Stop or loop

The idea

Dividing the numerator of a rational number by its denominator either terminates with remainder 0 or repeats; only finitely many remainders are possible, so one recurs and the division loops.

What can happen?

Divide the top of a fraction by its bottom and you get a decimal. For 3/8 the decimal 0.375 stops. For 5/11 the decimal 0.4545… goes on and on. What can happen?

Divide 1 by 7. What will the digits do?

• Stop after a few digits
• Go on forever in a repeating pattern
• Go on forever with no pattern

Long division

Do the long division one digit at a time. Watch the remainders: the blue chips show which ones have appeared.

Stop or loop

If you divide the numerator of a rational number by its denominator, exactly one of two things happens: it terminates (a remainder of 0) or it repeats (the digits loop). Only finitely many remainders are possible, so one must recur, and then the division loops.

When you divide by 7 the remainders can only be 1, 2, 3, 4, 5 or 6. After at most 6 steps, one of them must come back.

Notes

The decimal of a rational number either terminates (remainder 0) or repeats. Only finitely many remainders are possible, so one must recur and the division loops.

Check yourself

You divide by 7. How many different non-zero remainders are possible?

Answer: 6

The remainders can only be 1, 2, 3, 4, 5, 6: 6 of them. So after at most 6 digits a remainder must come back.

3/8 = 0.375, and the decimal stops. Why?

While dividing 5 by 11, a remainder appears for the second time. What happens next?

Which kind of decimal can a rational number not have?

• A remainder of 0 appeared — correct. Yes. With a remainder of 0 there is nothing left to divide.
• A remainder appeared twice. That makes the digits repeat. Here the remainder was 0.
• 8 is a small number. The size of 8 does not matter. The remainder became 0.
• The decimal stops. It stops only when the remainder is 0.
• The digits start to repeat — correct. Yes. The same remainder gives the same next digit, and the division loops.
• The digits never repeat. The same remainder always leads to the same digits again.
• One that stops, such as 0.375. Terminating decimals are rational numbers.
• One that repeats, such as 0.4545…. Repeating decimals are rational numbers.
• One that never ends and never repeats — correct. Yes. A rational number's decimal always terminates or repeats.