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Difficulty: EasyClassification of Numbers

Read the following three statements concerning the properties of numbers:

1. The integer 00 is considered a positive even number.
2. The fraction 227\frac{22}{7} is a rational number, whereas the constant π\pi is an irrational number.
3. The product of any two irrational numbers always results in an irrational number.

Which of the statements given above is/are mathematically correct?

  1. 2 onlyAnswer
  2. B
    1 and 2 only
  3. C
    2 and 3 only
  4. D
    1, 2, and 3

Answer

Only the second statement is mathematically accurate: the fraction 22/7 is rational, and pi is irrational.
The correct answer is '2 only' because it is the only mathematically sound statement. The number 0 is neither positive nor negative, invalidating statement 1. The product of two irrational numbers can be rational (such as multiplying two identical square roots), invalidating statement 3. Statement 2 correctly identifies that any ratio of integers (like 22/7) is rational, while the mathematical constant pi is irrational.

Step-by-Step Solution

1
Evaluate Statement 1: 'The integer 0 is considered a positive even number.'
Statement 1 is incorrect.
By mathematical definition, the integer 0 is an even number (since it is divisible by 2 with no remainder), but it serves as the boundary between positive and negative numbers. It is strictly neutral, neither positive nor negative.
2
Evaluate Statement 2: 'The fraction 22/7 is a rational number, whereas the constant pi is an irrational number.'
Statement 2 is correct.
A rational number is any number that can be expressed as a fraction p/q where p and q are integers and q is not 0. Since 22 and 7 are integers, 22/7 is definitely rational. The mathematical constant pi represents a non-terminating, non-repeating decimal and cannot be expressed exactly as a simple fraction, making it irrational.
3
Evaluate Statement 3: 'The product of any two irrational numbers always results in an irrational number.'
Statement 3 is incorrect.
The set of irrational numbers is not closed under multiplication. For example, multiplying the irrational number sqrt(2) by another irrational number sqrt(2) yields 2, which is a rational integer.
4
Synthesize the evaluations to find the correct choice.
Since only statement 2 is mathematically valid, the correct option is '2 only'.
Matching our findings with the provided options confirms the final answer.

Key Concept

Classification of Real Numbers: Properties of Rational and Irrational Numbers
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Topics

Classification of Numbers

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