Question

Prove √3-√5 is irrational

Answer 1

ANSWER:

• √3-√5 is an Irrational number.

GIVEN:

• Number = √3 - √5

TO PROVE:

• (√3 - √5) is an Irrational number.

SOLUTION:

Let ( √3 - √5 ) be a rational number which can be expressed in the form of p/q where p and q have no other Common factor than 1.

$\implies \: \sqrt{ 3} - \sqrt{5} = \dfrac{p}{q} \\ \\ \implies \: \sqrt{3} = \dfrac{p}{q} + \sqrt{5} \\ \: \: \: squaring \: both \: sides \: we \: get. \\ \\ \implies \: 3 = \dfrac{p {}^{2} }{q {}^{2} } + 5 + \dfrac{2 \sqrt{5} p}{q} \\ \implies - \dfrac{2 \sqrt{5} p}{q} = \dfrac{p {}^{2} }{q {}^{2} } + 5 - 3 \\ \\ \implies \: \dfrac{ - 2 \sqrt{5}p }{q} = \dfrac{p {}^{2} }{q {}^{2} } + 2 \\ \\ \implies \: \dfrac{ - 2 \sqrt{5} p}{q} = \dfrac{p {}^{2} + 2q {}^{2} }{q {}^{2} } \\ \\ \implies \: \sqrt{5} = \dfrac{ - p {}^{2} -2 q {}^{2} }{2pq}$

Here:

• (-p²-2q²)/2pq is rational while √5 is irrational.
• Thus our contradiction is wrong .
• √3-√5 is an Irrational number.

Answer 2

√3-√5 is the irrational number .

thank you ...