Find the repeating sequence of digits in the nonterminating decimal fraction representation of:(

step 1 Hi guys, let's solve problem 45. Find the repeating sequence of digits in the non -terminating d

Find the repeating sequence of digits in the nonterminating...

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Key Concepts

Rational Numbers

Rational numbers are numbers expressed as the quotient of two integers. Their decimal representations either terminate or have a repeating pattern, which results from the finite number of possible remainders when one integer is divided by another.

Repeating Decimals

A repeating decimal is a nonterminating decimal in which a specific sequence of digits repeats indefinitely. This repeating portion, often called the repetend or period, arises uniquely from the division process for rational numbers.

Long Division Method

Long division is a systematic technique for converting fractions to decimals. By performing division, one can detect the onset of a repeating pattern once the remainders begin to recur, thereby identifying the repeated sequence.

Periodicity

Periodicity in the context of decimal representations refers to the regular repetition of a block of digits. The period length is the number of digits in the repeating sequence, which is a consequence of the underlying properties of division in the base-ten system.

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