Question

Prove that conjugate of 2+√3 is irrational

Answer 1

Conjugate of 2 + √3 = 2 -√3

To prove: 2-√3 is irrational number

Proof : Let 2-√3 be not an irrational number

Which implies, 2-√3 is a rational number

Then, 2-√3 = a / b : a,b ∈ Z and b ≠0 and

a/b =1

=> 2-√3 = a / b

(2b-a)/ b =√3

L.H.S ∈ Q , But R.H.S ∈ R - Q

Hence, this cannot be possible

=> our assumption that 2-√3 is rational is wrong

=> 2-√3 is irrational

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