No.
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No because the square root of 2 is an irrational number that can't be expressed as a fraction.
Square root of 2; square root of any natural number that is not a perfect square; the number pi; the number e.
Yes, it's 2
No. For two reasons: 1) Negative numbers are not natural numbers 2) All natural numbers are rational numbers but the square root of 6 is an irrational number and thus cannot be a natural number.
The same reasoning you may have seen in high school to prove that the square root of 2 is not rational can be applied to the square root of any natural number that is not a perfect square.