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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