What you’ll learn
Recognize terminating and repeating decimals.
Key idea
A nonzero remainder repeating forces later division digits to repeat; zero means the expansion stops.
Zero remainder stops division; recurrence of a nonzero remainder forces repeated digits and does not make the fraction irrational.
Worked example
Classify examples and non-examples for the concept 'Recognize terminating and repeating decimals'.
Classify the usual decimal expansion of 1/17 as terminating or nonterminating repeating, with division evidence.
- For 1/17, repeated decimal division produces remainder record [1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12, 1].
- The ending record shows nonzero remainder 1 repeats, so subsequent digits cycle; classify the usual decimal expansion as nonterminating repeating.
Answer: Nonterminating repeating — nonzero remainder 1 repeats, so subsequent digits cycle.
Quick Check
Classify examples and non-examples for the concept 'Recognize terminating and repeating decimals'.
Classify the usual decimal expansion of 2/25 as terminating or nonterminating repeating, with division evidence.