Question

❤️Heyaa❤️ How do you show that √7 is irrational ? {Prove} Time limit : 7 mins ❗️Don’t spam ✔️5 points✔️ Answers to be given accordingly of class 10 .

Answer 1

$\huge\mathfrak{Answer}$

To prove that √7 is irrational

Solution :-

Let us assume that √7 is rational

Then there exists integers a and b such that b not equals to 0 and a,b are co primes

√7 = a/b where b not equals to and (a,b)=1

√7b = a

Squaring on both sides,

(√7b)^2 = a^2

=> 7|a^2 (since ,7|b^2 and a^2 = 7b^2)

=>7|a ——-(1)

So a can be written as

a = 7c for some integer ‘c’

=> a^2 = 7c^2

=>7b^2 = 49 c^2 (since ,a^2 = 7b^2)

=>b^2 = 7c^2

=>7|b^2 [since ,7|7c^2 and b^2 = 7c^2]

=> 7|b ——-(2){since ,by theorem}

From (1) and (2) we have ,

7|a and 7|b .This shows that 7 is a common factor for a and b.

This contradicts the fact that a and b are co-primes

This contraction has arisen because of our wrong assumption that √7 is rational

This our assumption is wrong

Hence √7 is irrational

Answer 2

$\huge\mathfrak\pink{Hey--dear}$ ❤♥

$YOUR \ ANSWER \ IS \ HERE$ ⤵

Proof : If possible, ʟᴇᴛ √3 ʙᴇ ʀᴀᴛɪᴏɴᴀʟ ᴀɴᴅ ʟᴇᴛ ɪᴛꜱ ꜱɪᴍᴩʟᴇꜱᴛ ꜰᴏʀᴍ ʙᴇ $\frac{a}{b}$

ᴛʜᴇɴ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ɪɴᴛᴇɢᴇʀꜱ ʜᴀᴠɪɴɢ ɴᴏ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ ᴏᴛʜᴇʀ ᴛʜᴀɴ 1 ᴀɴᴅ ʙ ɪꜱ ɴᴏᴛ 0.

Now, √3 =
$\frac{a}{b} = > \: 3 = \: \frac{a {}^{2} }{ {b}^{2} } \: {on \: squaring \: both \: sides} \\ \\ = > {3ab}^{2} = \: {a}^{2} ..............(i) \\ \\ = > 3 \: divides \: {a}^{2} \: \: \: ( \: 3 \: divides \: {3b}^{2} ) \\ \\ = > \:3 \: divides \: a \:$
[ 3 is prime and 3 divides a² => 3 divides a ] .

let a = 3 c for some integer c.
putting a = 3c in (i) , we get :
3b² = 9c² => b² = 3c²
=> 3 divides b² [ 3 divides 3c²]
=> 3 divides b

ᴛʜᴜꜱ, 3 ɪꜱ ᴀ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ ᴏꜰ ᴀ ᴀɴᴅ ʙ.

ʙᴜᴛ, ᴛʜɪꜱ ᴄᴏɴᴛʀᴀᴅɪᴄᴛꜱ ᴛʜᴇ ꜰᴀᴄᴛ ᴛʜᴀᴛ ᴀ ᴀɴᴅ ʙ ʜᴀᴠᴇ ɴᴏ ᴄᴏᴍᴍᴏɴ ꜰᴀᴄᴛᴏʀ ᴏᴛʜᴇʀ ᴛʜᴀɴ 1.

ᴛʜᴇ ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ ᴀʀɪꜱᴇꜱ ʙyᴇ ꜱɪ ᴜ ᴍᴇᴀɴ ᴛʜᴀᴛ √ 3 ɪꜱ ʀᴀᴛɪᴏɴᴀʟ.

ʜᴇɴᴄᴇ √3 ɪꜱ ɪʀʀᴀᴛɪᴏɴᴀʟ.

$<marquee > Hope it's help you$ ✌

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