Question

(a) Prove that 5 + 5 is an irrational number.
1
(b) Prove that is not a rational number.
1/2-
$\sqrt{3}$
​

Answer 1

❶ GIVEN:

• 5 + √5

TO FIND:

• Prove that 5 + √5 is an irrational number.

SOLUTION:

Let 5+√5 be a rational number, which can be written in the form of p/q,where p and q are integers and q ≠ 0

$\sf{\hookrightarrow 5 + \sqrt{5} = \dfrac{p}{q}}$

$\sf{\hookrightarrow \sqrt{5} = \dfrac{p}{q} - 5 }$

$\sf{\hookrightarrow \sqrt{5} = \dfrac{p - 5q}{q} }$

✎ Since, p and q are integers, so we get p –5q/q is rational, and so √5 is rational.

✎ But this contradicts the fact that √5 is irrational.

Hence, 5 + √5 is an irrational number.

❷ GIVEN:

• 1/2 –√3

TO FIND:

• Prove that 1/2 –√3 is an irrational number.

SOLUTION:

 Let 1/2 –√3 be a rational number, which can be written in the form of p/q and q ≠ 0

$\sf{\hookrightarrow \dfrac{1}{2} - \sqrt{3} = \dfrac{p}{q} }$

$\sf{\hookrightarrow \dfrac{1}{2} - \dfrac{p}{q} = \sqrt{3} }$

$\sf{\hookrightarrow \dfrac{q-2p}{2q} = \sqrt{3} }$

✎ Since, p and q are integers, so we get q–2p/2q is rational, and so √3 is rational.

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

Thus, 1/2 –√3 is not a rational number.

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