Question

3. Prove that the following are irrationals

$(1)1 \div \sqrt{5}$
​

Answer 1

$let \: assume \: that \: \frac{1}{ \sqrt{2} } is \: rational \: number$

$hence \: \frac{1}{ \sqrt{2} } can \: be \: writen \: in \: the \: form \: of$

$\frac{a}{b} \: where \: a \: b$

(b ≠0) are co prime

$\frac{1}{ \sqrt{2} } = \frac{a}{b}$

${b}^{a} = \sqrt{2}$

$but \: here \: \sqrt{2 \:} is \: irrational \: and \: \frac{a}{b} is \: rational$

As RATIONAL ≠IRRATIONAL

$this \: is \: a \: contradiction \: so \: \frac{1}{ \sqrt{2} } is \: a \: irrational \: number$

Answer 2

Answer:

let \: assume \: that \: \frac{1}{ \sqrt{2} } is \: rational \: numberletassumethat

2

1

isrationalnumber

hence \: \frac{1}{ \sqrt{2} } can \: be \: writen \: in \: the \: form \: ofhence

2

1

canbewritenintheformof

\frac{a}{b} \: where \: a \: b

b

a

whereab

(b ≠0) are co prime

\frac{1}{ \sqrt{2} } = \frac{a}{b}

2

1

=

b

a

{b}^{a} = \sqrt{2}b

a

=

2

but \: here \: \sqrt{2 \:} is \: irrational \: and \: \frac{a}{b} is \: rationalbuthere

2

isirrationaland

b

a

isrational

As RATIONAL ≠IRRATIONAL

this \: is \: a \: contradiction \: so \: \frac{1}{ \sqrt{2} } is \: a \: irrational \: numberthisisacontradictionso

2

1

isairrationalnumber