Show that root 2 is irrational number

1.	Show that root 2 is irrational number
Answer» ᴀɴSᴡᴇƦ :- ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √2 ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. ᴛʜᴇʀᴇғᴏʀᴇ, $\frac{ \sqrt{2} }{1} = \frac{a}{b}$ ᴡʜᴇʀᴇ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ɪɴᴛᴇɢᴇʀs ᴀɴᴅ ʙ ɪs ɴᴏᴛ ᴇǫᴜᴀʟs ᴛᴏ 0. ɴᴏᴡ, $\sqrt{2 } b = a \\$ Sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs. $2 {b}^{2} = {a}^{2} \\ \\ {b}^{2} = \frac{ {a}^{2} }{2}$ ɪғ 2 ᴅɪᴠɪᴅᴇs ${a}^{2}$ ɪᴛ ᴀʟsᴏ ᴅɪᴠɪᴅᴇs ᴀ ɴᴏᴡ, $a = 2c$ Sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs, ${a}^{2} = 4 {c}^{2} \\ \\ 2 {b}^{2} = 4 {c}^{2} \\ \\ {b}^{2} = 2 {c }^{2} \\ \\ {c}^{2} = \frac{ {b}^{2} }{2} \\ \\$ ɪғ 2 ᴅɪᴠɪᴅᴇs ${b}^{2}$ 2 ᴀʟsᴏ ᴅɪᴠɪᴅᴇs ʙ ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜɪs ɪs ᴀ ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ, ʜᴇɴᴄᴇ, √2 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ, ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ʏᴏᴜ. ${\huge{\pink{\ddot{\smile}}}}$

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