Question

Prove that √3 is an irrational number.​

Answer 1

$\huge\bold{Answer:}$

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

ɪᴛ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇꜱꜱᴇᴅ ɪɴ ᴛʜᴇ ꜰᴏʀᴍ ᴏꜰ ᴘ/Q

ᴡʜᴇʀᴇ ᴘ ᴀɴᴅ Q ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇꜱ ᴀɴᴅ Q≠ 0.

⇒ √3 = ᴘ/Q

⇒ 3 = ᴘ2/Q2 (ꜱQᴜᴀʀɪɴɢ ᴏɴ ʙᴏᴛʜ ᴛʜᴇ ꜱɪᴅᴇꜱ)

⇒ 3Q2 = ᴘ2………………………………..(1)

ɪᴛ ᴍᴇᴀɴꜱ ᴛʜᴀᴛ 3 ᴅɪᴠɪᴅᴇꜱ ᴘ2 ᴀɴᴅ ᴀʟꜱᴏ 3 ᴅɪᴠɪᴅᴇꜱ ᴘ ʙᴇᴄᴀᴜꜱᴇ ᴇᴀᴄʜ ꜰᴀᴄᴛᴏʀ ꜱʜᴏᴜʟᴅ ᴀᴘᴘᴇᴀʀ ᴛᴡᴏ ᴛɪᴍᴇꜱ ꜰᴏʀ ᴛʜᴇ ꜱQᴜᴀʀᴇ ᴛᴏ ᴇxɪꜱᴛ.

ꜱᴏ ᴡᴇ ʜᴀᴠᴇ ᴘ = 3ʀ

ᴡʜᴇʀᴇ ʀ ɪꜱ ꜱᴏᴍᴇ ɪɴᴛᴇɢᴇʀ.

⇒ ᴘ2 = 9ʀ2………………………………..(2)

ꜰʀᴏᴍ ᴇQᴜᴀᴛɪᴏɴ (1) ᴀɴᴅ (2)

⇒ 3Q2 = 9ʀ2

⇒ Q2 = 3ʀ2

ᴡʜᴇʀᴇ Q2 ɪꜱ ᴍᴜʟᴛɪᴘʟʏ ᴏꜰ 3 ᴀɴᴅ ᴀʟꜱᴏ Q ɪꜱ ᴍᴜʟᴛɪᴘʟᴇ ᴏꜰ 3.

ᴛʜᴇɴ ᴘ, Q ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ ᴏꜰ 3. ᴛʜɪꜱ ʀᴜɴꜱ ᴄᴏɴᴛʀᴀʀʏ ᴛᴏ ᴛʜᴇɪʀ ʙᴇɪɴɢ ᴄᴏ-ᴘʀɪᴍᴇꜱ. ᴄᴏɴꜱᴇQᴜᴇɴᴛʟʏ, ᴘ / Q ɪꜱ ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. ᴛʜɪꜱ ᴅᴇᴍᴏɴꜱᴛʀᴀᴛᴇꜱ ᴛʜᴀᴛ √3 ɪꜱ ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

Answer 2

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

ɪᴛ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇꜱꜱᴇᴅ ɪɴ ᴛʜᴇ ꜰᴏʀᴍ ᴏꜰ ᴘ/Q

ᴡʜᴇʀᴇ ᴘ ᴀɴᴅ Q ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇꜱ ᴀɴᴅ Q≠ 0.

⇒ √3 = ᴘ/Q

⇒ 3 = ᴘ2/Q2 (ꜱQᴜᴀʀɪɴɢ ᴏɴ ʙᴏᴛʜ ᴛʜᴇ ꜱɪᴅᴇꜱ)

⇒ 3Q2 = ᴘ2………………………………..(1)

ɪᴛ ᴍᴇᴀɴꜱ ᴛʜᴀᴛ 3 ᴅɪᴠɪᴅᴇꜱ ᴘ2 ᴀɴᴅ ᴀʟꜱᴏ 3 ᴅɪᴠɪᴅᴇꜱ ᴘ ʙᴇᴄᴀᴜꜱᴇ ᴇᴀᴄʜ ꜰᴀᴄᴛᴏʀ ꜱʜᴏᴜʟᴅ ᴀᴘᴘᴇᴀʀ ᴛᴡᴏ ᴛɪᴍᴇꜱ ꜰᴏʀ ᴛʜᴇ ꜱQᴜᴀʀᴇ ᴛᴏ ᴇxɪꜱᴛ.

ꜱᴏ ᴡᴇ ʜᴀᴠᴇ ᴘ = 3ʀ

ᴡʜᴇʀᴇ ʀ ɪꜱ ꜱᴏᴍᴇ ɪɴᴛᴇɢᴇʀ.

⇒ ᴘ2 = 9ʀ2………………………………..(2)

ꜰʀᴏᴍ ᴇQᴜᴀᴛɪᴏɴ (1) ᴀɴᴅ (2)

⇒ 3Q2 = 9ʀ2

⇒ Q2 = 3ʀ2

ᴡʜᴇʀᴇ Q2 ɪꜱ ᴍᴜʟᴛɪᴘʟʏ ᴏꜰ 3 ᴀɴᴅ ᴀʟꜱᴏ Q ɪꜱ ᴍᴜʟᴛɪᴘʟᴇ ᴏꜰ 3.

ᴛʜᴇɴ ᴘ, Q ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ ᴏꜰ 3. ᴛʜɪꜱ ʀᴜɴꜱ ᴄᴏɴᴛʀᴀʀʏ ᴛᴏ ᴛʜᴇɪʀ ʙᴇɪɴɢ ᴄᴏ-ᴘʀɪᴍᴇꜱ. ᴄᴏɴꜱᴇQᴜᴇɴᴛʟʏ, ᴘ / Q ɪꜱ ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. ᴛʜɪꜱ ᴅᴇᴍᴏɴꜱᴛʀᴀᴛᴇꜱ ᴛʜᴀᴛ √3 ɪꜱ ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.