Question

prove that √11 is an irrational number

Answer 1

To Prove : $\tt{\sqrt{11}}$ is irrational.

Proof :

Assume that $\tt{\sqrt{11}}$ is rational.

So,

$\tt{\sqrt{11}= \dfrac{a}{b}} \: \: \: \gray{ \rm{(a \: and \: b \:are \: co - prime \: numbers)}}$

$\tt{\longrightarrow} \: \sqrt{11}b = a$

Squaring both the sides

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

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

Substitute the value of a in equation 1.

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

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

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, $\tt{\sqrt{11}}$ is irrational.

Answer 2

Answer:

To Prove,

√11 is an irrational number

Recall the concept:

An irrational number is a number that can not be expressed in the form $\frac{p}{q}$, where p and q are integers and q≠0

A rational number is a number that can be expressed in the form $\frac{p}{q}$, where p and q are integers and q≠0

Solution:

We need to prove that √11 is an irrational number.

Let us assume that  √11 is a rational number.

Then as per the definition of rational numbers, there exists two  'p' and 'q' such that

√11 = $\frac{p}{q}$,  and q≠0

Here,  $\frac{p}{q}$ is in its simplest form, that is p and q are coprime.

√11 = $\frac{p}{q}$ ----------(1)

Squaring on both sides

11 = $\frac{p^2}{q^2}$

11q² = p²------------(2)

⇒11 divides p²

⇒Since 11 divides p² and 11 is a prime we have 11 divides p

Let us take p = 11r

Substituting the value of 'p' in equation(2) we get

11q² = (11r)² = 121r²

q² = 11r²

⇒11 divides q²

⇒Since 11 divides q² and 11 is a prime we have 11 divides q

Hence we have, 11 divides both p and q, which contradicts the statement that p and q are coprime.

Hence our assumption is wrong.

∴√11 is an irrational number

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