Question

prove that √3 is irrational​

Answer 1

it is irrational because it's root is not a whole number so it's a irrational number but it can be represented on the number line

Answer 2

Answer :

• √3 is irrational

To prove:

• √3 is irrational

Solution:

Let us assume that √3 is rational so,

√3 can be written in the form of a/b :√3 = a/b

where , a and b are co primes so their hcf is 1

● √3 = a / b

● √3 b = a

squaring on both sides we get,

● (√3 b)² = a²

● 3b² = a²

● a² / 3 = b²

● 3 divides a²

So 3 divides a is (1)

● a / 3 = c ( where c is integer)

● a = 3c

As we know that ,

● 3b² = a²

Now putting value a = 3c

● 3b² = (3c)²

● 3b² = 9c²

● b² = 1/3 × 9c²

● b² = 3c²

● b² / 3 = c²

● Again 3 divides b²

So 3 divides also b is equation (2)

Now by equation (1) and (2) we get :

• 3 divides both a and b

so 3 is a factor of a and b then a and b have a factor of 3 so, a and b are not co primes

• our assumptions is wrong by contradiction that √3 is irrational

So, √3 is irrational

Theorem:

• if p is prime factor a and prime factor (p) divides a² , then prime factor (p)divides a , where a is positive number