Question

Prove that√3 is an irrational number hence prove 3-√5 is also an irrational number?

Answer 1

Let √3 is rational
√3=p/q(p,q are rational numbers, q#0)(prime numbers)
√3= p/q
S. O. B
3=p*2/q*2
q*2=p*2/3
P=3d
q*2=9d/3=3d
so,p,q are rational and prime numberd √3 is irrational
So 3-√5 is also irrational because √5 is also irrational

Answer 2

Step-by-step explanation:

Let us assume that √3 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

So,

√3 = p/q { where p and q are co- prime}

√3q = p

Now, by squaring both the side

we get,

(√3q)² = p²

3q² = p² ........ ( i )

So,

if 3 is the factor of p²

then, 3 is also a factor of p ..... ( ii )

=> 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 & q both

So, our assumption that p & q are co- prime is wrong

hence,. √3 is an irrational number