Question

Prove that the root 11 is an irrational number​

Answer 1

Step-by-step explanation:

If √11 is irrational, how can you prove it?

By the method of contradiction..

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.

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Answer 2

Answer:

let us assume that √ 11 is rational

√11= p/q , where pand q are some integers.where q not equal to 0 . p and q r coprime

(√11) square= ( p/ q) whole square

11= p square/ q square

p square= 11 q square ➡️1 eq

p square is a multiple of 11

therefore p is a multiple of 11

p= 11 m

sub in 1 eq

(11m) square = 11 q square

121 m square= 11 q

11 q square= 121 m square

q square= 121 m square/ 11 = 11 m square

q square= 11 square

p and q are multiples of 11 .

p and q are coprime .

therefore this is a contradiction .

this contradiction came because of our assumption.

our assumption is wrong

therefore √11 is irrational

Step-by-step explanation:

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