Question

prove √5 in irrational number or not​

Answer 1

Answer:

because 5 is not a square of any basic number

Step-by-step explanation:

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Answer 2

To prove:

√5 is an irrational number.

Proof:

Let us assumed that √5 is a rational number.

It can be expressed in the from of p/q where q ≠ 0 and p and q are co-prime integers.

⟹ √5 = p/q

Squaring both sides,

⟹ (√5)² = (p/q)²

⟹ 5 = p²/q²

⟹ 5q² = p² ...1

⟹ q² = p²/5

Here, 5 is dividing p and we know that p is a multiple of 5. So, p² will be multiple of 5.

⟹ p = 5m

⟹ p² = (5m)²

⟹ p² = 25m² ...2

Now, from eq1 and eq2, we get:

⟹ 5q² = 25m²

⟹ q² = 25/5m²

⟹ q² = 5m²

So, here q is a multiple of 5. So, q² is also a multiple of 5.

This, contradicts our assumption that they are co-prime because co-prime numbers do not have a common factor except 1 but they also have common factor 5. Therefore, p/q is not a rational number.

Therefore,

√5 is an irrational number

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