Question

show that 1/√2 is irrational​

Answer 1

ANSWER:

• 1/√2 is an irrational number.

GIVEN:

• Irrational number = 1/√2

TO PROVE:

• 1/√2 is an irrational number.

SOLUTION:

Let 1/√2 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.

$\implies \: \dfrac{1}{ \sqrt{2} } = \dfrac{p}{q} \\ \\ \implies \: \sqrt{2} p = q \\ \\ \: \: \: \: squaring \: both \: the \: sides \: we \: get. \\ \\ \implies \: ( \sqrt{2} p) {}^{2} = q {}^{2} \\ \\ \implies \: 2p {}^{2} = q {}^{2} ......(i) \\ \\ \: \: \: \: 2 \: divides \: q {}^{2} \: \: \: therefore : \\ \: \: \: 2 \: divides \: q \: \: \: .....(ii) \\ \\ \: \: \: \: let \: \: q = 2m \: \: in \: equation \: (i) \\ \\ \implies \: 2p {}^{2} = (2m) {}^{2} \\ \\ \implies \: 2p {}^{2} = 4m {}^{2} \\ \\ \implies \: p {}^{2} = 2m {}^{2} \\ \\ \: \: \: here \: 2 \: divides \: p {}^{2} \: therefore : \\ \: \: \: 2 \: divide \: p \: .......(iii)$

From eq (ii) and (ii)

• 2 is the common factor of p and q.
• Thus our supposition was wrong so 1/√2 is an irrational number.

NOTE:

• This method of proving and irrational number is called contradiction method.