Temperatures below freezing, a diver far under the surface, a bank account in the red: all of these need numbers on both sides of zero. In this chapter you will learn to add and subtract positive and negative numbers with confidence, using the number line as your map.
1. Positive and negative numbers
The integers are the whole numbers and their opposites: \( \dots, -3, -2, -1, 0, 1, 2, 3, \dots \). Numbers greater than 0 are positive, numbers less than 0 are negative, and 0 is neither. Negative numbers appear whenever a quantity can go below a starting level: a temperature of \( -8^\circ\text{F} \), a depth of \( -30 \) feet, a debt of 25 dollars, \( -25 \).
On a horizontal number line, numbers increase from left to right. So \( -9 < -2 \) because \( -9 \) is farther to the left. The same idea works for decimals and fractions such as \( -2.5 \) or \( -\dfrac{3}{4} \): together with the integers they form the rational numbers, and every rule in this chapter works for them too.
2. Opposites and absolute value
Two numbers are opposites if they are the same distance from 0 but on different sides of 0. The opposite of \( n \) is written \( -n \). The opposite of 0 is 0.
The absolute value of a number is its distance from 0 on the number line. We write it \( |n| \). A distance is never negative, so \( |n| \ge 0 \) always.
In the figure, A is at \( -4 \) and B is at \( 4 \). Both are 4 units from zero, so \( |-4| = 4 \) and \( |4| = 4 \). Notice that the opposite of \( -4 \) is \( -(-4) = 4 \).
Absolute value does not simply “erase the sign” of an expression. \( |-3 + 1| = |-2| = 2 \), but it is not \( 3 + 1 \). Calculate inside the bars first.
3. Adding on the number line
Think of addition as a walk. Start at the first number. Adding a positive number means moving to the right. Adding a negative number means moving to the left.
Compute \( 3 + (-5) \). Start at 3 and move 5 units left. You land on \( -2 \). So \( 3 + (-5) = -2 \).
Compute \( -4 + (-3) \). Start at \( -4 \) and move 3 more units left: \( -7 \). Adding two negatives gives a negative whose absolute value is the sum of the absolute values.
4. Rules for adding integers
- Same signs: add the absolute values and keep the common sign.
- Different signs: subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
- Opposites always give 0.
| Sum | Signs | Work | Result |
|---|---|---|---|
| \( -6 + (-9) \) | same | \( 6 + 9 = 15 \), sign \( - \) | \( -15 \) |
| \( 12 + (-5) \) | different | \( 12 - 5 = 7 \), sign \( + \) | \( 7 \) |
| \( -12 + 5 \) | different | \( 12 - 5 = 7 \), sign \( - \) | \( -7 \) |
| \( -8 + 8 \) | opposites | cancel | \( 0 \) |
5. Subtraction as adding the opposite
Subtracting a number is the same as adding its opposite: \( a - b = a + (-b) \).
Why does \( 2 - (-3) \) equal \( 2 + 3 \)? Subtracting \( -3 \) removes a debt of 3, and taking away a debt makes you richer by 3. On the line you move 3 units to the right, landing on 5.
\( -4 - 7 = -4 + (-7) = -11 \) and \( -4 - (-7) = -4 + 7 = 3 \).
\( 1.5 - 4.25 = 1.5 + (-4.25) = -2.75 \), and \( -\dfrac{1}{2} - \left(-\dfrac{3}{4}\right) = -\dfrac{2}{4} + \dfrac{3}{4} = \dfrac{1}{4} \).
On my home planet we say: “Keep, change, change.” Keep the first number, change the subtraction to addition, change the second number to its opposite!
6. Distance between two numbers
The distance between two numbers on the number line is how far apart they are. To find it, subtract and take the absolute value: the distance between \( a \) and \( b \) is \( |a - b| \), which equals \( |b - a| \).
The distance between \( -6 \) and \( 5 \) is \( |5 - (-6)| = |11| = 11 \). When the numbers are on opposite sides of zero, add their absolute values: \( 6 + 5 = 11 \). When they are on the same side, subtract: the distance between \( -9 \) and \( -2 \) is \( 9 - 2 = 7 \).
7. The additive inverse
The additive inverse of \( n \) is the number that makes a sum of 0: \( n + (-n) = 0 \). It is the same thing as the opposite of \( n \).
This is why a number and its opposite are called a zero pair: \( 7 + (-7) = 0 \), \( -2.5 + 2.5 = 0 \). Zero pairs help you simplify long sums. For example, in \( -9 + 4 + 9 \), the \( -9 \) and \( 9 \) cancel and the answer is simply 4. Subtraction also fits: the opposite of \( a - b \) is \( b - a \).
8. Real-world contexts
A sign tells you a direction: up or down, gained or lost, above or below sea level. To model a word problem, decide what 0 stands for, then write each change as a signed number.
At 6 a.m. it is \( -7^\circ\text{F} \). By noon it rises 19\( ^\circ\text{F} \) and by 9 p.m. it falls 14\( ^\circ\text{F} \). Noon: \( -7 + 19 = 12 \). Night: \( 12 - 14 = -2 \). The temperature at 9 p.m. is \( -2^\circ\text{F} \). In metric units, a change of 19\( ^\circ\text{F} \) is about 10.6\( ^\circ\text{C} \).
Another example: a hiker at \( -120 \) ft (below sea level) climbs to a ridge at \( +340 \) ft. The climb is \( 340 - (-120) = 460 \) ft, about 140 meters.
Key takeaways
- Opposites are the same distance from 0 on different sides; \( |n| \) is the distance from 0 and is never negative.
- Adding a positive moves right; adding a negative moves left.
- Same signs: add and keep the sign. Different signs: subtract and keep the sign of the larger absolute value.
- \( a - b = a + (-b) \): subtraction is adding the opposite.
- The distance between \( a \) and \( b \) is \( |a - b| \).
- The additive inverse of \( n \) is \( -n \), and \( n + (-n) = 0 \).
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