Question

prove that the following are irrational :1/√5-√3​

Answer 1

Answer:

prove :

3

+

5

is irrational.

Let us assume it to be a rational number.

Rational numbers are the ones that can be expressed in

q

p

form where p,q are integers and q isn't equal to zero.

3

+

5

=

q

p

3

=

q

p

−

5

squaring on both sides,

3=

q

2

p

2

−2.

5

(

q

p

)+5

⇒

q

(2

5

p)

=5−3+(

q

2

p

2

)

⇒

q

(2

5

p)

=

q

2

2q

2

−p

2

⇒

5

=

q

2

2q

2

−p

2

.

2p

q

⇒

5

=

2pq

(2q

2

−p

2

)

As p and q are integers RHS is also rational.

As RHS is rational LHS is also rational i.e

5

is rational.

But this contradicts the fact that

5

is irrational.

This contradiction arose because of our false assumption.

so,

3

+

5

irrational.