Another Peek Beyond the Point · Lesson 11 of 13
Does This Ever End?
“Use remainders to understand terminating and repeating decimals, and reason about quotient size.”
• Distinguish a decimal that terminates from one that repeats. • Explain how a repeated remainder creates repeated quotient digits. • Predict whether division by a positive number makes a quantity larger or smaller. • Explore decimal patterns for fractions with powers of 2 or 5 as denominators.
When division continues indefinitely
In 1325 ÷ 4, the remainder eventually became zero. That allowed us to stop with the exact answer 331.25. A different division may keep producing non-zero remainders, no matter how many zeroes we append to the dividend. We need a way to describe such an answer exactly instead of pretending that the written digits have ended.
A decimal whose expansion ends after a finite number of decimal places, such as 0.125. Extra trailing zeroes may be written without changing it.
A decimal in which one digit or a block of digits repeats endlessly, such as 0.333… or 0.142857142857…. The dots indicate continuation, not an instruction to stop at the displayed digits.
Problem
Write 10 ÷ 3 as a decimal.
- 1.3 goes into 10 three times, leaving 1 one. So the whole-number part is 3.
- 2.Regroup the remainder as 10 tenths. Divide by 3: 3 tenths each, with 1 tenth left.
- 3.That remainder becomes 10 hundredths, giving 3 hundredths each and 1 hundredth left.
- 4.The same situation repeats at every place. Thus 10 ÷ 3 = 3.333… exactly.
- 5.3.33 is only a shortened value. It is not equal to 10/3 because 3.33 × 3 = 9.99, rather than 10.
Follow a longer cycle of remainders
Long division is determined by the current remainder and the next digit brought down. Once the original digits have been used, each new brought-down digit is zero. If the same remainder appears again, the next step repeats too, and so do all following steps. This explains a repeating block rather than just observing that some digits look familiar.
Problem
Find the decimal expansion of 1 ÷ 7.
- 1.The whole-number part is 0 because 1 is smaller than 7. Start with remainder 1.
- 2.10 ÷ 7 gives digit 1 and remainder 3. Then 30 ÷ 7 gives digit 4 and remainder 2.
- 3.20 ÷ 7 gives digit 2 and remainder 6. Then 60 ÷ 7 gives digit 8 and remainder 4.
- 4.40 ÷ 7 gives digit 5 and remainder 5. Then 50 ÷ 7 gives digit 7 and remainder 1.
- 5.We are back at remainder 1. The six digits 142857 repeat: 1/7 = 0.142857142857….
| Remainder before step | After × 10 | Next quotient digit | New remainder |
|---|---|---|---|
| 1 | 10 | 1 | 3 |
| 3 | 30 | 4 | 2 |
| 2 | 20 | 2 | 6 |
| 6 | 60 | 8 | 4 |
| 4 | 40 | 5 | 5 |
| 5 | 50 | 7 | 1 — cycle restarts |
For division by a positive whole number such as 7, possible remainders are 0, 1, 2, 3, 4, 5, and 6. If zero appears, the decimal terminates. If it never appears, eventually a non-zero remainder must be used again because there are only finitely many possible remainders. The repeated remainder makes a repeated block. Decimal divisors can first be scaled to whole numbers, so the same reasoning applies.
142857 × 1 = 142857; × 2 = 285714; × 3 = 428571; × 4 = 571428; × 5 = 714285; × 6 = 857142. Each product uses a rotation of the same six digits. Multiplying by 7 gives 999999. Such a repeating-block number is called cyclic when its specified multiples produce these rotations. Use actual multiplication to verify the pattern rather than assuming every repeating block behaves this way.
The repeating block of 1/17 gives another long pattern to explore. The chapter connects cyclic-number patterns with a conjecture proposed by Emil Artin in 1927. A conjecture is a mathematical claim that needs proof; checking several examples alone does not prove an unlimited claim. You do not need advanced number theory to investigate the multiplication patterns here.
Will the quotient be smaller or larger?
Use positive dividends and positive divisors for these comparisons. Dividing by a number greater than 1 splits the amount into more than one full equal share, so a single share is smaller than the original amount. Dividing by a number between 0 and 1 asks how many smaller groups fit into the amount, so the numerical quotient is larger. Dividing by exactly 1 leaves the amount unchanged.
| Positive divisor | Quotient compared with positive dividend | Example |
|---|---|---|
| Greater than 1 | Smaller | 8 ÷ 2 = 4 |
| Equal to 1 | Equal | 8 ÷ 1 = 8 |
| Between 0 and 1 | Larger | 8 ÷ 0.5 = 16 |
Problem
Find the missing divisors in 25 ÷ □ = 0.025, 25 ÷ □ = 250, and 25 ÷ □ = 2.5.
- 1.Divisor = dividend ÷ quotient, because divisor × quotient = dividend.
- 2.For 0.025, use 25 ÷ 0.025 = 25000 ÷ 25 = 1000. Check: 0.025 × 1000 = 25.
- 3.For 250, use 25 ÷ 250 = 0.1. Check: 250 × 0.1 = 25.
- 4.For 2.5, use 25 ÷ 2.5 = 250 ÷ 25 = 10. Check: 2.5 × 10 = 25.
- 5.The divisor below 1 produces a quotient greater than 25; the two divisors above 1 produce smaller quotients.
The reciprocal link gives useful shortcuts: dividing by 10 is multiplying by 0.1; dividing by 0.1 is multiplying by 10; dividing by 0.01 is multiplying by 100. These are different operations even though the same digits occur in their written numbers. Division by zero is undefined: no number multiplied by 0 can give a positive dividend.
Why powers of two and five give terminating decimals
A power is repeated multiplication of the same number: 2³ means 2 × 2 × 2 = 8, and 5³ means 5 × 5 × 5 = 125. Since 2 × 5 = 10, each factor of 2 can be paired with a factor of 5 to make a factor of 10. This lets us convert fractions with these denominators into decimal fractions and explains their terminating patterns.
| n | 1/(2ⁿ) | Decimal | 1/(5ⁿ) | Decimal |
|---|---|---|---|---|
| 1 | 1/2 | 0.5 | 1/5 | 0.2 |
| 2 | 1/4 | 0.25 | 1/25 | 0.04 |
| 3 | 1/8 | 0.125 | 1/125 | 0.008 |
| 4 | 1/16 | 0.0625 | 1/625 | 0.0016 |
| 5 | 1/32 | 0.03125 | 1/3125 | 0.00032 |
Problem
Explain why 1/16 and 1/625 terminate.
- 1.16 = 2⁴. Multiply numerator and denominator of 1/16 by 5⁴ = 625.
- 2.16 × 625 = 10000, so 1/16 = 625/10000 = 0.0625.
- 3.625 = 5⁴. Multiply numerator and denominator of 1/625 by 2⁴ = 16.
- 4.1/625 = 16/10000 = 0.0016. Both expansions end because each is an exact decimal fraction.
A few displayed digits do not show that a division has ended. Stop for an exact finite answer only when the remainder is zero. If you stop a non-terminating calculation early, identify it as an approximation; do not use an equals sign to claim that the truncated digits are exact.
Quiz
What makes the decimal digits of 1/7 repeat?
Which is an exact statement?
For a positive amount a, how does a ÷ 0.25 compare with a?
Which divisor leaves every positive dividend unchanged?
What is 1/32 as a decimal?
Which remainder allows an exact long division to stop?
Practice Problems
- Use long division to find the repeating blocks of 10/9 and 100/11. Record the repeated remainder in each.
- Repeat the remainder table for 1/7 without looking at the worked example.
- Verify all six multiples of 142857 and explain the word rotation in this pattern.
- Explore 1/17. Record remainders until a remainder repeats, and identify the repeating block.
- Convert 2/5, 13/4, 4/50, and 5/8 to terminating decimals by equivalent fractions.
- Without full division, compare 6.4 ÷ 4, 6.4 ÷ 1, and 6.4 ÷ 0.4 with 6.4. Then calculate them.
- Using 156 × 12 = 1872, find 15.6 × 1.2, 187.2 ÷ 1.2, 18.72 ÷ 15.6, and 0.156 × 0.12.
- Find the length of each of five equal pieces cut from 4 m of wood; the side of a regular 12-sided figure (all sides equal) with perimeter 208.8 cm; and each of eight equal shares of 3 L of juice.
- Explain why 1/(2⁶) and 1/(5⁶) must terminate, then find both decimals.
Key Takeaways
• An exact decimal terminates when its division remainder becomes zero. • A repeated non-zero remainder produces a repeating block of digits. • Dots or a bar show that repetition continues indefinitely. • For positive numbers, division by a divisor above 1 makes the quotient smaller, and division by a divisor below 1 makes it larger. • Division by 1 leaves a number unchanged; division by 0 is undefined. • Powers of 2 and 5 can be paired to make powers of 10 and explain terminating patterns.