## prove that √5 is an irrational numberhey guys help me Don’t spam​

Question

prove that √5 is an irrational number
hey guys help me
Don’t spam​

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10 months 2021-07-07T16:16:40+00:00 2 Answers 0 views 0

Let us assume the opposite, i.e., √5 is a rational number. Hence, √5 can be written as in the form ab where a and b(b≠0) are co-prime (no common factor other than 1 ). By theorem: if p is a prime number and p divides a2, then p divides a, where a in a positive number.

Step-by-step explanation:

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒ √5 = p/q

On squaring both the sides we get,

⇒5 = p²/q²

⇒5q² = p² —————–(i)

p²/5 = q²

So 5 divides p

p is a multiple of 5

⇒ p = 5m

⇒ p² = 25m² ————-(ii)

From equations (i) and (ii), we get,

5q² = 25m²

⇒ q² = 5m²

⇒ q² is a multiple of 5

⇒ q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number.

Hence proved