As a domain expert in mathematics, I find the question of whether the square root of a number is rational or irrational to be a fundamental one in the study of number theory. To address this question, let's first define what we mean by rational and irrational numbers.

Rational numbers are those that can be expressed as the quotient or fraction `\(p/q\)` of two integers, where `p` is the numerator and `q` is the denominator, and `q` is not zero. A key characteristic of rational numbers is that they can be represented as a finite decimal or a repeating decimal.

Irrational numbers, on the other hand, are numbers that cannot be expressed as a simple fraction of two integers. They have decimal representations that are non-repeating and infinite. A classic example of an irrational number is the square root of 2, as you've mentioned.

Now, to determine whether the square root of a number is rational or irrational, we must consider the nature of the number in question. If the number is a perfect square, such as 4 or 9, then its square root is a rational number because it can be expressed as an integer. For instance, the square root of 4 is 2, and the square root of 9 is 3.

However, if the number is not a perfect square, its square root will be irrational. This is because there is no pair of integers that can be multiplied together to produce a non-perfect square. The proof of this statement can be approached by contradiction. Assume that there are two integers `m` and `n` such that `m/n` equals the square root of a non-perfect square `x`. Squaring both sides of the equation would give `m^2/n^2 = x`, which implies that `m^2 = x * n^2`. If `x` is not a perfect square, then `m^2` would also not be a perfect square, leading to a contradiction because `m^2` would have to be an integer (as it's the square of an integer `m`), and `x * n^2` would be a product of two integers, thus a perfect square if `x` were rational.

The proof by contradiction shows that if `x` is not a perfect square, then it cannot be expressed as a ratio of two integers, and therefore, its square root is irrational.

In conclusion, the determination of whether the square root of a number is rational or irrational hinges on whether the original number is a perfect square. If it is, the square root is rational; if not, the square root is irrational.

Now, let's proceed to the next steps as per your instructions.

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Alternatively, an irrational number is any number that is not rational. It is a number that cannot be written as a ratio of two integers (or cannot be expressed as a fraction). For example, the square root of 2 is an irrational number because it cannot be written as a ratio of two integers.read more >>