Question

prove that √5 is a irrational number​

Answer 1

$\huge \mathtt \blue{ƛƝƧƜЄƦ}$

Given: √5

We need to prove that √5 is irrational number

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

ʜᴇɴᴄᴇ, ᴘ,ǫ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ . ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴏᴜʀ ᴀssᴜᴍᴘᴛɪᴏɴ ᴛʜᴀᴛ ᴛʜᴇʏ ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇs. ᴛʜᴇʀᴇғᴏʀᴇ, ᴘ/ǫ ɪs ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ

√5 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ

ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ!!

Answer 2

$\geqslant$

We need to prove that √5 is irrational number

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

ʜᴇɴᴄᴇ, ᴘ,ǫ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ . ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴏᴜʀ ᴀssᴜᴍᴘᴛɪᴏɴ ᴛʜᴀᴛ ᴛʜᴇʏ ᴀʀᴇ ᴄᴏ-ᴘʀɪᴍᴇs. ᴛʜᴇʀᴇғᴏʀᴇ, ᴘ/ǫ ɪs ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ......

22❣️- inboz