Question

only prove that √3 is irrational
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Answer 1

Answer:

firstly assume it as a rational number and then by contradiction method prove that it can't be expressed as rational number so it is irrational.

Answer 2

let us assume that✓3 is rational. Then we can find intgers a and b such that

√3=a/b, where a and b are co-prime

√3b = a/ , b not equal to 0

Squaring both sides

(√3)^2= (a/b] ^2 = 3= a^2/b^2

3b^2=a^2

b^2= a^2/3 -----------1

=> 3 divides a^2

=> 3 divides a

So we can write a = 3c ------------- 2

where c is some integer

substituting 2 in 1

b^2 = 3c^2/3 , b^2=9c^2/3 = 3c^2/1 = b^2=3c^2

OR c^2=b^2/3

=> 3 divides b^2

=> 3 divides b

Therefore, a and b have atleast 3 as a common factor. This condradicts that a and b are co-prime. this is because of our worng assumption that √3 is rational .

=> √3 is irational