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Repeating Decimal Period

A division that never terminates still has a rule: the remainders cycle, so the digits after the point repeat in a fixed block. Once a child finds the block, any far-off decimal place is a remainder question rather than a long division -- the 18th or the 20th digit costs the same as the 3rd. This is where 'the division does not end' turns from a dead end into a pattern. In grade 8 the same block runs the other way: a k-digit repeating block is that block over k nines, so sums, reversals and digit conditions on the block become fraction arithmetic.

Builds on: Divisibility and Remainder ReasoningRepeating Cycle Patterns

Progression (5)