Which option correctly describes the nature of (2.3454545...)?
Answer and explanation
Correct answer: Rational number
The governing concept is the decimal characterization of rational numbers. A decimal is rational when it terminates or eventually repeats a finite block of digits. In 2.3454545..., the digits after the initial 3 continue with the repeating block 45: 2.3\overline{45}. Therefore it is a recurring, non-terminating decimal and must represent a rational number. It could be converted into a fraction by the standard place-value subtraction method, although that calculation is not needed to classify it. It is not irrational because irrational decimals never repeat periodically. It is real, so option C is impossible, and it is not an integer because it has a nonzero fractional part. Hence option A is correct.
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What is the correct answer to this question?
Rational number
Why is this the correct answer?
The governing concept is the decimal characterization of rational numbers. A decimal is rational when it terminates or eventually repeats a finite block of digits. In 2.3454545..., the digits after the initial 3 continue with the repeating block 45: 2.3\overline{45}. Therefore it is a recurring, non-terminating decimal and must represent a rational number. It could be converted into a fraction by the standard place-value subtraction method, although that calculation is not needed to classify it. It is not irrational because irrational decimals never repeat periodically. It is real, so option C is impossible, and it is not an integer because it has a nonzero fractional part. Hence option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.
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