prove that √ 5 is an irrational number
Answers
ɢɪᴠᴇɴ: √5
ᴡᴇ ɴᴇᴇᴅ ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜᴀᴛ √5 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ
ᴘʀᴏᴏғ:
ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √5 ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.
sᴘ ɪᴛ ᴛ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇssᴇᴅ ɪɴ ᴛʜᴇ ғᴏʀᴍ ᴘ/ϙ ᴡʜᴇʀᴇ ᴘ,ϙ ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇ ɪɴᴛᴇɢᴇʀs ᴀɴᴅ ϙ≠0
⇒√5=ᴘ/ϙ
ᴏɴ sϙᴜᴀʀɪɴɢ ʙᴏᴛʜ ᴛʜᴇ sɪᴅᴇs ᴡᴇ ɢᴇᴛ,
⇒5=ᴘ²/ϙ²
⇒5ϙ²=ᴘ² —————–(ɪ)
ᴘ²/5= ϙ²
sᴏ 5 ᴅɪᴠɪᴅᴇs ᴘ
ᴘ ɪs ᴀ ᴍᴜʟᴛɪᴘʟᴇ ᴏғ 5
⇒ᴘ=5ᴍ
⇒ᴘ²=25ᴍ² ————-(ɪɪ)
ғʀᴏᴍ ᴇϙᴜᴀᴛɪᴏɴs (ɪ) ᴀɴᴅ (ɪɪ), ᴡᴇ ɢᴇᴛ,
5ϙ²=25ᴍ²
⇒ϙ²=5ᴍ²
⇒ϙ² ɪs ᴀ ᴍᴜʟᴛɪᴘʟᴇ ᴏғ 5
⇒ϙ ɪs ᴀ ᴍᴜʟᴛɪᴘʟᴇ ᴏғ 5
ʜᴇɴᴄᴇ, ᴘ,ϙ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ 5. ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴏᴜʀ ᴀssᴜᴍᴘᴛɪᴏɴ ᴛʜᴀᴛ ᴛʜᴇʏ ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇs. ᴛʜᴇʀᴇғᴏʀᴇ, ᴘ/ϙ ɪs ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ
√5 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ
ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ
have a great day ahead
Answer:
Done by Aritra kar.
Hope you are looking for this answer.
Step-by-step explanation:
®Proved by Method of Contradiction.
Given: √5
We need to prove that √5 is irrational
Proof:
Let us assume that √5 is a rational number.
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
Hence proved