Question

veify that root 2 is an irrational number​

Answer 1

Let root 2=a/b where a and b are non zero co prime no

Root 2 =a/b

Squaring both sides

2=a^2/b^2

2b^2 =a^2 ¹

So if 2 divided a square it should also divide a

Now a=2c for any integer c

Squaring both side

a^2=4c^2

From ¹

2b^2=4c^2

b^2 =2c^2

So 2 also divides b so 2 divide both a b but it contradics us that a and b are co prime hence root is irrational

Answer 2

Step-by-step explanation:

Let us assume root 2 a rational number.

so, root 2=p/q ( p and q are co primes)

squaring both the sides

(root 2)^2 = (p/q)^2

2=p^2/q^2

q^2=p^2/2

this shows that p^2 is divisible by 2

thus p is also divisible by 2.

as 2 is a factor of p

we can write p as 2m

now,

q^2= (2m)^2/2

q^2=4m^2/2

q^2=2m^2

q= 2m

so q is also divisible by 2.

now, as p and q were co primes , our assumption was wrong and root 2 is irrational.