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Number Theory: Interesting Results

Prove that ...

Question

Prove that $3 + 2 \sqrt{5}$ is irrational .

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Solution

$L e t u s a s s u m e t h a t$ $3 + 2 \sqrt{5}$ $i s a r a t i o n a l n u m b e r .$
$H e r e,$ $p, q, \in z, q \neq 0$
$3 + 2 \sqrt{5} = \frac{p}{q}$
$2 \sqrt{5} = \frac{p}{q} - 3$
$\sqrt{5} = \frac{p - 3 q}{2 q}$
$\sqrt{5}$ $i s a n i r r a t i o n a l n u m b e r .$
$S i n c e p, q a r e i n t e g e r s h e n c e \frac{p - 3 q}{2 q}$ $i s r a t i o n a l n u m b e r$
$B u t$ $\sqrt{5}$ $i s n o t a r a t i o n a l n u m b e r .$
$T h i s c o n t r a d i c t s t h e f a c t t h a t,$
$∴ 3 + 2 \sqrt{5}$ $i s a n i r r a t i o n a l n u m b e r .$

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Similar questions

\sqrt{2}

is irrational, prove that

(5 + 3 \sqrt{2})

is an irrational number.

3 + 2 \sqrt{5}

is an irrational.

\sqrt{2}

is irrational and hence prove that

\frac{5 - 3 \sqrt{2}}{7}

\sqrt{2}

is on irrational number and also prove that

3 + 5 \sqrt{2}

is irrational number.

Q. Question 2
Prove that

3 + 2 \sqrt{5}

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