Question

Prove that root 3 is an irrational number and also prove that 2 + root 3 is also irrational number

Answer 1

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Answer 2

Step-by-step explanation:

First let us look at the laws:

Irrational+rational (and vice versa) = irrational

Rational + rational = rational

Let us assume √3 to be rational.

==> √3 =p/q. (Where q#0 , gcd of p and q is 1)

==> 3 = P²/q²

==> 3q² = p². ————1

Therefore p² is a multiple of 3.

Hence p is also a multiple of 3. ———2

Let p = 3x (for some x)

==>(3x)² =p²

By 1

(3x)² = 3q²

==>3x= q²

Therefore q² is a multiple of 3.

Hence q is also a multiple of 3. ———3

By the notes 2 and 3 we get a contradictory statement as both p,q are divisible by 3. This means GCD of p and q #1.

Therefore √3 is irrational.

‘#’ means ‘not equal”