Question

prove that under root 3 is irrational with explanation​

Answer 1

Step-by-step explanation:

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Answer 2

ANSWER:

• √3 is an irrational number.

TO PROVE:

• √3 is an irrational number.

SOLUTION:

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

$\implies \: \sqrt{3} = \dfrac{p}{q} \\ \implies \: \sqrt{3} q = p \: \: \: \: ...(i)$

Squaring both the sides we get in eq(i)

$\implies \: ( \sqrt{3} q) {}^{2} = (p) {}^{2} \\ \implies \: 3q {}^{2} = p {}^{2} \: \: \: ....(ii)$

This shows that :

• 3 divides p²
• therefore. 3 divides p .....(iii)

Let p = 3m in eq(ii) we get;

$\implies \: 3q {}^{2} = (3m) {}^{2} \\ \implies \: 3q {}^{2} = 9m {}^{2} \\ \implies \: q {}^{2} = 3m {}^{2}$

This shows that

• 3 divides q²
• therefore 3 divides q. .......(iv)

From eq(iii) and eq(iv)

• p and q have 3 as a common factor.
• Thus our contradiction is wrong
• So √3 is an irrational number