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Question

Which of the following is an irrational number?

i) 23

ii) 0.143¯¯¯¯¯¯32

iii) 5.¯¯¯¯¯¯46

iv) 5


A
Only iv)
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B
ii) and iii)
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C
i) and iv)
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D
All of the above
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Solution

The correct option is C i) and iv)

i) 23 is product of 2 and 3. We know that 3 is an irrational number and cannot be expressed in the form of pq where p and q are integers and q0. When we multiply a rational number with an irrational number, the result is an irrational number. Therefore, 23 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

1000x=143.323232323232...100000x=14332.3232..

Subtracting the two equations we get,

99000x=14159x=1415999000

which is in the form of pq where p and q are integers and q0. Hence it is a rational number.

iii) 5.¯¯¯¯¯¯46 is again a recurring decimal.

Let x=5.4646464646¯46100x=546.¯¯¯¯¯¯4699x=541

Therefore, x=54199

which is in the form of pq where p and q are integers and q0.Hence is a rational number.

iv) 5 We know that 5 is an irrational number since it cannot be represented in the form of pq where p and q are integers and q0. Its value is 2.23606797749979... which is a non-terminating and non-recurring decimal which is the very definition of an irrational number.


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