Question

prove that root11 is irrational​

Answer 1

FIRST METHOD:-

root 11

after long division method we get that

10.10 value

here 10 is a recurring no

So we can say that root 11 is a irrational no

SECOND METHOD:-

Let as assume that √11 is a rational number.

A rational number can be written in the form of p/q where q ≠ 0 and p , q are non negative number.

√11 = p/q ....( Where p and q are co prime number )

Squaring both side !

11 = p²/q²

11 q² = p² ......( i )

p² is divisible by 11

p will also divisible by 11

Let p = 11 m ( Where m is any positive integer )

Squaring both side

p² = 121m²

Putting in ( i )

11 q² = 121m²

q² = 11 m²

q² is divisible by 11

q will also divisible by 11

Since p and q both are divisible by same number 11

So, they are not co - prime .

Hence Our assumption is Wrong √11 is an irrational number .

Hope it helps u^_^

Answer 2

Hola

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Let us assume √11 as a rational number

So we have

$\sqrt{11} \: = \: \frac{a}{b} \\ \\ square \: on \: both \: sides \\ \\ ( \sqrt{11) {}^{2} } \: = \: ( \frac{a}{b} ) {}^{2} \\ \\ 11b {}^{2} \: = \: a {}^{2}$

So , √11 is divisible by both A and B

Hence , our assumption was wrong √11 is irrational

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Hope you understand