Practice question · Multiple choice
Every fraction has a decimal expansion that either terminates or eventually repeats, and no irrational number does either. Why does that difference follow from what a fraction is?
Hints
- Do the long division for 1/7 and watch the remainders. How many different ones can there be?
- What happens the moment a remainder appears for the second time?
Show the answer
B. Because division by b has only b remainders, so one must recur.
Why
Long division by b has only b possible remainders, so within b steps one must recur, and from there the digits repeat because the division is identical. A remainder of 0 is the terminating case. The converse completes it: a repeating expansion converts back to a fraction, so 0.101001000100001… is irrational by construction.
Practise Number Systems
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