Question

√3 is irrational proof​

Answer 1

Step-by-step explanation:

$\sqrt{3} is \: irrtional \: because \: 3 \: is \: not \: a \: perfect \: square$

Answer 2

We can prove that √3 is irrational by the method of contradiction:

Let us assume that √3 is a rational number.  

Then, we know that a rational number should be in the form of p/q  where p and q are co- prime numbers.

 So,  √3 = p/q { where p and q are co- primes}

√3q = p   Squaring both the sides

(√3q)² = p²

3q² = p² ........ ( Eq 1 )  

So,  if 3 is the factor of p²  then, 3 is also a factor of p ..... ( Theorem 10.1 )

 => Let p = 3m { where m is any integer }

Squaring both sides  p² = (3m)²

p² = 9m²

Putting the value of p² in  (Equation i )

3q² = (p² ) (3q² ) = 9m²

q² = 3m²  

So,  if 3 is factor of q²  then, 3 is also factor of q   Since  3 is factor of p and q . Then, our assumption that p & q are co- prime is wrong. Hence,. √3 is an irrational number.

Hence Proved