Question

Prove that √3 is an irrational number. Hence prove that √3-5 is also an irrational number

Answer 1

take root 3 as rational no. first..
do as i did..
then prove that 3 is also a factor of q.
this would contradict the fact that they are coprime..
so our assumption is wrong and
$\sqrt{3}$
is a irrational no.
for root 3 -5 ..
we know the sum of irrational and rational no. is a irrational no. so it will be an irrational no.

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