Question

prove that √11-√6 is irrational​

Answer 1

Step-by-step explanation:

Let √11 be rational , then there should exist √11=p/q ,where p & q are coprime and q≠0(by the definition of rational number). So,

√11= p/q

On squaring both side, we get,

11= p²/q² or,

11q² = p². …………….eqñ (i)

Since , 11q² = p² so ,11 divides p² & 11 divides p

Let 11 divides p for some integer c ,

so ,

p= 11c

On putting this value in eqñ(i) we get,

11q²= 121p²

or, q²= 11p²

So, 11 divides q² for p²

Therefore 11 divides q.

So we get 11 as a common factor of p & q but we assumpt that p & q are coprime so it contradicts our statement. Our supposition is wrong and √11 is irrational.

Answer 2

Answer:

Step-by-step explanation:

Let as assume that √11 is a rational number.

A rational number can be written in the form of p/q where q ≠ 0 and p , q are non negative number.

√11 = p/q ....( Where p and q are co prime number )

Squaring both side !

11 = p²/q²

11 q² = p² ......( i )

p² is divisible by 11

p will also divisible by 11

Let p = 11 m ( Where m is any positive integer )

Squaring both side

p² = 121m²

Putting in ( i )

11 q² = 121m²

q² = 11 m²

q² is divisible by 11

q will also divisible by 11

Since p and q both are divisible by same number 11

So, they are not co - prime .

Hence Our assumption is Wrong √11 is an irrational number

and similarly prove √6 is irrational

difference of two irrational no. is irrational