Number Play · Lesson 2 of 8
Number Lines, Place Value, and Digit Sums
“Read number scales and investigate what the digits of a number reveal.”
• Read equally spaced number-line positions by identifying the size of each step. • Compare and place numbers using place value and their positions on a number line. • Find how many positive numbers have a specified number of digits. • Construct numbers with a given digit sum and explain patterns in digit sums. • Count occurrences of a digit systematically without missing repeated appearances.
The spacing on a number line carries information
A number line shows order and distance together. Numbers increase as you move to the right, and equal physical gaps represent equal numerical changes. You cannot assume that each short gap means one. A line used to show thousands may have a very different scale from a line used to show individual numbers.
First read two labelled positions. Count the intervals between them, then find the change represented by one interval. For instance, if 2010 and 2020 are two intervals apart, the increase of 10 is shared equally between those two intervals. Each interval represents 5. Count intervals between positions, not the number of tick labels you see.
The numerical change represented by a chosen distance or by one interval between equally spaced ticks.
Problem
A line has ten equally spaced positions. Position 5 is 2010 and position 7 is 2020. Label all ten positions.
- 1.The move from position 5 to position 7 spans two intervals. The numerical increase is 2020 − 2010 = 10.
- 2.Each interval is therefore 10 ÷ 2 = 5. Move left from 2010 four times: 2005, 2000, 1995, 1990.
- 3.Move right in steps of 5. The ten labels are 1990, 1995, 2000, 2005, 2010, 2015, 2020, 2025, 2030, 2035.
- 4.The smallest is the leftmost value, 1990, and the largest is the rightmost, 2035.
| Known neighbouring labels | Step | Complete ten-position sequence |
|---|---|---|
| 9996 and 9997 | 1 | 9993, 9994, 9995, 9996, 9997, 9998, 9999, 10,000, 10,001, 10,002 |
| 15,077 and 15,078 | 1 | 15,077, 15,078, 15,079, 15,080, 15,081, 15,082, 15,083, 15,084, 15,085, 15,086 |
| 86,705 and 87,705 | 1000 | 83,705, 84,705, 85,705, 86,705, 87,705, 88,705, 89,705, 90,705, 91,705, 92,705 |
The change from 9999 to 10,000 is still just one. The extra written digit does not make a bigger jump on the line. Similarly, 86,705 to 87,705 changes the thousands part while leaving 705 unchanged. To check any completed line, subtract successive labels and make sure the differences agree.
Problem
Place 2180, 2754, 1500, 3600, 9950, 9590, 1050, 3050, 5030, 5300, and 8400 on a line from 1000 to 10,000.
- 1.Order them first: 1050, 1500, 2180, 2754, 3050, 3600, 5030, 5300, 8400, 9590, 9950.
- 2.Use the two surrounding thousands for each. For example, 2180 lies between 2000 and 3000, only 180 beyond 2000, so it belongs near 2000.
- 3.2754 is 754 beyond 2000, so it belongs nearer 3000. 1500 is halfway between 1000 and 2000. 9950 is just 50 short of 10,000.
- 4.3050 and 5030 look similar but their thousands digits differ. Their positions near 3000 and 5000 make the difference visible.
Putting 2180 somewhere to the left of 2754 gets their order right, but placing them equally far from 2000 would give incorrect distances. On an equally scaled line, 180 beyond 2000 occupies a smaller fraction of the interval than 754 beyond 2000.
How many numbers have each number of digits?
Consider the positive counting numbers 1, 2, 3, and so on. The one-digit members run from 1 to 9. Two-digit numbers begin at 10 and end at 99; three-digit numbers begin at 100 and end at 999. Finding these endpoints lets you count an entire range without writing every number.
Subtracting the first value from the last counts the gaps between them. To count the values themselves, include the first one as well. From 10 to 12, for example, there are two gaps but three numbers: 10, 11, 12. The same reasoning applies to larger ranges.
Problem
How many positive four-digit numbers are there?
- 1.The smallest four-digit number is 1000 and the largest is 9999.
- 2.The difference 9999 − 1000 = 8999 counts the one-step gaps.
- 3.Include the first number: 8999 + 1 = 9000. Thus there are 9000 four-digit numbers.
| Number of digits | Range of positive counting numbers | Count |
|---|---|---|
| 1 | 1–9 | 9 |
| 2 | 10–99 | 90 |
| 3 | 100–999 | 900 |
| 4 | 1000–9999 | 9000 |
| 5 | 10,000–99,999 | 90,000 |
The first digit of a positive number has nine choices, from 1 to 9. Each later digit has ten choices, from 0 to 9. Another place therefore multiplies the count by ten. Zero itself is written with one digit, but the table is specifically counting positive numbers starting at 1.
A digit sum ignores position but not the digits
The value of 68 is sixty-eight because 6 is in the tens place. Its digit sum, however, is simply 6 + 8 = 14. This is a new question about the same written number. A digit sum adds the digits as individual numbers; it does not add their place values. The numbers 176 and 545 also have digit sum 14 even though their values are quite different.
The sum of the individual digits in the usual written form of a number.
Problem
Find the smallest positive number with digit sum 14 and the largest five-digit number with that digit sum.
- 1.A single digit cannot be 14, so the smallest number needs at least two digits.
- 2.If the tens digit were 1, 2, 3, or 4, the units digit would have to exceed 9. The smallest possible tens digit is therefore 5, paired with 9. The smallest number is 59.
- 3.To maximise a five-digit number, put as much of the sum as possible in the leftmost place: 9. The remaining sum is 5, so put 5 in the next place and zeros after it.
- 4.The largest five-digit number is 95,000. It has digit sum 14 and exceeds every other five-digit arrangement with that sum.
There is no largest number with digit sum 14 when the number of digits is unrestricted. Appending a zero to 59 makes 590, then 5900, without changing the digit sum. Each number is larger than the previous one. The phrase largest five-digit number includes an essential restriction that makes a largest answer possible.
| Number range or example | Digit sums | Observation |
|---|---|---|
| 40–49 | 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | They rise by 1 within this block. |
| 50–59 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | The units restart at 0. |
| 60–69 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | The tens contribution is now 6. |
| 70 | 7 | The sum drops when 69 changes to 70. |
| 123, 234, 345, 456, 567, 678, 789 | 6, 9, 12, 15, 18, 21, 24 | Each digit rises by 1, so the sum rises by 3. |
Across a change such as 49 to 50, the digit sum drops from 13 to 5 even though the number increases by 1. One digit rises, but another changes from 9 to 0. For three consecutive ascending digits, the middle digit has one neighbour one smaller and one larger. Together, the three digits have a sum equal to three times the middle digit. This explains the multiples of 3. The written pattern 123 through 789 cannot continue as three single decimal digits beyond 9.
Count digit appearances by place
Counting numbers containing 7 and counting appearances of 7 are different tasks. The number 77 is one number, but it contributes two appearances. An organised count separates the units, tens, and hundreds positions. This avoids both omissions and accidental double counting of the wrong thing.
Problem
How many times is 7 written in the numbers 1 through 100?
- 1.The units digit is 7 in 7, 17, 27, 37, 47, 57, 67, 77, 87, 97: ten appearances.
- 2.The tens digit is 7 in 70 through 79: another ten appearances.
- 3.Add the positions: 10 + 10 = 20. Counting 77 in both lists is correct because it contains two written 7s. The endpoint 100 adds none.
Problem
How often is 7 written from 1 through 1000?
- 1.Think of 000 through 999 as three-place labels. The added leading zeros do not create or remove any 7s, so they are safe for this particular count.
- 2.In each chosen position, fixing 7 leaves ten choices for each of the other two positions. Thus each position contributes 10 × 10 = 100 appearances.
- 3.The units, tens, and hundreds contributions total 100 + 100 + 100 = 300. The extra endpoint 1000 contains no 7. Therefore the answer is 300.
Quiz
Two labelled ticks are 2010 and 2020, with one unlabelled tick between them. What is one tick interval?
Which number is closest to 10,000?
How many positive three-digit numbers are there?
Which is the smallest number with digit sum 14?
What happens to the digit sum when a zero is appended to 59?
How many appearances of 7 does the number 777 contribute?
Practice Problems
- Complete a line of six positions whose first two labels are 9997 and 9998.
- A line labels 15,077 and 15,083 six equal intervals apart. What is its step?
- Place 3050 and 5030 on a line labelled at each thousand. Explain their positions.
- Find the number of positive five-digit numbers in two different ways.
- Construct three different numbers with digit sum 14, including one six-digit number.
- Explain why 95,000 is the largest five-digit number with digit sum 14.
- Find the digit sums of 58, 59, 60, and 61. Explain the change at 60.
- How many appearances of 7 occur from 1 through 79? Count separately by position.
9997, 9998, 9999, 10,000, 10,001, 10,002. The step stays 1 through the change of written length.
Key Takeaways
• Read the scale of a number line before placing or naming positions. • An inclusive range has last minus first plus one members. • Positive one- through five-digit counts are 9, 90, 900, 9000, and 90,000. • A digit sum adds individual digits and can decrease when a number increases. • Count digit occurrences by position; a number can contribute more than one occurrence.