Try to find the decimal expansions of and . What do you observe about the repetition of the digits after the decimal point?

Question

KnowledgeBoat · Accepted Answer

Decimal expansion of : By long division : ⇒ The digit 3 repeats infinitely after the decimal point. Decimal expansion of : By long division: ⇒ After the decimal point, "91" appears once and then the digit 6 repeats infinitely. Observations : 1\. Both decimal expansions are non-terminating but repeating. 2\. In , the repetition starts immediately after the decimal point (pure repeating decimal). 3\. In , there are some non-repeating digits before the repeating block begins (general/mixed repeating decimal). This happens because the denominator 3 (in ) has only the prime factor 3, and the denominator 12 = 22 × 3 (in ) has both 2 and 3 as factors. The presence of a prime factor other than 2 and 5 in the denominator makes the decimal repeating. Hence, and . Both have repeating digits after the decimal point.

Mathematics

Try to find the decimal expansions of and . What do you observe about the repetition of the digits after the decimal point?

Whole Numbers

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