Question

prove that underrote 5 is a irratioal number​

Answer 1

Solution:

To Prove: √5 is irrational.

Proof:

★ Let us assume that √5 is a rational number.

Then, we can write:

→ √5 = p/q where p and q are integers, q ≠ 0 and p, q have no common factors (except 1)

→ 5 = p²/q²

→ p² = 5q² — (i)

As 5 divides 5q² so 5 divides p² but 5 is prime.

→ 5 divides p

Let p = 5k where k is some integer.

Substituting the value of p in (i), we get:

→ (5k)² = 5q²

→ 25k² = 5q²

→ 5k² = q²

As 5 divides k², 5 divides q² but 5 is prime.

→ 5 divides q

Therefore:

→ 5 divides p and q both.

Thus, p and q have common factor 5. So, this contradicts the fact that p, q have no common factors except 1.

Therefore, √5 is irrational number (Proved)

Learn More:

Rational Number: Any number that can be expressed in the form of p/q where p and q are integers, q ≠ 0 and p, q have no common factors except 1 is called Rational Number. Example: 2, 3, 4.5, 6 etc

Irrational Number: Any number that can not be expressed in the form of p/q where p and q are integers, q ≠ 0 and p, q have no common factors except 1 is called Irrational Number. Example: √2, √3, 1.01001000100001...