Assertion: Five is a Rational Number
Five is classified as a rational number because it can be expressed as a fraction of two integers. Specifically, it can be written as:
- 5 = 5/1
This representation confirms that five meets the criteria of a rational number, which is any number that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).
Reason: The Square Roots of Positive Integers are Irrational Numbers
The assertion that the square root of positive integers are irrational is not accurate for all positive integers. However, it holds true specifically for certain integers. Here’s a detailed explanation:
- Definition of Irrational Numbers: An irrational number cannot be expressed as a fraction of two integers. It has a non-repeating, non-terminating decimal expansion.
- Square Roots of Perfect Squares: The square roots of perfect squares (e.g., 1, 4, 9, 16) are rational numbers. For example:
- √4 = 2 (rational)
- √9 = 3 (rational)
- Square Roots of Non-Perfect Squares: In contrast, the square roots of non-perfect squares (e.g., 2, 3, 5) are irrational. For example:
- √2 ≈ 1.414 (irrational)
- √3 ≈ 1.732 (irrational)
- Proof of Irrationality: The irrationality of square roots of non-perfect squares can be proven using contradiction. Assuming \( \sqrt{n} \) is rational leads to a contradiction regarding the prime factorization of integers.
In conclusion, while five is a rational number, the square roots of non-perfect squares are irrational, illustrating the distinction between different types of integers and their properties.