Question

Prove that 2 + √5 is irrational.​

Answer 1

Answer:

possible, let us assume 2+ √5 is a rational number.

2+ √5 = p/q where p,q∈z,q



=0

2−

p/q =− √5

q

2q−p

=−

√5

⇒−

√5

is a rational number

∵

q

2q−p

is a rational number

But −

√5

is not a rational number.

∴ Our supposition 2+

√5

is a rational number is wrong.

⇒2+

√5

is an irrational number.

Answer 2

let us assume that 2 + √5 is not an irrational number.

if 2 + √5 is rational then let 2 + √5 = p upon q.

Therefore we get √5 = p upon q - 2.

In this equation left side is an irrational number and right side rational number which is contradictory, so 2 + √5 is not an rational but it is an irrational number.