0.090909...=0.09,Given real number is expressed in decimal form. Find whether they are rational or not. If rational, express them in the form p/q. Comment on factors of q.
Answers
REAL NUMBERS :
a number which is either rational or irrational is called a real number.
RATIONAL NUMBERS :
A number that can be expressed as p/q , where p,q are Integers and q≠0 is called a rational number.
IRRATIONAL NUMBERS :
A number that cannot be expressed as p/q , where p,q are Integers and q≠0 is called a irrational number.
NON TERMINATING REPEATING DECIMAL EXPANSION :
The number which does not terminate but repeat the particular number again and again in the process of division , is said to be a non terminating repeating decimal expansion.
SOLUTION :
Given : 0.090909...=0.09
0.09 non terminating recurring ,so it is a rational number.
Let x = 0.090909……(1)
Multiply equation 1 by 100
100 × x = 100 × 0.090909…
100x = 9.0909….. … (2)
Subtract equation 1 from 2
99 x = 9
x = 9/99
x =1/11
Hence, 0. 090909...=0.09 is a rational number.
HOPE THIS ANSWER WILL HELP YOU...
0.090909.... = 0.09 bar ( bar on 09 )
Period = 09
periodicity = 2
0.909090... is a retional number.
Since , it is a non terminating
repeating decimal .
ii ) Let x = 0.090909.... ---( 1 )
multiply equation ( 1 ) with 100 , we get
100x = 9.090909..... ( 2 )
Subtract equation ( 1 ) from ( 2 ) ,
we get
99x = 9
x = 9/99
x = 1/11 [ p/q form ]
Therefore ,
x = 0.0909.... = 1/11
I hope this helps you.
: )