Question

Prow that √11 is a irrational number.


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Answer 1

Answer:

rational number can be written in the form of p/q where q ≠ 0 and p , q are non negative number. Squaring both side ! So, they are not co - prime . Hence Our assumption is Wrong √11 is an irrational number .

Answer 2

To prove = √11 is rational number

Proof:

Assume that √11 is rational.

$\tt \sqrt{11} = \frac{a}{b} \color{grey}(a \: and \: b \: are \: co \: - prime \: numbers)$

$\tt⟶\sqrt{11} b = a$

Squaring both the sides.

$\tt \sqrt{11} {b}^{2} = {a}^{2} - - -( 1)$

As 11b² = a², so a = 11c. 11 is a factor of a.

Substitute the value of a in equation 1.

$\tt{ \longrightarrow} \sqrt{11} {b}^{2} = {(11c)} ^{2}$

$\tt{ \longrightarrow} \sqrt{11} {b}^{2} = {121c ^{2} }$

Divide both the sides by 11.

$\tt{ \longrightarrow} {b}^{2} = {11c}^{2}$

11 is a factor of b.

Both a and b have 11 as their common factor.

This is a contradiction to the assumption of considering a and b co-prime numbers.

The contradiction arisen due to wrong assumption.

Therefore, √11 is irrational.