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Difficulty: MediumClassification of Numbers

When evaluating the fundamental categories and properties of the real number system, which of the following statements are mathematically correct?

  1. The square root of any prime number inherently belongs to the set of irrational numbers.Answer
  2. B
    Zero is classified as a positive integer that is neither odd nor even.
  3. Every integer can be expressed as a rational number in the form pq\frac{p}{q}, where qq is a non-zero integer.Answer
  4. D
    The mathematical constant π\pi is exactly equivalent to the ratio 227\frac{22}{7}, meaning it belongs to the rational numbers.

Answer

The mathematically correct statements are that the square root of any prime number is irrational, and every integer can be expressed as a rational number.
The correct statements accurately identify that prime numbers lack perfect square roots, making their roots inherently irrational, and that all integers are essentially a subset of rational numbers since any integer can be expressed as a fraction with a denominator of one.

Step-by-Step Solution

1
Analyze the statement regarding prime numbers and irrationality.
Since prime numbers only have two distinct positive divisors, they are never perfect squares. Therefore, their square roots cannot be simplified into rational fractions, making the statement correct.
To verify the relationship between primes and algebraic irrationality.
2
Evaluate the properties of zero given in the second statement.
Zero is determined to be neutral (neither positive nor negative) and even (since 0=2×00 = 2 \times 0). Therefore, the statement claiming it is positive and neither odd nor even is false.
To check for misconceptions regarding the classification and parity of zero.
3
Assess the definition of rational numbers applied to integers.
An integer zz can always be written as z1\frac{z}{1}. Since 11 is a non-zero integer, this satisfies the pq\frac{p}{q} definition of rational numbers, making the statement correct.
To confirm the subset relationship between integers and rational numbers.
4
Examine the classification of π\pi.
While 227\frac{22}{7} is a common approximation, π\pi cannot be written as a simple fraction of integers. It is irrational, making the statement false.
To differentiate between an exact mathematical constant and its heuristic approximation.

Key Concept

Classification of Numbers and Fundamental Properties
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Topics

Classification of Numbers

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