What type of number is (0.\overline{45})?
Answer and explanation
Correct answer: Rational real number
The notation 0.\overline{45} means 0.454545…, where the two-digit block 45 repeats indefinitely. A repeating decimal is rational because it can be written as a fraction of integers. Let x = 0.454545…; then 100x = 45.454545…. Subtracting gives 99x = 45, so x = 45/99 = 5/11. Therefore the number is rational, and every rational number is also a real number. It is not irrational or undefined. It is also not an integer: 5/11 is between 0 and 1, whereas an integer has no nonzero fractional part. Hence option C is the complete and precise classification.
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What is the correct answer to this question?
Rational real number
Why is this the correct answer?
The notation 0.\overline{45} means 0.454545…, where the two-digit block 45 repeats indefinitely. A repeating decimal is rational because it can be written as a fraction of integers. Let x = 0.454545…; then 100x = 45.454545…. Subtracting gives 99x = 45, so x = 45/99 = 5/11. Therefore the number is rational, and every rational number is also a real number. It is not irrational or undefined. It is also not an integer: 5/11 is between 0 and 1, whereas an integer has no nonzero fractional part. Hence option C is the complete and precise classification.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Decimal representation.
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