Question

prove that√3 is irrational?​

Answer 1

Answer:

yes it is irrational

Step-by-step explanation:

Let us assume to the contrary that √3 is a rational number. where p and q are co-primes and q≠ 0. ... Consequently, p / q is not a rational number. This demonstrates that √3 is an irrational number

Answer 2

⭐ $\rm\underline\purple{ Proof \: \: :- }$ Let us assume that

√3 is not irrational

⟶ √3 is a rational

⟶ √3 = $\frac{p}{q}$ where p, q € Z, q ≠ 0 and p, q are co-primes

• Squaring on both sides

⟶ (√3) ² = ($\frac{p}q{}$) ²

⟶ 3 = $\frac{p²}{q²}$

⟶ $\sf\fbox{p² \: = \: 3q²}$ ⟶ ①

⟶ p² is divisible by 3

⟶ p is also divisible by 3

Let $\fbox{p \: \: = \: \: 3k }$

from equation ①

⟶ (3k) ² = 3q²

⟶ 9k² = 3q²

⟶ 3k² = q² / q² = 3k²

⟶ q² is divisible by 3

⟶ q is also divisible by 3

:. 3 are the common factor for both p and q

But, p, q are co-primes

♨️ It is contradiction to our Assumption ♨️

:. Our assumption is Wrong

:. √3 is an irrational