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Does This Ever End? Never-Ending Decimals

Find out why 10 ÷ 3 and 1 ÷ 7 never end, meet the magic number 142857, and see which fractions give ending decimals.

10 micro-lessons

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A Division That Never Stops

10 ÷ 3 leaves a remainder of 1 at every step.

Try 10 ÷ 3 (Textbook pp. 84–85)

10 Ones ÷ 3 → 3, remainder 1.
Regroup 1 One as 10 Tenths ÷ 3 → 3, remainder 1.
Regroup as 10 Hundredths ÷ 3 → 3, remainder 1.
… and again, and again.

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The Remainders of 1 ÷ 7

Watch the remainders go round in a circle.

Problem. Divide 1 by 7. Does it end? (Textbook p. 85)

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Will It End?

Decide whether each division ends and what repeats.

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$10 \div 9 = ?$

A Magic Number

The repeating block of 1 ÷ 7 does tricks when multiplied.

142857 (Textbook pp. 85–86)

Take 142857, the repeating block of $1 \div 7$ . Multiply it by 1, 2, 3, 4, 5 and 6. What do you notice?

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Multiplying 142857 by 1–7

142857 × 1 to 7, and the surprise at 7.

Problem. Multiply 142857 by 1, 2, 3, 4, 5, 6 and then 7. (Textbook p. 86)

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An Unsolved Puzzle

Emil Artin's 1927 guess is still open today.

Did you know?

Cyclic numbers like 142857 (from 1/7) and 0588235294117647 (from 1/17) come from fractions whose repeating block is as long as it can possibly be.

In 1927, Emil Artin conjectured that there are infinitely many of them. Nearly 100 years later it is still unsolved: a real open question in mathematics that you can understand in Class 7.

Cyclic Number Check

Quick questions on 142857.

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$142857 \times 3 = ?$

← previousDividing By A Decimal next →When Division Makes Numbers Bigger

Will It End?

Decide whether each division ends and what repeats.

1 / 4

$10 \div 9 = ?$

A Magic Number

The repeating block of 1 ÷ 7 does tricks when multiplied.

142857 (Textbook pp. 85–86)

Take 142857, the repeating block of $1 \div 7$ . Multiply it by 1, 2, 3, 4, 5 and 6. What do you notice?

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Cyclic Number Check

Quick questions on 142857.

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$142857 \times 3 = ?$