Directed Numbers

Numbers below zero in exam questions: temperatures that rise and fall, the difference between two temperatures, the lowest value, and signs when you calculate.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 57 real past-paper questions below

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What Are Directed Numbers?

A directed number has a sign. Positive numbers are above zero, negative numbers are below zero. A temperature of $- 5^{\circ} C$ is 5 degrees below zero.

Picture a number line. Right is bigger, left is smaller.

You see	It means
rises, warmer, higher	move right: add
falls, colder, lower	move left: subtract
difference	the gap between two numbers

Rule

The further left a number is, the smaller it is. So $- 5$ is smaller than $- 1$ , even though 5 is bigger than 1.

Key idea

In the year 628, the Indian mathematician Brahmagupta wrote down rules for negative numbers. He called positive numbers fortunes and negative numbers debts. One rule: a debt taken away from zero is a fortune, just like $0 - (- 3) = 3$ .

A Temperature Rises or Falls

The most common question, for 1 or 2 marks: a temperature goes up or down, or one place is so many degrees lower than another. Find the new temperature.

It is $- 4^{\circ} C$ in Oslo. Cairo is 19 degrees higher. Moscow is 6 degrees lower than Oslo.

Worked example

1
$- 4 + 19 = 15$ — Cairo is 19 degrees higher, so add 19 to Oslo's −4. Up 4 reaches zero, then 15 more: Cairo is 15°C.
2
$- 4 - 6 = - 10$ — Moscow is 6 degrees lower, so take 6 away. You are already below zero, so you go further down: Moscow is −10°C.

The 2-mark question adds a step. At 8 pm it is $5^{\circ} C$ . It falls 3 degrees every hour. What is it at midnight?

Worked example

1
$12 - 8 = 4$ — Count the hours from 8 pm to midnight (12 o'clock). That is how many falls there are.
2
$4 \times 3 = 12$ — It falls 3 degrees each hour, so 4 hours means a total fall of 12 degrees.
3
$5 - 12 = - 7$ — Take the whole fall from the start. 5 down reaches zero, then 7 more takes you below: −7°C.

Common mistake

Answering $7$ instead of $- 7$ . A fall of 12 from 5 goes 7 below zero, so the sign matters.

The Difference Between Two Temperatures

The 1-mark question, just as common: the difference between two temperatures, or how much lower one is. Count the gap on the number line.

Find the difference between $6^{\circ} C$ and $- 9^{\circ} C$ .

Worked example

1
$- 9 \to 0 : 9$ — Start at the lower temperature. It takes 9 degrees to climb up to zero.
2
$0 \to 6 : 6$ — Then 6 more degrees to climb from zero up to 6.
3
$9 + 6 = 15$ — Add the two climbs to get the whole gap. The difference is 15 degrees.
4
$6 - (- 9) = 6 + 9 = 15$ — Check with bigger minus smaller. Taking away −9 is adding 9, and it gives the same 15.

Halfway between two temperatures: add them and halve. Halfway between $7^{\circ}$ and $- 1 1^{\circ}$ :

Worked example

1
$7 + (- 11) = - 4$ — Add the two temperatures. Adding −11 means going down 11 from 7, which lands on −4.
2
$- 4 \div 2 = - 2$ — Halve the total to find the middle. A negative divided by a positive stays negative: −2°.
3
$7 - (- 2) = 9 and - 2 - (- 11) = 9$ — Check: −2 is 9 degrees from each end, so it really is halfway.

Common mistake

Answering $3$ for the gap between 6 and $- 9$ . That is $9 - 6$ . When one is below zero, the sizes add.

Decimals and Two-Step Changes

Exams also use decimals, and some 2-mark questions chain two changes together. The method is the same: move up or down, past zero if needed.

Find the difference between $- 9. 4^{\circ} C$ and $16. 8^{\circ} C$ .

Worked example

1
$- 9.4 \to 0 : 9.4$ — Start at the lower temperature. It takes 9.4 degrees to climb up to zero.
2
$0 \to 16.8 : 16.8$ — Then 16.8 more to climb from zero up to 16.8.
3
$9.4 + 16.8 = 26.2$ — Add the two climbs, lining up the decimal points. The difference is 26.2 degrees.

A cabin is $1 8^{\circ} C$ . The cabin is $23. 5^{\circ} C$ warmer than the top of a hill. The top of the hill is $4. 2^{\circ} C$ colder than the valley. Find the temperature in the valley.

Worked example

1
$top = 18 - 23.5$ — The cabin is the warmer one, so the top of the hill is colder. Take 23.5 away from the cabin's 18.
2
$top = - 5.5$ — 18 down to zero, then 5.5 more below zero: the top is −5.5°C.
3
$valley = - 5.5 + 4.2$ — The top is colder than the valley, so the valley is warmer. Add 4.2.
4
$valley = - 1.3$ — Going up 4.2 from −5.5 does not reach zero, so the valley is −1.3°C.

Common mistake

Adding 23.5 to 18 because the question says “warmer”. Ask which place is warmer: the cabin is, so the top is 23.5 degrees lower.

Key idea

Show every line. On a 2-mark question the first line, like 18 − 23.5, earns a method mark even if a later slip spoils the answer.

Lowest, Highest and Signs

The 1-mark question: which place was the coldest, or which sign goes in the box. The lowest is the one furthest left on the number line.

Place	Temperature
Oslo	$- 4^{\circ} C$
Moscow	$- 1 0^{\circ} C$
Cairo	$1 5^{\circ} C$
Quebec	$- 1 3^{\circ} C$
Lima	$0^{\circ} C$

Worked example

1
$- 13 < - 10 < - 4 < 0 < 15$ — Write the temperatures in order, smallest first. The bigger the number after a minus, the further left it sits.
2
$- 13$ — The lowest is the first in the order. The question asks for the place, so write Quebec, not the number.
3
$- 5 < - 1$ — −5 is further left than −1, so it is smaller. The sign opens towards the bigger number and points at the smaller one.
4
$0 > - 6$ — Zero is to the right of every negative number, so zero is the bigger one.

Common mistake

Ignoring the minus sign when you compare. 13 is bigger than 10, but $- 13$ is colder than $- 10$ : it is further below zero.

Signs When You Calculate

Without a calculator you may need to add, take away, multiply or divide with negatives. Two rules do all the work.

Rule

Taking away a negative is the same as adding. Adding a negative is the same as taking away.

Worked example

1
$4 - (- 6) = 4 + 6$ — Taking away a negative is the same as adding, so the two minus signs become one plus.
2
$4 + 6 = 10$ — Now it is an ordinary addition.
3
$- 3 + (- 5) = - 3 - 5$ — Adding a negative is the same as taking away, so + (−5) becomes − 5.
4
$- 3 - 5 = - 8$ — Start at −3 and go down 5 more. You land on −8.

Rule

Multiply or divide: same signs give a positive answer, different signs give a negative answer.

Worked example

1
$(- 4) \times (- 6) = 24$ — 4 × 6 = 24. Both numbers are negative, same signs, so the answer is positive.
2
$- 20 \div 5 = - 4$ — 20 ÷ 5 = 4. One negative and one positive are different signs, so the answer is negative.
3
$(- 3) \times 5 = - 15$ — 3 × 5 = 15. Different signs again, so the answer is negative.

When a line has more than one operation, multiply or divide first, then add or take away. Keep each negative in brackets.

Worked example

1
$20 - 0.5 \times (- 8)$ — The question. Do the multiplication before the take-away.
2
$0.5 \times (- 8) = - 4$ — Different signs, so the answer is negative.
3
$20 - (- 4) = 20 + 4$ — Taking away a negative is adding.
4
$20 + 4 = 24$ — So the answer is 24.

Common mistake

Using the multiply rule when adding. $- 3 + (- 5)$ is $- 8$ , not $8$ . Two negatives make a positive only when you multiply or divide.

Common mistake

Squaring a negative. $(- 3)^{2} = (- 3) \times (- 3) = 9$ , but without brackets $- 3^{2}$ means $- (3 \times 3) = - 9$ . Only the 3 is squared.

Find the Missing Number

Some questions ask which number you add or take away to reach an answer. Write it with a box, then undo it.

Write down the number you add to $- 6$ to get $5$ .

Worked example

1
$- 6 + □ = 5$ — Write the question as a sum, with a box for the missing number.
2
$□ = 5 - (- 6)$ — Undo the adding by taking away: missing number = answer − start.
3
$□ = 5 + 6 = 11$ — Taking away −6 is the same as adding 6.
4
$- 6 + 11 = 5$ — Check by putting 11 back in the box. It gives 5, so 11 is right.

Write down the number you take away from $- 7$ to get $- 2$ .

Worked example

1
$- 7 - □ = - 2$ — Write the question as a take-away, with a box for the missing number.
2
$□ = - 7 - (- 2)$ — The amount taken away is the gap from the start to the answer: missing number = start − answer.
3
$□ = - 7 + 2 = - 5$ — Taking away −2 is the same as adding 2.
4
$- 7 - (- 5) = - 7 + 5 = - 2$ — Check by putting −5 back in the box. It gives −2, so −5 is right.

Key idea

Going from −7 up to −2 is a move up, yet the question says “take away”. The only way to go up by taking away is to take away a negative.

Common mistake

Answering 5 for the second one. Check it: −7 − 5 = −12, not −2. Always put your answer back in the box.

Biggest and Smallest Products

A list question: find the highest or lowest product (the answer when you multiply). Two negatives multiply to a positive, so always try them too.

Rule

Count the minus signs when you multiply. An even number of them gives a positive answer; an odd number gives a negative answer.

Use this list: $- 8, - 5, - 2, 0, 1, 3, 4, 7$ .

1. The largest number times the smallest number.

Worked example

1
$7 \times (- 8) = - 56$ — The largest is 7 and the smallest is −8, the one furthest left. Different signs, so the answer is negative.

2. The highest product of two numbers.

Worked example

1
$7 \times 4 = 28$ — Try the two biggest positive numbers first.
2
$(- 8) \times (- 5) = 40$ — Now try the two most negative numbers. Same signs, so the answer is positive.
3
$40 > 28$ — Pick the bigger answer. The highest product of two is 40.

3. The lowest product of three numbers.

Worked example

1
$(- 8) \times 7 \times 4$ — For a big negative answer, use one minus sign: the most negative number with the two biggest positives.
2
$(- 8) \times 7 = - 56$ — Multiply the first two. Different signs, so negative.
3
$- 56 \times 4 = - 224$ — Multiply by the third. Still one minus sign, so still negative.
4
$(- 8) \times (- 5) \times (- 2) = - 80$ — Also try the three negatives: three minus signs is odd, so this is negative too.
5
$- 224 < - 80$ — −224 is further left on the number line, so the lowest product of three is −224.

Common mistake

Answering 28 for the highest product of two. $(- 8) \times (- 5) = 40$ is bigger, because two negatives make a positive.

Which Method? Read the Question

The words in the question tell you which way to move.

You see	You do
higher, rises, warmer	add
lower, falls, colder	subtract, and go past zero if needed
difference, how much lower	count the gap: up to zero, then on
halfway between	add the two, then halve
lowest, coldest	the one furthest left
a sign in the box	the sign points at the smaller one
A is warmer than B, B is colder than C	work out each place in turn: the warmer one is higher
the number you add to get…	answer − start
the number you take away to get…	start − answer
highest or lowest product	try the negatives too, and count the minus signs
write these in order	Ordering Values
whole numbers, leftovers	Integers

Key idea

Sketch a quick number line with zero on it. Most mistakes with negatives vanish once you can see which way you are moving.

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