Show that (2 + √3) is an irrational number.

1.	Show that (2 + √3) is an irrational number.
Answer» Assume that 2 +√3 is a rational number, therefore it can be write in \(\frac{p}{q}\) form, where q ≠ 0 and p & q have no common factor other than 1. Therefore, 2 + √3 = \(\frac{p}{q}\) . ⇒ √3 = \(\frac{p}{q}\) – 2 = \(\frac{ p− 2q}{q}\), q ≠ 0. But √3 is an irrational number and \(\frac{ p− 2q}{q}\) is a rational number. (∵ p − 2q, q ∈ I & q ≠ 0, ∴ \(\frac{ p− 2q}{q}\) is a rational number. ) It mean rational = irrational which is not possible. (contradiction) Hence, our assumption is wrong. Therefore, 2 +√3 is an irrational number. Let us assume that 2+√3 is a rational number. A rational number can be written in the form of p/q. 2+√3=p/q √3=p/q-2 √3=(p-2q)/q p,q are integers then (p-2q)/q is a rational number. But this contradicts the fact that √3 is an irrational number. So,our assumption is false. Therefore,2+√3 is an irrational number. Hence proved.

Answer»

Assume that 2 +√3 is a rational number, therefore it can be write in \(\frac{p}{q}\) form, where q ≠ 0 and p & q have no common factor other than 1.

Therefore, 2 + √3 = \(\frac{p}{q}\) .

⇒ √3 = \(\frac{p}{q}\) – 2 = \(\frac{ p− 2q}{q}\), q ≠ 0.

But √3 is an irrational number and \(\frac{ p− 2q}{q}\) is a rational number.

(∵ p − 2q, q ∈ I & q ≠ 0, ∴ \(\frac{ p− 2q}{q}\) is a rational number. )

It mean rational = irrational which is not possible. (contradiction)

Hence, our assumption is wrong.

Therefore, 2 +√3 is an irrational number.

Let us assume that 2+√3 is a rational number.
A rational number can be written in the form of p/q.
2+√3=p/q
√3=p/q-2
√3=(p-2q)/q
p,q are integers then (p-2q)/q is a rational number.
But this contradicts the fact that √3 is an irrational number.
So,our assumption is false.
Therefore,2+√3 is an irrational number.
Hence proved.

Show that (2 + √3) is an irrational number.

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