The decimal form of some real numbers are given below. In each case, decide whether the number is rational or not. If it is rational, and expressed in form p/q , what can you say about the prime factors of q?
(i) 43.123456789 (ii) 0.120120012000120000… (iii) 43.123456789 (all the decimal numbers are recurring)
Answers
Answered by
237
Hi ,
********************************************
i ) Let x = p/q be a rational number , such
that the prime factorisation of q is of the
form 2ⁿ × 5^m , where n and m are
non - negative integers . Then x has
a decimal expansion which terminates.
ii ) The number which is non - terminating and
non - repeating is called an
irrational number.
*******************************************
i ) x = 43.123456789
is a rational .
x = 43123456789/( 1000000000 )
= 43123456789/( 10^9 )
= 43123456789/( 2 × 5 )^9
= 43123456789/( 2^9 × 5^9 )
Here , q = 2^9 × 5^9 ( 2ⁿ × 5^m form )
43.123456789 is a terminating decimal.
ii ) 0.120120012000120000....
is non - terminating and non - repeating
decimal .
Therefore , it is an irrational number.
iii ) 43.123456789123456789....
is a non - terminating , repeating
decimal. So it is a rational number.
x = 43.123455789123456789....---( 1 )
10^9 x = 43123456789.123456789....--(2 )
subtracting ( 1 ) from ( 2 ) , we get
10^9 x = 43123456746
x = 43123456746/10^9
x = 43123456746/( 2 × 5 )^9
x = 43123456746/( 2^9 × 5^9 )
Therefore ,
q = 2^n × 5^m form
Given number is a rational .
I hope this helps you.
: )
********************************************
i ) Let x = p/q be a rational number , such
that the prime factorisation of q is of the
form 2ⁿ × 5^m , where n and m are
non - negative integers . Then x has
a decimal expansion which terminates.
ii ) The number which is non - terminating and
non - repeating is called an
irrational number.
*******************************************
i ) x = 43.123456789
is a rational .
x = 43123456789/( 1000000000 )
= 43123456789/( 10^9 )
= 43123456789/( 2 × 5 )^9
= 43123456789/( 2^9 × 5^9 )
Here , q = 2^9 × 5^9 ( 2ⁿ × 5^m form )
43.123456789 is a terminating decimal.
ii ) 0.120120012000120000....
is non - terminating and non - repeating
decimal .
Therefore , it is an irrational number.
iii ) 43.123456789123456789....
is a non - terminating , repeating
decimal. So it is a rational number.
x = 43.123455789123456789....---( 1 )
10^9 x = 43123456789.123456789....--(2 )
subtracting ( 1 ) from ( 2 ) , we get
10^9 x = 43123456746
x = 43123456746/10^9
x = 43123456746/( 2 × 5 )^9
x = 43123456746/( 2^9 × 5^9 )
Therefore ,
q = 2^n × 5^m form
Given number is a rational .
I hope this helps you.
: )
Answered by
31
:-
) = / ,
2ⁿ × 5^ ,
- .
.
) -
-
.
*******************************************
) = 43.123456789
.
= 43123456789/( 1000000000 )
= 43123456789/( 10^9 )
= 43123456789/( 2 × 5 )^9
= 43123456789/( 2^9 × 5^9 )
, = 2^9 × 5^9 ( 2ⁿ × 5^ )
43.123456789 .
) 0.120120012000120000....
- -
.
, .
) 43.123456789123456789....
- ,
. .
= 43.123455789123456789....---( 1 )
10^9 = 43123456789.123456789....--(2 )
( 1 ) ( 2 ) ,
10^9 = 43123456746
= 43123456746/10^9
= 43123456746/( 2 × 5 )^9
= 43123456746/( 2^9 × 5^9 )
,
= 2^ × 5^
.
.
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