The square root of 4 is 2, which can be expressed in the form (p)/(q) as (2)/(1), where p = 2 and q = 1 are integers and q ≠ 0. Therefore, 2 is a rational number.

Show that the squre roots of all positive integer are not irrational

Answer: The square roots of all positive integers are not irrational. For example, $\sqrt{4} = 2 = \frac{2}{1}$ , which is a rational number.

Step-by-step solution

Step 1: Understand the statement and define counterexample strategy

To disprove the universal claim that the square root of every positive integer is irrational, we only need to provide a single counterexample. A counterexample is a positive integer whose square root is a rational number.

Step 2: Select a perfect square as a counterexample

Consider the positive integer $4$ , which is a perfect square. Any perfect square like $1$ , $4$ , $9$ , or $16$ can serve as a valid counterexample.

Step 3: Evaluate the square root and verify rationality

The square root of $4$ is $2$ , which can be expressed in the form $\frac{p}{q}$ as $\frac{2}{1}$ , where $p = 2$ and $q = 1$ are integers and $q \neq 0$ . Therefore, $2$ is a rational number.

Step 4: State the conclusion

Since the square root of a perfect square like $4$ (or $1$ , $9$ , $16$ , etc.) is rational, it is not true that the square roots of all positive integers are irrational.

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