Question

no 5 one pls I need this urgently
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Answer 1

Let us consider 2 be a rational number, then

2=p/q, where ‘p’ and ‘q’ are integers, q=0 and p, q have no common factors (except 1).

So,

2=p2/q2

p2=2q2 …. (1)

As we know, ‘2’ divides 2q2, so ‘2’ divides p2 as well. Hence, ‘2’ is prime.

So 2 divides p

Now, let p=2k, where ‘k’ is an integer

Square on both sides, we get

p2=4k2

2q2=4k2 [Since, p2=2q2, from equation (1)]

q2=2k2

As we know, ‘2’ divides 2k2, so ‘2’ divides q2 as well. But ‘2’ is prime.

So 2 divides q

Thus, p and q have a common factor of 2. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that 2 is not a rational number.

2 is an irrational number.

Now, let us assume 3−2 be a rational number, ‘r’

So, 3−2=r

3–r=2

We know that, ‘r’ is rational, ‘3−r’ is rational, so ‘2’ is also rational.

This contradicts the statement that 2 is irrational.

So, 3−2 is an irrational number.

Hence proved