Question

Show that √5 is an
irrational no.
tence prove that 2√5-3 is
also irrational.​

Answer 1

Answer:

Hope it helps..

Step-by-step explanation:

Let us consider

5

be a rational number, then

5

=p/q, where ‘p’ and ‘q’ are integers, q



=0 and p, q have no common factors (except 1).

So,

5=p

2

/q

2

p

2

=5q

2

…. (1)

As we know, ‘5’ divides 5q

2

, so ‘5’ divides p

2

as well. Hence, ‘5’ is prime.

So 5 divides p

Now, let p=5k, where ‘k’ is an integer

Square on both sides, we get

p

2

=25k

2

5q

2

=25k

2

[Since, p

2

=5q

2

, from equation (1)]

q

2

=5k

2

As we know, ‘5’ divides 5k

2

, so ‘5’ divides q

2

as well. But ‘5’ is prime.

So 5 divides q

Thus, p and q have a common factor of 5. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that

5

is not a rational number.

5

is an irrational number.

Now, let us assume −3+2

5

be a rational number, ‘r’

So, −3+2

5

=r

−3–r=2

5

(−3–r)/2=

5

We know that, ‘r’ is rational, ‘(−3–r)/2’ is rational, so ‘

5

’ is also rational.

This contradicts the statement that √5 is irrational.

So, −3+2

5

is irrational number.

Hence proved.