

The denominator is of the form 5*n*.

Hence, the decimal expansion of ** **is terminating.

(ix)

The denominator is of the form 2*m* × 5*n*.

Hence, the decimal expansion of is terminating.

(x)** **

Since the denominator is not of the form 2*m* × 5*n*, and it also has 3 as its factors, the decimal expansion of ** **is non-terminating repeating.

**Question** 2:

Write down the decimal expansions of those rational numbers in **Question** 1 above which have terminating decimal expansions.

**Answer**:

(viii)

**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 factor of *q*?

(i) 43.123456789 (ii) 0.120120012000120000… (iii)

**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

i.e., the prime factors of *q* will be either 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 recurring, the given number is a rational number of the form and *q* is not of the form i.e., the prime factors of *q* will also have a factor other than 2 or 5.

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