Question

prove that 11-root 3 is an irrational number ​

Answer 1

Answer:

Let us assume that

11 - √3 is not an irrational

11 - √3 becomes rational

11 - √3 = p / q Where p , q belongs to integers and q ≠ 0

√3 = p / q - 11 / 1

√3 = p - 11q / q

LHS = √3 is an irrational because "3" is not a perfect square .

RHS = p - 11q / q becomes rational where p , q belongs to integers and q ≠ 0

But LHS ≠ RHS

It is contradiction to our Assumption .

Our assumption is wrong .

11 - √3 is an irrational .

Answer 2

let 11-root3 be rational number

11-root3 = a/b ( a and b are co prime )

root3 =a + 11b/b

so , a+11b/b and root 3 is rational number.

root 3 = p/k ( p and k are co prime )

3 = p^2/k^2

3k^2 = p^2 .............1

p^2 is divisible by 3

p is also divisible by 3

p = 3c .............2

Put equation 2 in 1

3k^2 = 9c^2

k^2 = 3c^2

k^2 is divisible by 3

k is also divisible by 3

it contradict that p and k are co prime numbers.

So, our supposition was wrong.

11-root3 is an irrational number