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Difficulty: MediumClassification of Numbers

Consider the following statements regarding the properties of numbers:

Statement 1: The integer 00 is neither positive nor negative, but it is classified as an even rational number.
Statement 2: The addition of any rational number and any irrational number always results in an irrational number.
Statement 3: The fraction 227\frac{22}{7} is an irrational number because it represents the exact value of π\pi.

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

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

Answer

The correct answer includes Statement 1 and Statement 2 only.
The correct answer correctly identifies that Statement 1 and Statement 2 are mathematically sound truths, while Statement 3 is fundamentally flawed. Statement 1 holds because 00 satisfies the definition of an even number (2n2n) and a rational number (pq\frac{p}{q}). Statement 2 holds due to mathematical contradiction proofs regarding sums. Statement 3 is false because 227\frac{22}{7} is a rational number by definition, despite being an approximation for the irrational π\pi.

Step-by-Step Solution

1
Evaluate the mathematical properties of 00 in Statement 1.
Statement 1 is valid.
Zero has no sign (neither positive nor negative). It is an even integer because it can be expressed as 2×02 \times 0. It is a rational number because it can be written as 01\frac{0}{1}.
2
Analyze the closure property of addition for Statement 2.
Statement 2 is valid.
If a rational number rr is added to an irrational number xx, the sum must be irrational. (If r+x=qr + x = q where qq is rational, then x=q−rx = q - r, meaning xx would be rational, which contradicts the premise).
3
Examine the classification of 227\frac{22}{7} in Statement 3.
Statement 3 is invalid.
The fraction 227\frac{22}{7} is the ratio of two integers, which strictly defines it as a rational number. It is merely a common numerical approximation for π\pi, not the exact irrational value of π\pi itself.

Key Concept

Core properties of rational/irrational numbers and fundamental definitions of integers like zero.
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Topics

Classification of Numbers

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