0.9999......=0.9,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.
Given : 0.9999......= 0.9
ANSWER :
0.9999 is non terminating recurring,so it is a RATIONAL NUMBER.
Let x= 0.9999 = 0.9…….
x = 0.99999….. ----(1)
Multiply eq 1 by 10
10 × x = 10 × 0.99999
10x = 9.9999….. -----(2)
Subtract eq 1 from 2
9x = 9
x = 9/9 = 1
x = 1
Hence, 0.9999 = 0.9…….is a RATIONAL NUMBER.
HOPE THIS ANSWER WILL HELP YOU...
Let x = 0.999.....-----( 1 )
multiply equation ( 1 ) with 10 , we get
10x = 9.999.... ---( 2 )
subtract equation ( 1 ) from (2 )
we get ,
9x = 9
x = 9/9
x = 1
Therefore ,
x = 0.999.... = 1 [ p/q form ]
o.9999... = 1 is a rational number.
I hope this helps you.
: )