Q2:

Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Step 1: Theoretical Foundation & Definitions

To rigorously evaluate the proposition, we must first establish the mathematical definitions of the sets of numbers involved:

• Positive Integers ($\mathbb{Z}^+$): The set of whole numbers greater than zero, denoted as $\{1, 2, 3, 4, 5, \dots\}$.
• Rational Numbers ($\mathbb{Q}$): Any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$) and $q \neq 0$. Their decimal expansions are either terminating or non-terminating but repeating.
• Irrational Numbers ($\mathbb{I}$): Real numbers that cannot be expressed as a simple fraction $\frac{p}{q}$. Their decimal expansions are non-terminating and non-repeating [Per the fundamental properties of real numbers].

Step 2: Evaluating the Proposition

The question poses a universal statement: "The square roots of all positive integers are irrational."

In formal logic, to disprove a universal statement ("for all $x$"), one only needs to provide a single valid counterexample ("there exists an $x$ such that..."). If we can find even one positive integer whose square root is a rational number, the entire proposition is proven false [By the principle of Proof by Counterexample].

Step 3: Constructing the Counterexample

Let us examine the positive integer $n = 4$.

Taking the principal square root of $4$:

$ \sqrt{4} = 2 $

Now, we must determine if the result, $2$, is a rational number. We apply the definition of a rational number ($\frac{p}{q}$):

$ 2 = \frac{2}{1} $

Here, $p = 2$ (which is an integer) and $q = 1$ (which is an integer and $1 \neq 0$). Because $2$ perfectly satisfies the conditions of a rational number, $\sqrt{4}$ is rational. This directly contradicts the claim that the square roots of all positive integers are irrational.

Step 4: Generalization of the Counterexample

This phenomenon is not isolated to the number $4$. It applies to all perfect squares. A perfect square is an integer that is the square of an integer.

Let $k$ be any positive integer. If we define a positive integer $n$ such that $n = k^2$, then:

$ \sqrt{n} = \sqrt{k^2} = k $

Since $k$ is an integer, it can always be written as $\frac{k}{1}$, making it a rational number. We can observe this pattern in the table below:

Step 5: Visualizing Rational vs. Irrational Square Roots

The geometric representation below maps the square roots of positive integers onto a real number line, demonstrating the precise locations where square roots yield rational integers versus irrational values.

Final Solution: No, the square roots of all positive integers are not irrational. A definitive counterexample is the positive integer $4$. The square root of $4$ is $2$, which is a rational number because it can be expressed exactly as the fraction $\frac{2}{1}$. This holds true for all perfect squares (e.g., $\sqrt{9}=3$, $\sqrt{16}=4$, $\sqrt{25}=5$).