Distance from zero
−5 and 5 are on opposite sides of zero, but they are the same distance from it. From 0 to 5 is 5 steps to the right, and from 0 to −5 is 5 steps to the left.
Two hops of 5 from zero, one each way, land on 5 and on −5.
The absolute value
The distance of a number from zero is called its absolute value. It is written with a straight bar on each side of the number: |−5| = 5, read as "the absolute value of −5 is 5".
|5| = 5 as well, because 5 is the same distance from zero on the other side. |0| = 0, because zero is no distance from itself. A distance is never negative, so an absolute value is never below zero.
When there is a calculation inside the bars, work it out first, then take the distance from zero: |3 − 8| = |−5| = 5.
−7 is the smaller number, but it is further from zero, so |−7| = 7 is bigger than |4| = 4.
The distance between two numbers
The same idea measures the gap between two numbers. From −2 up to 6 is 2 steps up to zero and 6 more beyond it, so −2 and 6 are 2 + 6 = 8 apart.
The hop from −2 to 6 crosses zero and is 8 steps long.
Subtract either way round
Subtracting finds the gap: 6 − (−2) = 6 + 2 = 8. Subtracting the other way round gives (−2) − 6 = −8, the same size with the opposite sign.
The absolute value keeps only the size, so the distance between two numbers p and q is |p − q|, whichever way round you subtract: |6 − (−2)| = |(−2) − 6| = 8.
A cliff top is 30 m above sea level, and the sea bed below it is at −25 m. The distance between them is |30 − (−25)| = 30 + 25 = 55 m.
The usual mistakes
Stopping before the bars. |3 − 8| is not −5. The subtraction gives −5, and the bars then give its distance from zero, 5.
Treating a negative number as positive. The gap between −2 and 6 is not 6 − 2 = 4. That treats −2 as if it were 2. The two numbers are on opposite sides of zero, so their distances from zero add: 2 + 6 = 8.
Worked example: Heights Above and Depths Below Sea Level
Question The lamp of a lighthouse is 45 m above sea level. Directly below it, a diver is at −18 m and a wreck lies on the sea bed at −62 m, where a negative height is a depth below sea level. (a) Find the vertical distance between the lamp and the diver. (b) How much further must the diver descend to reach the wreck?
1.Write the heights as signed numbers on a vertical number line: the lamp is at 45, sea level is 0, the diver is at −18 and the wreck is at −62.
Sea level is 0. The lamp is at 45, the diver at −18 and the wreck at −62. 2.The distance between the lamp and the diver is |45 − (−18)|. Subtracting −18 is adding 18, so this is |45 + 18| = 63.
The distance between the lamp and the diver is |45 − (−18)| = |45 + 18|. 3.(a) The lamp and the diver are 63 m apart. On the number line this is 45 m down to sea level and 18 m more below it.
(a) 45 m down to sea level and 18 m more: 63 m. 4.The distance between the diver and the wreck is |−18 − (−62)| = |−18 + 62| = |44| = 44.
The distance between the diver and the wreck is |−18 − (−62)| = |−18 + 62|. 5.(b) The diver must descend 44 m further. Check: 63 + 44 = 107, and the lamp is 45 − (−62) = 107 m above the wreck.
(b) The diver must descend 44 m further.
Answer: (a) 63 m; (b) 44 m
Common mistakes
- Subtracting the numbers without their signs in part (a), 45 − 18 = 27. The lamp and the diver are on opposite sides of sea level, so the two distances from zero are added: 45 + 18 = 63.
- Adding in part (b), 18 + 62 = 80. The diver and the wreck are both below sea level, so the distance between them is the difference of the two depths: 62 − 18 = 44.
More negative numbers and linear equations problems, worked step by step →