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Negative numbers

Numbers · level 4, lesson 6 of 7

A question to start

What is \(5 - 8\)?

For a long time, many people said: "You can't take 8 from 5. It is impossible." Even some mathematicians a few hundred years ago called answers like this "absurd". But a thermometer has no trouble with it. If it is 5 °C and the temperature drops by 8 degrees, it is now −3 °C: three degrees below zero.

And here is a stranger one: what is \(3 - (-2)\)? Taking away a negative? By the end of this lesson you will know the answer (it is 5) and, more importantly, why.

The idea

Numbers below zero

Some things can go below a starting point:

  • Temperature: −10 °C is ten degrees below freezing. The average temperature on Mars is about −60 °C.
  • Height: the shore of the Dead Sea is more than 400 m below sea level.
  • Floors: in many buildings, the floor below the ground floor is floor −1.
  • Money: if you owe 20 dollars, your balance is −20 dollars.

Each of these has a zero that is a starting point, not "nothing": 0 °C is when water freezes, sea level is a chosen height. Numbers above it are positive, numbers below it are negative. We write a negative number with a minus sign in front: −3, read "negative three". Positive numbers can be written with a plus sign (+3) but usually are not. Zero is neither positive nor negative.

The whole numbers together with their negatives, ..., −3, −2, −1, 0, 1, 2, 3, ..., are called the integers.

The number line, both ways

Extend the number line to the left of 0. The negative numbers are a mirror image of the positive ones.

4 steps4 steps −6−5−4−3−2−10123456

Opposites. 4 and −4 are the same distance from 0 (4 steps) but on opposite sides. They are called opposites. The opposite of 7 is −7; the opposite of −7 is 7; the opposite of 0 is 0. Writing a minus sign in front of a number means "the opposite of", so \(-(-7) = 7\): the opposite of the opposite takes you back where you started, like turning around twice.

The distance of a number from 0 is called its absolute value, written with two bars: \(|-4| = 4\) and \(|4| = 4\).

Comparing: further left is smaller

On the number line, numbers get bigger to the right and smaller to the left. So:

\[-7 < -3\]

even though 7 is bigger than 3. Think of temperature: −7 °C is colder than −3 °C. Or money: owing 7 dollars is worse than owing 3 dollars. Among negative numbers, the one that is further from zero is the smaller one.

Every negative number is smaller than 0, and every positive number is bigger than every negative number: \(-1000 < -1 < 0 < 1\).

Adding with chips

Here is a model that makes adding and subtracting integers easy to see. Use two kinds of chips:

  • an orange chip (+) stands for +1,
  • a blue chip (−) stands for −1.

One orange and one blue chip together make 0: +1 and −1 cancel out. A pair like this is called a zero pair. You can add or remove zero pairs whenever you like, because adding 0 changes nothing.

Adding means putting chips together.

  • \(5 + (-3)\): put 5 orange and 3 blue together. Make 3 zero pairs and remove them. 2 orange are left: 2.
  • \(-4 + (-2)\): 4 blue and 2 blue make 6 blue: −6.
  • \(-5 + 3\): 5 blue and 3 orange. 3 zero pairs cancel, and 2 blue are left: −2.

So: adding two negatives gives a bigger negative. Adding numbers with different signs, the chips cancel, and whichever kind has more chips "wins", by the difference.

Subtracting with chips

Subtracting means taking chips away.

\(6 - 2\): start with 6 orange, take away 2 orange, 4 are left. Nothing new.

\(2 - 5\): start with 2 orange. We need to take away 5 orange, but there are only 2. So add zero pairs until there are enough. Add 3 zero pairs: now there are 5 orange and 3 blue, which is still worth 2. Take away the 5 orange. Left: 3 blue, so \(2 - 5 = -3\).

+++++ −−− 3 zero pairs added take away 5 positives left: 3 negatives so 2 − 5 = −3

\(3 - (-2)\): start with 3 orange. Take away 2 blue, but there are none. Add 2 zero pairs: now 5 orange and 2 blue (still worth 3). Take away the 2 blue. Left: 5 orange. So \(3 - (-2) = 5\).

Look at what happened. To take away 2 blue chips, we had to bring in 2 zero pairs, which also brought in 2 orange chips. Taking away the blue left those 2 extra orange behind. So taking away −2 has the same effect as adding +2:

\[3 - (-2) = 3 + 2 = 5\]

The same thing happens with \(2 - 5\): taking away 5 orange leaves 5 extra blue behind, so it is the same as adding −5. In general:

\[a - b = a + (\text{the opposite of } b)\]

Subtracting a number is the same as adding its opposite. It makes sense in real life too: if someone takes away a debt of 2 dollars that you owed, you are 2 dollars better off.

On the number line

Think of adding as moving:

  • adding a positive number moves you right;
  • adding a negative number moves you left;
  • subtracting a number moves you the opposite way from adding it.
−2 + 5 = 3 3 + (−5) = −2 −6−5−4−3−2−10123456

Subtracting as a difference. \(a - b\) also answers "how far is it from \(b\) up to \(a\)?" For \(-3 - (-7)\): from −7 up to −3 is 4 steps to the right, so \(-3 - (-7) = 4\). Check with the rule: \(-3 + 7 = 4\). ✓ This is how we find the change in temperature: from −7 °C to −3 °C is a rise of 4 degrees.

Worked examples

1. Compare. Order from smallest: 2, −5, 0, −1, −12. On the number line from left to right: −12, −5, −1, 0, 2.

2. Add. \(8 + (-12)\). 8 orange and 12 blue: 8 zero pairs cancel, 4 blue left. −4.

3. Add. \(-15 + 7\). 7 zero pairs cancel; 8 blue left. −8.

4. Subtract. \(-6 - 4\). Add the opposite: \(-6 + (-4) = -10\). On the line: start at −6, move 4 left.

5. Subtract a negative. \(-2 - (-9)\). Add the opposite: \(-2 + 9 = 7\).

6. Temperature on Mars. Suppose a rover's thermometer reads −78 °C at dawn and −19 °C in the afternoon. How much did it warm up? The change is \(-19 - (-78) = -19 + 78 = 59\) degrees. Check on the line: from −78 up to 0 is 78, but we stop 19 short of 0, so \(78 - 19 = 59\). ✓ Big daily swings like this are normal on Mars, because the thin air holds very little heat.

Try it

  1. Write the opposite of: (a) 9 (b) −15 (c) 0. What is \(-(-6)\)?
  2. Use \(<\) or \(>\): (a) −8 ☐ −2 (b) −1 ☐ −10 (c) −3 ☐ 1.
  3. Use chips to work out: (a) \(4 + (-6)\) (b) \(-3 + (-5)\) (c) \(-7 + 7\).
  4. Use chips (add zero pairs) to work out: (a) \(1 - 4\) (b) \(5 - (-3)\) (c) \(-2 - (-6)\).
  5. Work out using "add the opposite": (a) \(-9 - 3\) (b) \(10 - (-10)\) (c) \(-4 - (-1)\).
  6. A diver is at −18 m (18 m below the surface) and rises 7 m. Where is she now? Then she dives 12 m. Where now?
  7. The temperature at midnight was −63 °C and at noon −18 °C. What was the rise?
  8. Harder: find two integers whose sum is −2 and whose difference (bigger minus smaller) is 10.
Answers
  1. (a) −9 (b) 15 (c) 0. \(-(-6) = 6\).
  2. (a) \(-8 < -2\) (b) \(-1 > -10\) (c) \(-3 < 1\).
  3. (a) 4 zero pairs cancel, 2 blue left: −2. (b) 8 blue: −8. (c) 7 zero pairs: 0.
  4. (a) 1 orange; add 3 zero pairs (4 orange, 3 blue); take away 4 orange; left 3 blue: −3. (b) 5 orange; add 3 zero pairs; take away 3 blue; left 8 orange: 8. (c) 2 blue; add 4 zero pairs (6 blue, 4 orange); take away 6 blue; left 4 orange: 4.
  5. (a) \(-9 + (-3) = -12\) (b) \(10 + 10 = 20\) (c) \(-4 + 1 = -3\).
  6. \(-18 + 7 = -11\), so 11 m below the surface. Then \(-11 - 12 = -23\): 23 m below.
  7. \(-18 - (-63) = -18 + 63 = 45\) degrees.
  8. The two numbers are 10 apart and their middle is \(-2 \div 2 = -1\). So they are 5 either side of −1: 4 and −6. Check: \(4 + (-6) = -2\) ✓ and \(4 - (-6) = 10\) ✓.

Common mistake

Thinking −7 is bigger than −3 "because 7 is bigger than 3". On the number line, −7 is further left, so it is smaller. Ask: which is colder?

Two minus signs in a row. In \(5 - (-3)\), the first sign means "subtract" and the second means "negative". Together they make adding 3: the answer is 8, not 2. Rewrite as "add the opposite" before calculating.

Where it leads