Prove that root 3 is rational number

1.	Prove that root 3 is rational number
Answer» This is of root 2 Assume $\\sqrt{2}$ is rational, i.e. it can be expressed as a rational fraction of the form $\\frac{b}{a}$, where $a$ and $b$ are two relatively prime integers. Now, since $\\sqrt{2}=\\frac{b}{a}$, we have $2=\\frac{b^2}{a^2}$, or $b^2=2a^2$. Since $2a^2$ is even, $b^2$ must be even, and since $b^2$ is even, so is $b$. Let $b=2c$. We have $4c^2=2a^2$ and thus $a^2=2c^2$. Since $2c^2$ is even, $a^2$ is even, and since $a^2$ is even, so is a. However, two even numbers cannot be relatively prime, so $\\sqrt{2}$ cannot be expressed as a rational fraction; hence $\\sqrt{2}$ is irrational. $\\blacksquare$ Given in ncert examples

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