Which of the following is an irrational number?
i) 2√ is product of 2 and √. We know that √ is an irrational number and cannot be expressed in the form of pq where p and q are integers and q≠0. When we multiply a rational number with an irrational number, the result is an irrational number. Therefore, 2√ is an irrational number.
ii) 0.143¯¯¯¯¯¯32 is a recurring decimal
Let x=0.143¯¯¯¯¯¯32. In this case we see that 143 does not repeat itself but the block 32 repeats itself. Since two digits are repeating, we multiply x by 1000 to get
Subtracting the two equations we get,
which is in the form of pq where p and q are integers and q≠0. Hence it is a rational number.
iii) 5.¯¯¯¯¯¯46 is again a recurring decimal.
which is in the form of pq where p and q are integers and q≠0.Hence is a rational number.
iv) √ We know that √ is an irrational number since it cannot be represented in the form of pq where p and q are integers and q≠0. Its value is 2.23606797749979... which is a non-terminating and non-recurring decimal which is the very definition of an irrational number.