Why is root three an irrational number?

Question

Why is $\sqrt{3}$ an irrational number since it can be expressed as ratio of two numbers $(2\sqrt{3}+3)$ and $(2+\sqrt{3})$ ?

Answer 1

I would like to focus on the key issue here (the definition of rationality) as opposed to the actual proof (which is commonplace, straightforward, and available here anyways).

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Being rational or irrational is not a matter of being a ratio of two numbers, but instead of two integers.

If it was the former, every number $x$ would be rational: consider $\frac x 1$.

To prove that $x$ is irrational, you need to show that there exists no pair of integers $a,b$ such that $x = \frac a b$.

$2\sqrt 3 + 3$ and $2+\sqrt 3$ are not integers.

Answer 2

Square root of 3 is irrational because it can not be expressed as a ratio between two integers but what you are trying is just to express it as a ratio between two real numbers. If you are interested you can use the method of contradiction to prove that square root 3 is irrational number.