Question

1.Prove that 1/2-√5/3 is irrational.
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Answer 1

Proof

⇒1/2 - √5/3

⇒(3 - 2√5)/6

Let (3 - 2√5)/6 be a rational number in form of p/q.

⇒(3 - 2√5)/6 = p/q

⇒3 - 2√5 = 6p/q

⇒2√5 = 3 - 6p/q

⇒2√5 = (3q - 6p)/q

⇒√5 = (3q - 6p)/2q

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Thus, √5 is rational number due (3q - 6p)/2q i.e., rational form.

This contradicts fact that √5 is irrational.

Therefore 1/2 - √5/3 is irrational.

Q.E.D

Answer 2

$\Large{\mathfrak{\underline{\underline{Solution :-}}}}$

Since $\sf \sqrt{5}$ is irrational and We know that,

Any Number Multiplied, Divide, Added or Subtracted with Irrational Number is also a Irrational Number.

So, $\sf \dfrac{1}{2} - \sqrt{\dfrac{5}{3}}$ is Irrational Number.