Question

Prove that 2-√3 is irrational, given that √3 is irrational​

Answer 1

Step-by-step explanation:

Let us assume to the contrary that 2-√3 is rational.

2-√3=a/b ( here a&b co-prime integers)

-√3=a-2/b

-√3=a-2b/b (here a, b, -2 are rational but -√3 is irrational)

But this contradicts the fact that -√3 is irrational.

Hence 2-√3 is irrational.

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Answer 2

Let, 2-√3 be rational numbers

so,

=> 2- √3 = p/q

=> -√3 = p/q - 2

=> √3 = 2-p/q

=> √3 = (2q-p)/q (observe LHS and RHS)

(LHS=Irrational) | (RHS= Rational)

(Given) | All the Numbers are rational

since we know,

LHS is irrational and RHS is rational and

Rational Numbers does not equal to Irrational numbers so This is contradict to the fact that 2- √3 is a rational number

Thus,

2- √3 is a irrational number hence proved