Question

prove that the following are irrational ​

Answer 1

Answer:

1. under root 2 is an irrational so whether it is divided by 1 it remains irrational.

Answer 2

ᴛᴏ ᴘʀᴏᴠᴇ 1/√2 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ

ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √2 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ 

1/√2 = ᴘ/ϙ (ᴡʜᴇʀᴇ ᴘ ᴀɴᴅ ϙ ᴀʀᴇ ᴄᴏ ᴘʀɪᴍᴇ)

ϙ/ᴘ = √2

ϙ     = √2ᴘ

sϙᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs

ϙ²   = 2ᴘ²                                                  .....................(1)

ʙʏ ᴛʜᴇᴏʀᴇᴍ 

ϙ ɪs ᴅɪᴠɪsɪʙʟᴇ ʙʏ 2

∴ ϙ = 2ᴄ ( ᴡʜᴇʀᴇ ᴄ ɪs ᴀɴ ɪɴᴛᴇɢᴇʀ)

 ᴘᴜᴛᴛɪɴɢ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ϙ ɪɴ ᴇϙᴜɪᴛᴀᴛɪᴏɴ 1

2ᴘ² = ϙ² = 2ᴄ² =4ᴄ²

ᴘ² =4ᴄ² /2 = 2ᴄ²

ᴘ²/2 = ᴄ² 

ʙʏ ᴛʜᴇᴏʀᴇᴍ ᴘ ɪs ᴀʟsᴏ ᴅɪᴠɪsɪʙʟᴇ ʙʏ 2

ʙᴜᴛ ᴘ ᴀɴᴅ ϙ ᴀʀᴇ ᴄᴏᴘʀɪᴍᴇ

ᴛʜɪs ɪs ᴀ ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ ᴡʜɪᴄʜ ʜᴀs ᴀʀɪsᴇɴ ᴅᴜᴇ ᴛᴏ ᴏᴜʀ ᴡʀᴏɴɢ ᴀssᴜᴍᴘᴛɪᴏɴ

∴1/√2 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ