Starting below zero: separate position, displacement and distance on a number line

1. The number you add describes a directed movement

The first example is −7 + 12. Mark −7 and move 12 units right. Seven units bring you to zero; the remaining five bring you to 5. In symbols, −7 + (7 + 5) = (−7 + 7) + 5 = 5.

A negative term does not force a negative answer. Absolute value is distance from zero: |−7| = 7 and |12| = 12. When adding two numbers of opposite signs as here, movements in opposite directions cancel. Subtract the smaller absolute value, 7, from 12 and keep the positive sign belonging to 12, giving 5.

The option −5 has the wrong sign. The options 19 and −19 count the opposing movements as though they went the same way. Check the starting point too: move 12 units back from the answer 5 and you reach −7.

2. Subtracting a negative undoes a movement

The second example is −4 − (−9). The first minus belongs to the starting position −4. The middle minus is the subtraction operation. The minus inside the parentheses belongs to the number being subtracted, −9. Read those roles separately.

The answer to a − b is the number that returns to a when you add b. Here, adding −9 to the answer should give −4. Since 5 + (−9) = −4, the answer is 5. Undoing a movement of 9 units left requires a movement of 9 units right. That is why −4 − (−9) = −4 + 9.

For the other options, adding −9 to −5 gives −14, adding it to −13 gives −22, and adding it to 13 gives 4. None returns to −4. Using the opening numbers, −3 − (−8) = −3 + 8 = 5 works the same way. Check what you must add back rather than memorizing a visual instruction to erase two minus signs.

3. Distance between two points has no directional sign

The third example asks for the distance between −6 and 2. There are six unit intervals from −6 to zero and two from zero to 2, so the distance is 6 + 2 = 8. The spacing of −6, 0 and 2 in the diagram matches this count. The option 4 incorrectly subtracts the magnitudes despite the points lying on opposite sides of zero. The options −8 and −4 cannot be distances.

The displacement from −6 to 2 is 2 − (−6) = +8. In reverse it is −6 − 2 = −8. Displacement changes sign, but the distance is 8 in either direction. In general, the distance between a and b is |b − a|, the absolute value of their difference. Distance is nonnegative and is zero only when the two points coincide.

A minus sign does not mean you should always add absolute values. Both −6 and −2 lie to the left of zero, and their distance is |−2 − (−6)| = 4. Also distinguish the total length of a walk that turns back from the distance between its starting and finishing points.

4. Name the answer before moving on to coordinates

When you retry the three questions, make separate notes for the starting point and added displacement, the subtraction check by adding back, and the distance between two points. The first two answers are both 5, but the operations are read differently. In the third question, 8 describes a distance, not a position.

Return to Everyday calculations if you need practice naming what a number counts. Next, read Graphs and coordinates to extend the idea of position into horizontal and vertical directions. This article uses one line; coordinates also require you to name the axis a position belongs to.