Question

Prove that √3 is an irrational number.​

Answer 1

Answer:

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Answer 2

GIVEN:

• √3

TO FIND:

• Prove that √3 is an irrational number.

PROOF:

We assume that √3 is rational number and it can be written in the form of p/q, where p and q are coprimes and q ≠ 0

$\rm{\dashrightarrow \sqrt{3} = \dfrac{p}{q} }$

Squaring on both sides

$\rm{\dashrightarrow 3 = \dfrac{p^2}{q^2} }$

$\bf{\dashrightarrow p^2 = 3q^2 ....1) }$

$\bf{\dashrightarrow 3 \: divides \: p^2 }$

$\bf{\dashrightarrow 3 \: divides \: p }$

Let

$\rm{\dashrightarrow p = 3m}$

Squaring on both sides

$\rm{\dashrightarrow p^2 = 9m^2 }$

Putting the value of p² in equation 1), we have

$\rm{\dashrightarrow 9 m^2 = 3q^2 }$

Divide by '3' on both sides

$\rm{\dashrightarrow 3m^2 = q^2 }$

$\bf{\dashrightarrow 3 \: divides \: q^2 }$

$\bf{\dashrightarrow 3 \: divides \: q }$

Thus, 3 divides p and q

It means 3 is a common factor of p and q. This contradicts the assumption as there is no common factor of p and q.

Thus, √3 is not rational.

Hence, √3 is an irrational number.

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