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Difficulty: MediumClassification of Numbers

A student is reviewing fundamental number properties and evaluating several arithmetic scenarios. Identify which of the following assertions regarding number classification are mathematically valid. (Select all that apply)

  1. The product of any non-zero rational number and an irrational number is always an irrational number.Answer
  2. B
    The mathematical expression 2+3×52 + 3 \times 5 simplifies to a composite number.
  3. C
    The unit digit in the decimal expansion of 2402^{40} is classified as a perfect square.
  4. Every non-terminating, recurring decimal expansion can be expressed as a rational number.Answer

Answer

The mathematically valid assertions are that the product of a non-zero rational and an irrational number is always irrational, and that every non-terminating, recurring decimal is a rational number.
The assertions stating that the product of a non-zero rational and an irrational number is irrational, and that recurring decimals are rational, are fundamental and mathematically sound properties of the real number system.

Step-by-Step Solution

1
Evaluate the first assertion regarding the product of rational and irrational numbers.
The statement is determined to be valid.
Multiplying a ratio of integers (non-zero) by a non-repeating decimal inherently yields a non-repeating decimal.
2
Evaluate the arithmetic expression in the second assertion.
The expression equals 1717, which is a prime number.
Following order of operations, multiplication precedes addition (3×5=153 \times 5 = 15, then 15+2=1715 + 2 = 17).
3
Determine the unit digit for the third assertion.
The unit digit is 66, which is not a perfect square.
Because 4040 is a multiple of 44 (the cyclicity of base 22), the unit digit corresponds to the 44 th power in the cycle (24=162^4 = 16).
4
Evaluate the fourth assertion regarding recurring decimals.
The statement is determined to be valid.
All repeating decimals can be mathematically converted into a fraction format (e.g., 0.333...0.333... becomes 1/31/3), satisfying the definition of a rational number.

Key Concept

Classification of Rational and Irrational Numbers
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Topics

Classification of Numbers

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