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 (ii)17/8 (iii)64/455 (iv)15/1600 (v)29/343 (vi)23/2^3*5^2 (vii)129/2^2* 5^7*7^5 (viii)6/15 (ix)35/50 (x)77/210
(i) 13/3125
Factorize the denominator we get
3125 =5 x 5 x 5 x 5 x 5 = 5^5
So denominator is in form of 5^m so 13/3125 is terminating .
(ii) 17/8
Factorize the denominator we get
8 =2 x 2 x 2 = 2^3
So denominator is in form of 2^n so 17/8 is terminating .
(iii)64/455
Factorize the denominator we get
455 =5 x 7 x 13
There are 7 and 13 also in denominator so denominator is not in form of 2^n*5^m . hence 64/455 is not terminating.
(iv)15/1600
Factorize the denominator we get
1600 =2 x 2 x 2 x2 x 2 x 2 x 5 x 5 = 2^6 x 5^2
so denominator is in form of 2^n x5^m
Hence 15/1600 is terminating.
(v) 29/343
Factorize the denominator we get
343 = 7 x 7 x 7 = 7^3
There are 7 also in denominator so denominator is not in form of 2^nx5^m
Hence it is none - terminating.
(vi)23/(2^3 x 5^2)
Denominator is in form of 2^n x 5^m
Hence it is terminating.
(vii) 129/(2^2 x 5^7 x 7^5 )
Denominator has 7 in denominator so denominator is not in form of 2^n x 5^n
Hence it is none terminating.
(viii) 6/15
divide nominator and denominator both by 3 we get 2/5
Denominator is in form of 5^m so it is terminating.
(ix) 35/50 divide denominator and nominator both by 5 we get 7/10
Factorize the denominator we get
10=2 x 5
So denominator is in form of 2^n x5^m so it is terminating
(x) 77/210.
simplify it by dividing nominator and denominator both by 7 we get 11/30
Factorize the denominator we get
30=2 x 3 x 5
Denominator has 3 also in denominator so denominator is not in form of 2^n x 5^n
Hence it is none terminating.
1. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
 |
| terminating and non terminating |
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 p , q you say about the prime factors of q?
(i) 43.123456789
it has certain number of digits so they can be represented in form of p/q . Hence they are rational number
As they have certain number of digit and the number which has certain number of digits is always terminating number and for terminating number denominator has prime factor 2 and 5 only .
(ii) 0.120120012000120000. . .
In this problem repetitions number are not same so it is not a irrational number
so prime factor of denominator Q will has a value which is not equal to 2 or 5. And irrational number is always none terminating
(iii) 43.123456789
In this number 0.123456789 repeating again and again so it is a rational number and it is none terminating so that the prime factor has a value which is not equal to 2 or 5
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