Divisions that never end

The idea

In some divisions a remainder is left after every regrouping, so long division never ends and the quotient cannot be written with finitely many decimal digits; the remainders and quotient digits can repeat in a cycle.

10 ÷ 3

Divide 10 by 3 with long division. 3 ones go in 3 times, and 1 is left. Regroup it as 10 tenths: 3 tenths each, 1 tenth left. Regroup that as 10 hundredths …

Will this sharing ever come to an end?

• Yes, after a few more steps
• No, it goes on for ever
• We cannot tell

Does it end?

For each division, predict first. Then tap Regroup and share and watch the digits and the left-overs.

The left-over decides

Each step leaves a left-over smaller than the divisor. If it is 0, the division ends. If a left-over comes back, every step after it repeats the steps before, so the digits repeat in a cycle and the division never ends.

1 ÷ 7 gave the block 142857. Multiply it by 1, 2, 3, 4, 5, 6: 142857, 285714, 428571, 571428, 714285, 857142. The same digits come back, cycled around! And 142857 × 7 = 999999.

In some divisions a remainder is left after every regrouping. Then long division never ends, and the quotient cannot be written with finitely many decimal digits: for example 10 ÷ 3 = 3.333 … The remainders and the quotient digits can repeat in a cycle.

• Division | Left-overs | Result
• 1 ÷ 4 | 1, 2, 0 | 0.25 (ends)
• 10 ÷ 3 | 1, 1, 1 … | 3.333 …
• 1 ÷ 7 | 1, 3, 2, 6, 4, 5, 1 … | 0.142857 142857 …
• 100 ÷ 11 | 1, 10, 1, 10 … | 9.0909 …

Notes

If a remainder is left after every regrouping, the division never ends. A left-over that comes back makes the digits repeat in a cycle. A left-over of 0 means it ends.

Check yourself

Which of these divisions ends?

In 1 ÷ 7 the left-overs are 1, 3, 2, 6, 4, 5 and then 1 again. How many digits are in the repeating block?

Answer: 6 digits

Six different left-overs, so the block 142857 has 6 digits.

Why must 1 ÷ 7 go on for ever?

1 ÷ 7 = 0.142857 142857 … What is the 8th digit after the point?

Answer: 4

Digits: 1 4 2 8 5 7 | 1 4 … The 8th digit is the 2nd of the block: 4.

100 ÷ 11 = 9.0909 … What is the 10th digit after the point?

Answer: 9

The block 09 repeats. Even places (2nd, 4th, … 10th) are 9.

• 1 ÷ 8 — correct. The left-overs are 1, 2, 4, 0. When 0 is reached, the division ends: 0.125.
• 1 ÷ 6. The left-over 4 keeps coming back: 0.1666 … never ends.
• 1 ÷ 9. The left-over 1 keeps coming back: 0.111 … never ends.
• Because 7 is a prime number. 1 ÷ 5 ends (0.2) even though 5 is prime. What matters is whether a left-over comes back before reaching 0.
• Because 1 is smaller than 7. 1 ÷ 4 also starts with a dividend smaller than the divisor, and it ends: 0.25.
• Each left-over is one of 1 to 6, so after at most 6 steps one comes back, and then everything repeats — correct. Yes. A left-over that comes back starts the whole cycle again.