Classify each number as rational or irrational: 4.27 0.375 0.232342345 _ 462 1 Rational Irrational
The Correct Answer and Explanation is:
Classification:
Rational:
- 4.27
- 0.375
- 13/1
Irrational:
- 0.232342345…
- √62
Explanation:
Numbers can be categorized based on whether they can be expressed as a ratio of two integers. Rational numbers are those that can be written in fraction form with a nonzero denominator, while irrational numbers have decimal expansions that neither terminate nor repeat.
The number 4.27 is a finite decimal and can be written as a fraction, 427100\frac{427}{100}, which makes it rational. Similarly, 0.375 has a terminating decimal expansion and can be expressed as 3751000\frac{375}{1000}, confirming its rationality. The fraction 131\frac{13}{1} is already in the form of a ratio, clearly demonstrating it belongs to the rational category.
The number 0.232342345… has an unpredictable pattern in its decimal places, indicating a non-repeating, non-terminating nature. Since it cannot be rewritten as an exact fraction, it qualifies as irrational. The square root of 62 results in a decimal that continues indefinitely without repetition. Since the square root of a non-perfect square lacks an exact fractional representation, it also falls under the irrational classification.
Mathematical classifications such as these help in understanding numerical structures, essential for various applications in algebra, calculus, and real-world problem-solving. Recognizing whether a number is rational or irrational provides insight into its computational behavior, guiding mathematical operations with precision. Each type serves distinct purposes, from forming algebraic equations to defining continuous, infinite sets in advanced mathematics.
