Question

prove that
$\sqrt{3}$
is an irrational number

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Answer 1

it is not a other number

Explanation:

..

Answer 2

Answer:

$\huge\fbox\purple{❥Answer}$

ʟᴇᴛ ᴜꜱ ᴀꜱꜱᴜᴍᴇ ᴏɴ ᴛʜᴇ ᴄᴏɴᴛʀᴀʀʏ ᴛʜᴀᴛ √3 ɪꜱ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.  

ᴛʜᴇɴ, ᴛʜᴇʀᴇ ᴇxɪꜱᴛ ᴘᴏꜱɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀꜱ ᴀ ᴀɴᴅ ʙ ꜱᴜᴄʜ ᴛʜᴀᴛ

√3 = ᴀ/ʙ  ᴡʜᴇʀᴇ, ᴀ ᴀɴᴅ ʙ, ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇ ɪ.ᴇ. ᴛʜᴇɪʀ ʜᴄꜰ ɪꜱ 1

Now,

√3  = a/b

⇒ 3 = a²/b²

⇒ 3b² = a²

⇒ 3 divides a² [∵3 divides 3b² ]

⇒ 3 divides a...(i)

⇒ a = 3c for some integer c

⇒ a²= 9c²  

⇒ 3b²  =9c² [∵a² = 3b² ]

⇒ b² = 3c²

⇒ 3 divides b²  [∵3 divides 3c² ]

⇒ 3 divides b...(ii)

ꜰʀᴏᴍ (ɪ) ᴀɴᴅ (ɪɪ), ᴡᴇ ᴏʙꜱᴇʀᴠᴇ ᴛʜᴀᴛ ᴀ ᴀɴᴅ ʙ ʜᴀᴠᴇ ᴀᴛ ʟᴇᴀꜱᴛ 3 ᴀꜱ ᴀ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ. ʙᴜᴛ, ᴛʜɪꜱ ᴄᴏɴᴛʀᴀᴅɪᴄᴛꜱ ᴛʜᴇ ꜰᴀᴄᴛ ᴛʜᴀᴛ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇ. ᴛʜɪꜱ ᴍᴇᴀɴꜱ ᴛʜᴀᴛ ᴏᴜʀ ᴀꜱꜱᴜᴍᴘᴛɪᴏɴ ɪꜱ ɴᴏᴛ ᴄᴏʀʀᴇᴄᴛ

Hence, √3  is an irrational number.

explanation:

ᴘʟᴇᴀꜱᴇ ᴍᴀʀᴋ ᴍᴇ ᴀꜱ ʙʀᴀɪɴʟɪᴇꜱᴛ