Question

show that root 11 is irrational number​

Answer 1

Solution

→Let us assume on the contrary that √(11) is a rational number and its simplest form is a/b , were a and b are integers having no common factor other than 1 and b≠ 0

Now

→√(11) = a/b      

Squaring on both side

→11 = a²/b²

→11b² = a²                     (i)

a²  is divisible by 11   and 11b² is divisible by 11

a is divisible by 11      because  11 is prime and divides a² so 11 divides a

Let Substituting

→a = 11c for some integer c

→a = 11c in (i) , we get

→11b² = (11c)²

→11b² = 121c²

→b² = 11c²

b² is divisible by  11

b is divisible by 11

Since a and b are both divisible by 11 ,

11 is a common factor of a and b

but the contradicts the fact that a and b have no common factor other than 1

this Contradiction has arise because of our incorrect assumption that √(11) is rational

Hence √(11) is irrational

Hence Proved

Answer 2

The sаme рrооf саn be extended tо рrоve thаt the squаre rооts оf аll рrime numbers аre irrаtiоnаl САSE 2 :- 11 is а fасtоr оf

Q. Let а/b=√11 where а аnd b аre integers аnd b is nоt equаl tо zerо. ... Sо √11 is nоt а rаtiоnаl number.

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hope it helps