The Other Side of Zero · Lesson 4 of 10
The Full Number Line and Larger Integers
“Extend floor reasoning to an endless number line and calculate efficiently with larger integers.”
• Describe the integers and their unbounded number line. • Compare and order integers by their positions. • Represent larger changes on unmarked number lines. • Convert subtraction to addition of an additive inverse. • Convert addition to subtraction of an additive inverse. • Check calculations using starting and target positions.
Floors can become levels
A mineshaft may carry people and minerals far below ground. A mountainside lift may also reach above ground. We can label ground level 0 metres, positions above it with positive numbers, and positions below it with negative numbers. The same movement ideas now work with metres instead of floors.
Problem
Find (+40) + (+60) and (−90) + (−55).
- 1.Starting 40 m above ground and moving up 60 m gives 100 m above: +40 + (+60) = +100.
- 2.Starting 90 m below ground and going 55 m farther down gives 145 m below: −90 + (−55) = −145.
- 3.The numbers describe levels and changes measured in the same unit, metres.
Problem
Find the change from −50 m to +40 m, and from +40 m to −90 m.
- 1.For the first journey, move up 50 m to zero and 40 m more: +90 m. Thus +40 − (−50) = +90.
- 2.For the second, move down 40 m to zero and 90 m farther: −130 m. Thus −90 − (+40) = −130.
- 3.The sign describes the journey’s direction, not whether the target level itself is positive.
An endless line of integers
A real building or mine has limits, but the numbers do not. After any positive integer, we can go one higher. After any negative integer, we can go one lower. Together with zero, these positive and negative whole numbers form the integers.
The numbers …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … . They include negative whole numbers, zero, and positive whole numbers. Fractions such as 1/2 are not integers.
Imagine an infinite lift and turn its picture sideways. Upwards becomes rightwards, and downwards becomes leftwards. The ground-level zero remains the reference point. This completes the ray beginning at zero into a number line extending both ways. We often omit the + on positive numbers: 5 and +5 mean the same number.
A number farther right is greater. Thus −5 < −3 < 0 < 2 < 5. There is no greatest integer and no least integer. Between −7 and 0, the integers are −6, −5, −4, −3, −2, −1; we leave out the endpoints when the instruction says strictly between them.
Problem
Describe journeys 5 → 9, 9 → 3, and 3 → −2.
- 1.From 5 to 9, move +4: 5 + 4 = 9 and 9 − 5 = 4.
- 2.From 9 to 3, move −6: 9 + (−6) = 3 and 3 − 9 = −6.
- 3.From 3 to −2, move −5: 3 + (−5) = −2 and −2 − 3 = −5.
Unmarked number lines
Drawing every tick is unnecessary when numbers are large. An unmarked number line shows zero and only the positions needed for a calculation. You can use one long arrow for a large movement. Keep the relative order and spacing sensible so the picture agrees with the numbers.
Problem
Use an unmarked line to find 85 + (−60).
- 1.Start at 85, to the right of zero.
- 2.Adding −60 means moving 60 units left. Because 60 is less than 85, you remain to the right of zero.
- 3.85 − 60 = 25, so the target is +25.
Problem
Find (−100) − (+250).
- 1.Read −100 as the target and +250 as the start.
- 2.Move −250 to zero, then −100 to the target.
- 3.The required movement is −250 + (−100) = −350. Check: +250 + (−350) = −100.
Why subtraction can become addition
The journey from 2 to −3 can be described as −3 − 2. Splitting at zero gives a movement of −2, followed by −3. The total is −3 + (−2), also −5. The starting position is cancelled by adding its inverse, then the target position is reached.
This reasoning works for every pair of integers. Subtracting a number has the same effect as adding its additive inverse. A positive subtracted number becomes a negative addend; a negative subtracted number becomes a positive addend. Change the operation and replace the entire subtracted number by its inverse.
Problem
Evaluate +105 − (−55) and −99 − (−200).
- 1.For the first, replace subtracting −55 with adding +55: 105 + 55 = 160.
- 2.For the second, replace subtracting −200 with adding +200: −99 + 200 = +101.
- 3.Check the second as a journey: −200 + 101 = −99, so the movement from −200 to −99 is +101.
You can also reverse the conversion: adding a number has the same effect as subtracting its inverse. For example, 8 + (−2) = 8 − (+2), and −3 + (+8) = −3 − (−8). These are equal ways of describing the same result, not permission to drop signs at random.
| Expression | Equivalent addition | Result |
|---|---|---|
| 7 − 5 | 7 + (−5) | 2 |
| −3 − 8 | −3 + (−8) | −11 |
| 8 − (−2) | 8 + 2 | 10 |
| 6 − (−9) | 6 + 9 | 15 |
In 80 − (−150), the outer minus is an operation and the inner minus belongs to the number −150. Replacing subtraction by addition gives 80 + 150 = 230, not 80 − 150.
Quiz
Which number is not an integer?
Which statement is true?
What is −125 + (−30)?
What is +80 − (−150)?
Which expression equals −3 − 8?
What is −99 − (−200)?
Practice Problems
- Complete +40 + ? = +200; +40 + ? = −200; −50 + ? = +200; −50 + ? = −200.
- Evaluate −200 − (−40), +200 − (+40), and −200 − (+40).
- Calculate −125 + (−30), 105 − (−55), 105 + 55, 80 − (−150), 80 + 150, −99 − (−200), −99 + 200, and 1500 − (−1500).
- Draw unmarked number lines for 85 + (−60) and −100 − (+250).
- List the integers strictly between −8 and −15, and between −30 and −23, in increasing order.
- Calculate −5 + 0, 7 + (−7), −10 + 20, 10 − 20, 7 − (−7), and −8 − (−10).
- Evaluate 8 − 13, (−8) − 13, (−13) − (−8), (−13) + (−8), 8 + (−13), (−8) − (−13), 13 − 8, and 13 − (−8).
- Rewrite −6 + (+9) as a subtraction and 12 − (−4) as an addition. Explain each change.
+160, −240, +250, −150; each value is the movement needed to the target.
Key Takeaways
• Integers extend endlessly to both sides of zero. • Numbers farther right on a number line are greater. • Unmarked number lines make large movements easier to show. • Subtracting an integer is adding its additive inverse. • Adding an integer can be rewritten as subtracting its additive inverse. • The position–movement equations work for small and large integers.