Question

ʜᴇʏ ғʀᴇɪɴᴅs
ᴀɴsᴡᴇʀ ᴛʜɪs ᴏɴᴇ

ϙᴜᴇsᴛɪᴏɴ :) ᴘʀᴏᴏᴠᴇ ᴛʜᴀᴛ √2 ɪs ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

ᴛʜᴇ ᴀʙᴏᴠᴇ ᴍᴇɴᴛɪᴏɴᴇᴅ ϙᴜᴇsᴛɪᴏɴ ɪs ғʀᴏᴍ sᴜʙᴊᴇᴄᴛ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄs

sᴏ ᴘʟᴇᴀsᴇ ᴅᴏ ɴᴏᴛ sᴘᴀᴍ ᴀɴᴅ ɢɪᴠᴇ ᴏɴʟʏ ʀɪɢʜᴛ sᴏʟᴜᴛɪᴏɴs
-ɢᴜᴅ ᴀғᴛᴇʀɴᴏᴏɴ ​

Answer 1

Answer:

If √2 could be written as a rational number, the consequence would be absurd. So it is true to say that √2 cannot be written in the form p/q. Hence √2 is not a rational number. Thus, Euclid succeeded in proving that √2 is an Irrational number.

Answer 2

Answer:

Given √2

To prove: √2 is an irrational number.

Proof:

Let us assume that √2 is a rational number.

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

√2 = p/q

Here p and q are coprime numbers and q ≠ 0

Solving

√2 = p/q

On squaring both the side we get,

=>2 = (p/q)2

=> 2q2 = p2……………………………..(1)

p2/2 = q2

So 2 divides p and p is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ………………………………..(2)

From equations (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, p, q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√2 is an irrational number.