Mathematics

Mathematics

Problem

Is the number 0.123‾0.\overline{123} rational or irrational? Give a brief reason for your answer.
Intuitive explanation

Repeating decimals are always rational—that's the key insight. When a decimal pattern repeats forever, you can convert it to a fraction using algebra: multiply by a power of 10 to shift the pattern, then subtract to cancel the repeating part. Since 0.123‾0.\overline{123} has a finite repeating block, it can be expressed as a ratio of two integers, making it rational.

Intuitive explanation

Repeating decimals are always rational—that's the key insight. When a decimal pattern repeats forever, you can convert it to a fraction using algebra: multiply by a power of 10 to shift the pattern, then subtract to cancel the repeating part. Since 0.123‾0.\overline{123} has a finite repeating block, it can be expressed as a ratio of two integers, making it rational.

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