Question

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Prove that √3 is irrational​

Answer 1

Let us assume to the contrary that √3 is a rational number.

It can be expressed in the form of p/q

where p and q are co-primes and q≠ 0.

⇒ √3 = p/q

⇒ 3 = p²/q² (Squaring on both the sides)

⇒ 3q² = p²………………………………..(1)

It means that 3 divides p² and also 3 divides p because each factor should appear two times for the square to exist.

So we have p = 3r

where r is some integer.

⇒ p² = 9r²………………………………..(2)

from equation (1) and (2)

⇒ 3q² = 9r²

⇒ q² = 3r²

Where q² is multiply of 3 and also q is multiple of 3.

ᴛʜᴇɴ ᴘ, q ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ ᴏғ 3. ᴛʜɪs ʀᴜɴs ᴄᴏɴᴛʀᴀʀʏ ᴛᴏ ᴛʜᴇɪʀ ʙᴇɪɴɢ ᴄᴏ-ᴘʀɪᴍᴇs. ᴄᴏɴsᴇǫᴜᴇɴᴛʟʏ, ᴘ / q ɪs ɴᴏᴛ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. ᴛʜɪs ᴅᴇᴍᴏɴsᴛʀᴀᴛᴇs ᴛʜᴀᴛ √3 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

Answer 2

Answer:

Let us assume to the contrary that √3 is a rational number. where p and q are co-primes and q≠ 0. ... This demonstrates that √3 is an irrational number.