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Difficulty: MediumClassification of Numbers

Consider the following statements regarding the classification of numbers:

I. The number 00 is an even integer, but it is classified as neither positive nor negative.
II. The fraction 227\frac{22}{7} is an irrational number because it is commonly used as the value of the constant π\pi.
III. The product of any two distinct irrational numbers is always an irrational number.

Which of the statements given above is/are correct?

  1. A
    I and II only
  2. B
    I and III only
  3. I onlyAnswer
  4. D
    II and III only

Answer

Only the first statement is correct.
Only the first statement is true. Zero is an even integer because it is an integer multiple of 22 (0=2×00 = 2 \times 0), and it sits at the origin of the number line, meaning it is neither positive nor negative. The second statement is false because any number that can be expressed as the quotient of two integers, such as 227\frac{22}{7}, is rational by definition. The third statement is false because the product of distinct irrational numbers can be rational (for instance, 2×82=16\sqrt{2} \times 8\sqrt{2} = 16).

Step-by-Step Solution

1
Evaluate the parity and sign classification of zero in Statement I.
Zero is divisible by 22 without a remainder (0=2×00 = 2 \times 0), making it an even integer. It sits exactly between the negative and positive numbers on the number line, so it is strictly neither positive nor negative. Statement I is true.
To verify fundamental integer properties regarding the origin point.
2
Analyze the definition of the fraction in Statement II.
A rational number is any number that can be expressed as a ratio of two integers (pq\frac{p}{q}, where q≠0q \neq 0). Since 2222 and 77 are integers, 227\frac{22}{7} is a rational number. It is merely an approximation of π\pi, not its exact value. Statement II is false.
To test the distinction between a rational approximation and an irrational constant.
3
Test the closure property of irrational numbers under multiplication for Statement III.
Multiplying two distinct irrational numbers, such as (2−3)(2 - \sqrt{3}) and (2+3)(2 + \sqrt{3}), yields 4−3=14 - 3 = 1. Alternatively, 2×8=16=4\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4. Since 11 and 44 are rational numbers, the product of distinct irrationals is not always irrational. Statement III is false.
To determine if irrational numbers are closed under multiplication.

Key Concept

Classification of Real Numbers: Parity of Zero, Rational vs Irrational Definitions, and Closure Properties
Estimated Time:1m 15s
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Topics

Classification of Numbers

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