Repeating Decimals:
Nth Digit and Digit Sum
Lesson 12.2 · 循环小数中的远位数字与数字和
Discover a decimal cycle through long division, jump to a far-away digit with a remainder, and total thousands of digits without writing them all.
One repeating block can answer two different questions.
Read a repeating decimal precisely
A bar over digits means that the entire marked block repeats forever. The period is the number of digits in the smallest repeating block.
The digits after the decimal point are:
Three details to keep separate
Repeating block: 285714
Period: 6 digits
Position: the first digit after the decimal point is position 1.
Your turn
Watch long division create the cycle
A decimal digit is determined by the current remainder. Once the same remainder returns, the same future digits must repeat.
Remainder machine for 2 ÷ 7
At each step: multiply the remainder by 10, choose the next digit, and record the new remainder.
| Step | Start remainder | Remainder × 10 | Next digit | New remainder |
|---|
Remainders around the loop
The first remainder is 2. Follow it until 2 returns.
Find the 2,014th digit
Use the period to replace a huge position with one small position inside the repeating block.
Interpret the division
335 complete six-digit cycles come first.
The remainder 4 means the target lands on the fourth digit of 285714.
Record the result
Find the sum of the first 2,014 digits
For a digit sum, add the values of the individual digits, not the decimal numbers. Every complete cycle contributes the same cycle sum. The “leftover-prefix sum” means the sum of the first r digits of the repeating block.
Keep a leading zero inside the cycle
A zero immediately after the decimal point is still a digit and still occupies a position.
Long-division remainder chain
The first step produces digit 0 because 10 is smaller than 13. When remainder 1 returns, the six-digit block begins again.
Use the same 2,014 = 335×6+4 split
The fourth digit in 076923 is 9.
One block still sums to:
The first four digits still sum to 22, so the total is again 9,067.
Compare the two original examples
Two decimals can have different blocks and different distant digits while sharing the same period and the same total digit sum.
| Fraction | Repeating block | Period | Cycle sum | 2,014th digit | First 2,014 digit sum |
|---|---|---|---|---|---|
| 2/7 | 285714 | 6 | 27 | 7 | 9,067 |
| 1/13 | 076923 | 6 | 27 | 9 | 9,067 |
Use a repeating-decimal laboratory
Enter digits that repeat starting at decimal position 1 and a whole-number target position from 1 to 1,000,000,000. The laboratory reduces repeated copies to the shortest block before calculating. For example, entering 4545 uses block 45 and period 2. This lab does not include a nonrepeating prefix.
The fourth digit is highlighted.
Record two laboratory challenges
A. Block 36, N = 100
B. Block 142857, N = 2,025
Repeating-decimal error detective
Judge each statement. Pay special attention to leading zeros, the meaning of the remainder, and the difference between one digit and a digit sum.
1. In 0.076923, the first digit after the decimal point is 0.
2. Because 2014 ÷ 6 has remainder 4, the 2,014th digit of 0.285714 is 4.
3. To sum the first N digits, multiply N by the sum of one complete block.
4. If r = 0, the leftover-prefix sum is 0 and the Nth digit is the final digit of the block.
5. A repeating block may begin with 0; that zero still counts toward the period.
When long division stops — and when it repeats
For 1 ÷ 4: 10 ÷ 4 gives digit 2, remainder 2; 20 ÷ 4 gives digit 5, remainder 0. The decimal ends: 0.25. A zero remainder in the long-division calculation means the decimal terminates. This differs from a zero remainder when dividing a digit position by a block length, which locates the last digit of a block. A repeated nonzero long-division remainder starts the same future digits again.
original Exercise 1: a decimal with a warm-up
Find digit 2,016 of 3/35 and the sum of the first 2,016 digits. Long division begins with digit 0, remainder 30. Then the remainders are 30 → 20 → 25 → 5 → 15 → 10 → 30, producing digits 8,5,7,1,4,2.
Thus 3/35 = 0.0857142. The first zero is a nonrepeating prefix. The six-digit cycle starts at decimal position 2.
Hint: separate the prefix
Remove the first digit before dividing the remaining number of positions by six. For the sum, include the prefix's contribution too.
Worked solution
2,016 − 1 = 2,015 = 335 × 6 + 5. Read the fifth digit of 857142: 4. Each full block sums to 27; the first five digits sum to 25. Total = 0 + 335 × 27 + 25 = 9,070.
original Exercise 2: choose where to start on a digit ring
Read this ring clockwise forever: 8 → 3 → 4 → 3 → 9 → 8 → 9 → 7 → 3 → 9 → back to 8. Choose one digit for the integer part, put the decimal point immediately after it, and keep reading clockwise. Which starting point gives the greatest number? Which gives the least?
The order cannot be rearranged. A ring diagram is not needed to infer any extra dimensions: the arrows give the complete order.
Hint: compare from left to right
The greatest integer part must be 9, and the least must be 3. If two candidates share their integer part, compare decimal digits until they differ.
Worked solution
The largest starts at the fifth ring position, the 9 immediately after 4,3 and begins 9.8973983439…. The smallest starts at the second ring position, the 3 immediately before 4 and begins 3.4398973983…. The ten digits after each decimal point form its repeating block.
Independent workshop
Try each new problem before opening a hint. Correct all 8 answers to complete this mission.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
32 = 10 × 3 + 2. The target is block position 2, digit 4.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
32 = 10 × 3 + 2. Block sum 11; leftover sum 4. Total = 10 × 11 + 4 = 114.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
81 = 40 × 2 + 1. The target is block position 1, digit 2.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
81 = 40 × 2 + 1. Block sum 7; leftover sum 2. Total = 40 × 7 + 2 = 282.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
108 = 27 × 4 + 0. The target is block position 4, digit 4.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
108 = 27 × 4 + 0. Block sum 10; leftover sum 0. Total = 27 × 10 + 0 = 270.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
20 = 3 × 6 + 2. The target is block position 2, digit 7.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
20 = 3 × 6 + 2. Block sum 27; leftover sum 7. Total = 3 × 27 + 7 = 88.
Transfer exit ticket
Try each new problem before opening a hint. Correct all 5 answers to complete this mission.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
74 = 24 × 3 + 2. The target is block position 2, digit 3.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
74 = 24 × 3 + 2. Block sum 11; leftover sum 3. Total = 24 × 11 + 3 = 267.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
52 = 13 × 4 + 0. The target is block position 4, digit 2.
Hint
Keep any leading zero. Split into complete blocks and leftover digits.
Worked explanation
52 = 13 × 4 + 0. Block sum 13; leftover sum 0. Total = 13 × 13 + 0 = 169.
Hint
Can the entered block be made from copies of something shorter?
Worked explanation
4545 consists of two copies of 45. The smallest period is 2.
Optional reflection — not automatically graded
Lesson checkpoints completed
Repeating-Decimal Cycle Navigator
This certifies that the learner can discover a decimal cycle, locate a distant digit, preserve a leading zero, and sum many decimal digits by complete cycles and a leftover prefix.
Lesson 12.2 · Grade 5 Periodic Problems
Teaching notes
The long-division stepper, comparison table, general repeating-block laboratory, additional blocks 36 and 142857, feedback, workshop, and exit ticket are added self-contained teaching scaffolds.