Question

Prove that 1/√2 is irrational​

Answer 1

Answer:

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Step-by-step explanation:

this is the answer

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Answer 2

To prove : $\tt{\dfrac{1}{ \sqrt{2}}}$ is irrational

Proof:

Let's assume that $\tt{\dfrac{1}{ \sqrt{2}}}$ is rational number.

So,

$\tt{\longrightarrow{\dfrac{1}{\sqrt{2}}} = \dfrac{a}{b}}$

a and b are co prime numbers.

$\longrightarrow{\tt{b} = \sqrt{2}a}$

Squaring the both sides

$\longrightarrow{\tt{{b}^{2} } = 2 {a}^{2} \: - - - (eq.1)}$

b² = 2a², so 2 is a factor of b, b = 2c.

Substitute in eq. 1

$\longrightarrow{\tt {(2c)}^{2} = 2 {a}^{2} }$

$\longrightarrow{\tt {4c}^{2} = 2 {a}^{2} }$

$\longrightarrow{\tt {2c}^{2} = {a}^{2} }$

As 2c² = a², so 2 is a factor of a.

2 is the common factor of a and b. This is the contradiction to our assumption of a and b being co prime numbers. The contradiction was arisen due to wrong assumption.

Hence, $\tt{\dfrac{1}{ \sqrt{2}}}$ is irrational.