Year 8 · Unit 3 Integers · Lesson 2 of 6

How far apart?

Absolute value is the distance from zero, so it is never negative. Two distances to zero tell you the distance between any two integers.

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Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Write the opposite of −13.
    Answer13.
  2. Which is warmer, −11 °C or −4 °C?
    Answer−4 °C: it is higher up the thermometer, so −11 < −4.
  3. How many steps is it from 0 to −8 on a number line?
    Answer8 steps. Distance has no direction, so it is 8, not −8.
Words used in this lesson
absolute value
how far a number is from zero. For example: |−5| = 5.
distance
how far apart, with no direction. For example: the lift travels 10 floors.
between
in the space separating two things. For example: 0 is between −3 and 7.
same side
both above zero, or both below it. For example: −20 and −8.
level / floor
a storey of a building; below ground is negative. For example: level −3.

Distance has no direction

A lift in a tall building goes from the car park on level −3 up to floor 7. How many floors does it travel? Not 4, which is what you get from 7 and 3 if you ignore the signs. It climbs 3 floors to reach the ground floor, then 7 more: 10 floors.

The distance from zero to a number is called its absolute value, written with two upright bars: |−3| = 3 and |7| = 7. A distance is never negative, so an absolute value is never negative either.

|n| is the distance from n to zero. |−12| = 12, |12| = 12 and |0| = 0.

The distance between two integers: if zero is between them, add their distances from zero; if they are on the same side of zero, take the smaller distance from the larger one.

Set a and b anywhere. Before each move, say what the distance between a and b will be, then check.

−6
9
6|a|, distance of a from zero
9|b|, distance of b from zero
15distance between a and b

Zero is between −6 and 9, so the distance between them is 6 + 9 = 15.

Show me the steps
  1. a = −6, b = 9: |a| = 6, |b| = 9. Zero is between them, so the distance is 6 + 9 = 15.
  2. a = −12, b = −4: |a| = 12, |b| = 4. Both are left of zero, so the distance is 12 − 4 = 8.
  3. a = 3, b = 14: |a| = 3, |b| = 14. Both are right of zero, so the distance is 14 − 3 = 11.
  4. a = 0, b = −7: the distance is just |−7| = 7.

Two slips to watch for. |−7| is 7, not −7: the bars ask how far, and how far is never negative. And the distance from −20 to −8 is not 28: both are on the same side of zero, so the gap is 20 − 8 = 12.

Worked examples

The lift, from level −3 to floor 7.

  1. Is zero between them? Yes: one is below ground, one above.
  2. Add the distances from zero: |−3| + |7| = 3 + 7 = 10 floors.

At midnight it is −18 °C in Yakutsk; by noon it is −5 °C. How many degrees warmer is it?

  1. Is zero between them? No: both are below zero.
  2. Take the smaller distance from the larger: 18 − 5 = 13 degrees.

Your go. The first step is given. A research submarine is 250 m below the surface. A helicopter hovers 400 m above the surface, over the same spot. How far apart are they?

  1. As integers, the submarine is at −250 and the helicopter at 400, and zero (the surface) is between them.
  2. Finish it, then check.
Check the answer

250 + 400 = 650 m apart.

Check before you go on

1. What is |−12|?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Work out |−17|.

3. How far apart are −6 and 9 on a number line?

Level 2 · Building

4. How far apart are −20 and −8?

5. A lift goes from car park level −4 up to floor 11. How many floors does it travel?

6. Which two numbers have an absolute value of 8?

Level 3 · Stretch

7. Two different integers both have an absolute value of 6. How far apart are they?

8. Point P is at −3. Point Q is 10 units from P, to its left. Where is Q?

Something to think about

You can find the distance between two numbers on the same side of zero by subtracting their distances from zero. Does that still work if one of the numbers is zero? Try it with the figure.

One way to think about it

Yes. If b = 0 then |b| = 0, and taking 0 away from |a| leaves |a|, the distance from a to zero. Zero belongs to both sides, so either rule (add or subtract) gives the same answer.

Exit ticket

9. Which of these is the greatest: |−9|, |4|, −12 or |0|?