Question

prove that√3 is an irrational number


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Answer 1

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

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✏Given:

Number 3

✏To Prove:

Root 3 is irrational

✏Proof:

➜Let us assume the contrary that root 3 is rational. Then √3 = p/q, where p, q are the integers i.e., p, q ∈ Z and co-primes, i.e., GCD (p,q) = 1.

√3 = p/q

⇒ p = √3 q

By squaring both sides, we get,

p² = 3q²

p² / 3 = q² ------- (1)

(1) shows that 3 is a factor of p. (Since we know that by theorem, if a is a prime number and if a divides p2, then a divides p, where a is a positive integer)

Here 3 is the prime number that divides p2, then 3 divides p and thus 3 is a factor of p.

Since 3 is a factor of p, we can write p = 3c (where c is a constant). Substituting p = 3c in (1), we get,

(3c)2 / 3 = q²

9c2/3 = q²

3c²= q²

c² = q² /3 ------- (2)

Hence 3 is a factor of q (from 2)

Equation 1 shows 3 is a factor of p and Equation 2 shows that 3 is a factor of q. This is the contradiction to our assumption that p and q are co-primes. So, √3 is not a rational number. Therefore, the root of 3 is irrational.

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