Question

Show root 5 is irrational number

Answer 1

Let us assume that √5 is a rational number

{ for now }

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number

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Answer 2

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ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √5 ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ ᴛʜᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ɪɴ ᴛʜᴇ ғᴏʀᴍ ᴏғ ᴘ/ǫ ғᴏʀᴍ ᴡʜᴇʀᴇ ᴘ,ǫ ᴀʀᴇ ɪɴᴛᴇᴢᴇʀs.

sᴏ, √5 = ᴘ/ǫ

ᴘ = √5ǫ

ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ 'ᴘ' ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. sᴏ √5 ǫ ᴍᴜsᴛ ʙᴇ ʀᴀᴛɪᴏɴᴀʟ sɪɴᴄᴇ ɪᴛ ᴇǫᴜᴀʟs ᴛᴏ ᴘ

ʙᴜᴛ ɪᴛ ᴅᴏᴇsɴᴛ ᴏᴄᴄᴜʀs ᴡɪᴛʜ √5 sɪɴᴄᴇ ɪᴛs ɴᴏᴛ ᴀɴ ɪɴᴛᴇᴢᴇʀ

ᴛʜᴇʀᴇғᴏʀᴇ, ᴘ =/= √5ǫ

ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴛʜᴇ ғᴀᴄᴛ ᴛʜᴀᴛ √5 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ

ʜᴇɴᴄᴇ ᴏᴜʀ ᴀssᴜᴍᴘᴛɪᴏɴ ɪs ᴡʀᴏɴɢ ᴀɴᴅ √5 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

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