Question

show that 2+√3 is is an irrational number​

Answer 1

ɢɪᴠᴇɴ :

√3 is an irrational number.

ᴛᴏ ᴘʀᴏᴠᴇ

2+√3 is an irrational number.

ꜱᴏʟᴜᴛɪᴏɴ:

ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ ᴍᴇᴛʜᴏᴅ

Let 2+√3 be an rational number.

$➡ 2 + \sqrt{3} = \frac{p}{q } \\ \\ ➡ \sqrt{3} = \frac{p}{q} - 2 \\ \\ ➡ \sqrt{3} = \frac{p - 2q}{q}$

$➡ \frac{p - 2q}{q} \: is \: \: an \: \: rational \: \: number.$

➡ So ,√3 must be an rational number.

➡ But √3 is an irrational number.

Thus , our assumption is wrong.

2+√3 is an irrational number.

Hence proved.