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**Exercise 1.4****Ques 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:(i) 13/ 3125 **

(vi) 23/2

(i)

âˆµ 3125 = 5 Ã— 5 Ã— 5 Ã— 5 Ã— 5 = 5

= 2

which is of the form (2

âˆ´ 13/3125

i.e., 13/3125 will have a terminating decimal expansion.

(ii) 17/8

âˆµ 8 = 2 Ã— 2 Ã— 2 = 2^{3} = 1 Ã— 2^{3}

= 5^{0} Ã— 2^{3}, which is of the form 5^{m} Â· 2^{n}

âˆ´ 17/8 will have a terminating decimal expansion.

(iii) 64/455

âˆµ 455 = 5 Ã— 7 Ã— 13, which is not of the form 2^{n} Â· 5^{m}

âˆ´ 64/455 will have a non-terminating repeating decimal expansion.

(iv)** **15/1600

âˆµ 1600 = 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 5 Ã— 5

= 2^{6} Ã— 5^{2}, which is of the form 2^{n} Â· 5^{m}

âˆ´ 15/1600** **will have a terminating decimal expansion.

(v) 29/343

âˆµ 343 = 7 Ã— 7 Ã— 7 = 73, which is not of the form 2^{n} Â· 5^{m}.

âˆ´ 29/343** **will have a non-terminating repeating decimal expansion.

(vi) 23/2^{3}5^{2}

Let p/q = 23/2^{3}5^{2} ** **

i.e., q = 2^{3} Â· 5^{2}, which is of the form 2n Â· 5m.

âˆ´ 23/2^{3}5^{2}** ^{ }**will have a terminating decimal expansion.

(vii) 129/2^{2}5^{7}7^{5 }

Let p/q = 129/2^{2}5^{7}7^{5 }

i.e., q = 2^{2} Â· 5^{7} Â· 7^{5}, which is not of the form 2^{n} Â· 5^{m}.

âˆ´ 129/2^{2}5^{7}7^{5 }** ^{ }**will have a non-terminating repeating decimal expansion.

(viii) 6/15 ** **

âˆµ = which is of the form 2^{m}Â· 5^{n}.

âˆ´ 6/15 will have a terminating decimal expansion.

(ix)** **35/50 ** **

âˆµ 50 = 2 Ã— 5 Ã— 5 = 21 Ã— 52, which is of the form 2^{n} Â· 5^{m}.

âˆ´ 35/50 will have a terminating decimal expansion.

(x) 77/210 ** **

âˆµ 210 = 2 Ã— 3 Ã— 5 Ã— 7 = 2^{1} Â· 3^{1} Â· 5^{1} Â· 7^{1},

which is not of the form of 2^{n} Â· 5^{m}.

âˆ´ 77/ 210 will have a non-terminating repeating decimal expansion.

Fig: Decimal expression of rational numbers

**Ques 2: Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.****(i) 13/ 3125 ****(ii) 17/18 ****(iii) 64/455 ****(iv) 15/1600 ****(v) 29/343 ****(vi) ^{ } **

(i) 13/3125

Multiplying and dividing both sides, by 2

(ii) 17/18 ** **

Multiplying and dividing by 5^{3}, we have

(iii) 64/455 represents non-terminating repeating decimal expansion.

(iv) 15/1600

Multiplying and dividing by 5^{4}, we have

(v) 29/343** **, represents a non-terminating repeating decimal expansion.

(vi)** **

= |Multiplying and dividing by 5. = = 0.115

(vii) represents a non-terminating repeating decimal expansion.

(viii) 6/15

6/15 = 2/5

|Multiplying and dividing by 2,

= 4/10 = 0.4

(ix)** **35/50

35/50 = 0.7

(x)** **77/ 210** **, represents a non-terminating repeating decimal expansion.**Ques 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) ** **Sol: **

(i) 43.123456789

âˆµ The given decimal expansion terminates.

âˆ´ It is a rational of the form p/q

â‡’ p/q = 43.123456789

Here, p = 43.123456789 and

q = 2^{9} Ã— 5^{9}

âˆ´ Prime factors of q are 2^{9} and 5^{9}.

(ii) 0.120120012000120000 .....

âˆµ The given decimal expansion is neither terminating nor non-terminating repeating,

âˆ´ It is not a rational number.

(iii)

âˆµ The given decimal expansion is non-terminating repeating.

âˆ´ It is a rational number.

Let p/q = x = 43.123456789 ......(1)

Multiplying both sides by 1000000000, we have

1000000000 x = 43.123456789 ...(2)

Subtracting (1) from (2), we have:

(1000000000 x) â€“ x = (43.123456789) â€“ 43.123456789

â‡’ 999999999 x = 43.123456746

Here, p = 4791495194

and q = 111111111, which is not of form 2^{n}Â· 5^{m}

i.e., the prime factors of q are not of form 2^{n}Â· 5^{m}.

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