What are directed numbers in the context of Cambridge IGCSE Mathematics?

Directed numbers are numbers that have both magnitude and direction, typically represented on a number line. In Cambridge IGCSE Mathematics, they include both positive and negative numbers. Positive numbers are usually to the right of zero, while negative numbers are to the left. Understanding directed numbers is crucial for solving problems involving temperature changes, financial transactions, and elevation differences.

How do you add and subtract directed numbers?

To add directed numbers, you move along the number line. For example, adding a positive number means moving to the right, while adding a negative number means moving to the left. Subtraction can be thought of as adding the opposite. For instance, subtracting a positive number is the same as adding a negative number, and vice versa. This concept is essential for solving equations and real-world problems in the Cambridge IGCSE curriculum.

What is the significance of zero in directed numbers?

Zero is the neutral point on the number line for directed numbers. It acts as the dividing line between positive and negative numbers. In Cambridge IGCSE Mathematics, zero is crucial because it represents a balance point where there is no net gain or loss. Understanding zero's role helps in interpreting problems involving temperature, altitude, and financial balances.

How are directed numbers used in real-life scenarios?

Directed numbers are used in various real-life scenarios such as calculating temperature changes, financial gains or losses, and altitude differences. For example, a temperature drop can be represented by a negative number, while a rise is positive. In Cambridge IGCSE Mathematics, students learn to apply these concepts to solve practical problems, enhancing their analytical skills.

What are common mistakes to avoid when working with directed numbers?

Common mistakes include confusing the direction of movement on the number line and incorrectly applying the rules for addition and subtraction. Students often forget that subtracting a negative number is equivalent to adding a positive number. In the Cambridge IGCSE Mathematics curriculum, practicing these operations helps students avoid such errors and develop a strong foundation in handling directed numbers.

Why is "Mishandling double negatives" a common mistake in Directed Numbers?

Why it happens: 3 - (-2) looks confusing under pressure, leading to 1 instead of 5. How to avoid it: Always rewrite as 3 + 2 = 5 before evaluating. Source: 0580/12 — recurring across recent series

Why is "Treating $-3^{2}$ and $(-3)^{2}$ as the same" a common mistake in Directed Numbers?

Why it happens: Without the bracket, the negative is **not** squared, but students often square it anyway. How to avoid it: Read the brackets carefully: (-3)^2 = 9, -3^2 = -9. Source: 0580/22 examiner report

Why is "Getting the sign wrong when subtracting a larger number from a smaller one" a common mistake in Directed Numbers?

Why it happens: Treating subtraction as commutative. How to avoid it: Use a number line: 5 - 12 → start at 5, move 12 left → -7.

Why is "Multiplying three negatives and getting positive" a common mistake in Directed Numbers?

Why it happens: Students apply "two negatives make a positive" then forget the third. How to avoid it: Count the negatives: odd count → negative; even count → positive.

NumberCambridge IGCSE Maths 0580 Extended

Directed Numbers

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Directed Numbers — Cambridge IGCSE 0580 Maths Extended (2026)

Adding, subtracting, multiplying and dividing positive and negative numbers. The arithmetic is straightforward; the sign rules cause more lost marks on Paper 2 than any other topic in Number.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E1.7 — Use the four operations for calculations with whole numbers, decimals and fractions, including negative numbers, with and without a calculator.
Apply directed-number arithmetic to real-life contexts (temperature, debt, height above/below sea level).

Adding and subtracting directed numbers

Picture every operation as a movement on a number line. Right for positive, left for negative.

Imagine standing on the number line at the first number. The operation tells you which way and how far to move.

Adding a positive = move RIGHT. Adding a negative = move LEFT. Subtracting a positive = move LEFT. Subtracting a negative = move RIGHT (it's a "double negative").

Worked walkthroughs:

$-3 + 5$ → start at $-3$ , move 5 right → $2$ .
$4 + (-7)$ → start at $4$ , move 7 left → $-3$ .
$2 - 6$ → start at $2$ , move 6 left → $-4$ .
$-3 - (-8)$ → subtracting a negative becomes adding a positive: $-3 + 8 = 5$ .

Adding a positive moves right; −3 + 5 lands on 2.

The sign-rewrite trick. Two adjacent signs combine: $+(-) = -, \quad -(-) = +, \quad -(+) = -, \quad +(+) = +.$ So $5 - (-2) = 5 + 2 = 7$ and $5 + (-2) = 5 - 2 = 3$ .

ADD a positive → move right.
ADD a negative → move left.
SUBTRACT a negative → move right (cancels to a plus).
Two adjacent like signs ⇒ plus; two unlike signs ⇒ minus.

Multiplying and dividing directed numbers

Same sign → positive. Different signs → negative. The rule applies to both multiplication and division.

The sign of a product depends only on the signs of the factors:

First sign	Second sign	Product/quotient
$+$	$+$	$+$
$-$	$-$	$+$
$+$	$-$	$-$
$-$	$+$	$-$

$-4 \times -5 = 20$ (same signs).
$-4 \times 5 = -20$ (different signs).
$-12 \div -3 = 4$ (same signs).
$-12 \div 3 = -4$ (different signs).

For products of more than two numbers, count the number of negatives:

Even number of negatives → product is positive.
Odd number of negatives → product is negative.

So $(-2)(-3)(-4)(-5) = 120$ (four negatives → positive) and $(-2)(-3)(-4) = -24$ (three negatives → negative).

The same rule applies to division — only the signs decide the result.

Same sign → positive answer.
Different signs → negative answer.
Counts even? → positive. Odd? → negative.
Same rule for $\times$ and $\div$ .

BIDMAS with directed numbers

Brackets and indices first. The fastest way to lose marks is forgetting whether a square attaches to the negative.

The order of operations (BIDMAS / PEMDAS / BODMAS — same idea) is unchanged when negatives appear:

Brackets
Indices (powers / roots)
Division and Multiplication, left to right
Addition and Subtraction, left to right

The trap: how does a square attach to a negative?

$(-3)^{2}$ = $(-3) \times (-3) = 9$ . The negative is INSIDE the brackets, so it gets squared too.
$-3^{2}$ = $-(3^{2}) = -9$ . No brackets — by BIDMAS, the square applies to $3$ FIRST, then the negative is applied.

These are different by sign! Always check whether the negative sits inside brackets.

Worked example. $7 - (-2)^{3} \div 4 \times 2$ .

Indices: $(-2)^{3} = -8$ . Equation becomes $7 - (-8) \div 4 \times 2$ .
$\div$ and $\times$ left to right: $(-8) \div 4 = -2$ . Then $-2 \times 2 = -4$ . Equation becomes $7 - (-4)$ .
$7 - (-4) = 7 + 4 = 11$ .

BIDMAS doesn't change when negatives appear — sign tracking layers ON TOP.
$(-3)^{2} = 9$ vs $-3^{2} = -9$ — the brackets matter.
Resolve indices BEFORE multiplication.
Addition/subtraction last, left to right.

Real-world contexts

Cambridge loves dressing directed-number questions in temperature, finance, or sea-level clothing.

Most directed-number questions on the IGCSE wrap the arithmetic in a real-world story. Recognise the cue and the rest is mechanical.

Temperature. "The temperature is $-7\degree\text{C}$ at 6am and rises by $12\degree\text{C}$ by midday." → $-7 + 12 = 5\degree\text{C}$ .

Bank balance / debt. "An account is $45 overdrawn (i.e. balance $-45$ ). The account holder pays in $60." → $-45 + 60 = 15$ . New balance $15.

Altitude. "A submarine is $300\,\text{m}$ below sea level. It rises $120\,\text{m}$ ." → $-300 + 120 = -180\,\text{m}$ . Still $180\,\text{m}$ below.

Difference between two values. "Find the difference between $5\degree\text{C}$ and $-12\degree\text{C}$ ." → $5 - (-12) = 17\degree\text{C}$ . Subtract the smaller from the larger.

When questions ask "how much hotter / higher / further", expect a positive answer. The sign trick is to set up the calculation as (later/larger) $-$ (earlier/smaller).

Temperature rise → ADD the rise (signed).
Debt is negative; payment is positive.
'Below sea level' = negative altitude.
Difference = larger MINUS smaller (signed) → always positive.

How it’s examined

Directed numbers usually appear in 1-mark questions on Paper 2 (calculator) — a quick sign-rule check inside a longer problem. Paper 4 questions hide them inside word problems, often asking for a difference between two signed quantities. Examiner reports flag $(-3)^{2}$ vs $-3^{2}$ confusion every series — and the slip-up is almost always punished, even on follow-on parts.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E1.7); 0580/22 May/Jun 2024 — Q2 (signed-number arithmetic); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Directed Numbers

Step-by-step solutions to past-paper-style questions on directed numbers, written exactly the way a tutor would explain them at the board.

1Add and subtract directed numbers
Core• addition, subtraction
Question
Calculate (a) $-7 + (-4)$ , (b) $-3 - (-9)$ , (c) $5 + (-12)$ .
Step-by-step solution
1. Step 1
  (a) Adding a negative is the same as subtracting: $-7 + (-4) = -7 - 4 = -11$ .
2. Step 2
  (b) Subtracting a negative is the same as adding: $-3 - (-9) = -3 + 9 = 6$ .
3. Step 3
  (c) Add a negative — net result is subtraction: $5 + (-12) = 5 - 12 = -7$ .
Answer
(a) $-11$ (b) $6$ (c) $-7$
Examiner tip
Always rewrite double signs ( $+-$ or $--$ ) as a single sign before evaluating. Most arithmetic slips happen here.
2Multiply and divide with sign rules
Core• multiplication, division, sign rules
Question
Evaluate (a) $(-6) \times (-4)$ , (b) $\dfrac{-72}{8}$ , (c) $(-3)^{2}$ , (d) $-3^{2}$ .
Step-by-step solution
1. Step 1
  (a) Negative times negative is positive: $(-6) \times (-4) = 24$ .
2. Step 2
  (b) Negative divided by positive is negative: $\dfrac{-72}{8} = -9$ .
3. Step 3
  (c) The bracket means the whole $-3$ is squared: $(-3)^{2} = (-3)(-3) = 9$ .
4. Step 4
  (d) No bracket — the exponent applies only to $3$ , then negate: $-3^{2} = -(3^{2}) = -9$ .
Answer
(a) $24$ (b) $-9$ (c) $9$ (d) $-9$
Examiner tip
Parts (c) and (d) are a classic distinction students miss — write the brackets clearly to avoid losing accuracy marks.
3Apply directed numbers to temperature change
Core• Adapted from 0580/12 Feb/Mar 2024 Q2• applied, temperature
Question
The temperature at 6 a.m. was $-4{}^{\circ}\text{C}$ . By midday it had risen by $11{}^{\circ}\text{C}$ . By 9 p.m. it had fallen by $13{}^{\circ}\text{C}$ from midday. Find the temperature at 9 p.m.
Step-by-step solution
1. Step 1
  Midday: $-4 + 11 = 7{}^{\circ}\text{C}$ .
2. Step 2
  9 p.m.: $7 - 13 = -6{}^{\circ}\text{C}$ .
Answer
$-6{}^{\circ}\text{C}$
4Use directed numbers in a banking context
Core• applied, money
Question
Asha's bank balance is $-45$ dollars (i.e. $45 overdrawn). She deposits $120 and then pays a bill of $60. What is her new balance?
Step-by-step solution
1. Step 1
  Start: $-45$ . Deposit: $-45 + 120 = 75$ .
2. Step 2
  Pay bill: $75 - 60 = 15$ .
Answer
$15
5Evaluate an expression with brackets and mixed signs
Extended• Adapted from 0580/22 Oct/Nov 2023 Q2• brackets, addition, subtraction
Question
Calculate (a) $-8 + (-3) - (-11)$ , (b) $-(-5) + (-4) - (+7)$ , (c) $12 - (-6) + (-9)$ .
Step-by-step solution
1. Step 1
  (a) Rewrite double signs: $-8 - 3 + 11 = -11 + 11 = 0$ .
  $-8 + (-3) - (-11) = -8 - 3 + 11 = 0$
2. Step 2
  (b) Rewrite: $-(-5) = +5$ , $+(-4) = -4$ , $-(+7) = -7$ . So $5 - 4 - 7 = -6$ .
  $-(-5) + (-4) - (+7) = 5 - 4 - 7 = -6$
3. Step 3
  (c) Rewrite: $12 - (-6) = 12 + 6 = 18$ ; then $18 + (-9) = 18 - 9 = 9$ .
  $12 - (-6) + (-9) = 12 + 6 - 9 = 9$
Answer
(a) $0$ (b) $-6$ (c) $9$
Examiner tip
Mark schemes award method marks for explicit double-sign simplification on the working line. Skip that step and one arithmetic slip kills the whole question.
6Multiply three or more negative numbers
Extended• multiplication, sign rules
Question
Calculate (a) $(-2) \times (-3) \times (-5)$ , (b) $(-1)^{7}$ , (c) $(-4) \times (-2) \times (-3) \times (-5)$ .
Step-by-step solution
1. Step 1
  Count the negatives. Odd count → negative result; even count → positive result.
2. Step 2
  (a) Three negatives (odd): magnitude $2 \times 3 \times 5 = 30$ , sign negative. Result $-30$ .
3. Step 3
  (b) $(-1)^{7}$ is $(-1)$ multiplied by itself $7$ times — seven negatives, odd: $-1$ .
4. Step 4
  (c) Four negatives (even): magnitude $4 \times 2 \times 3 \times 5 = 120$ , sign positive. Result $+120$ .
Answer
(a) $-30$ (b) $-1$ (c) $120$
Examiner tip
The examiner report flags that candidates often apply 'two negatives make a positive' twice and forget the third. Count the negatives once at the end — the rule is parity.
7Divide a fraction with mixed signs
Extended• division, fractions, sign rules
Question
Evaluate $\dfrac{-\dfrac{3}{4}}{-\dfrac{9}{8}}$ , giving your answer as a fraction in lowest terms.
Step-by-step solution
1. Step 1
  Dividing by a fraction is the same as multiplying by its reciprocal.
  $\dfrac{-\frac{3}{4}}{-\frac{9}{8}} = -\dfrac{3}{4} \times -\dfrac{8}{9}$
2. Step 2
  Sign rule: negative $\times$ negative $=$ positive.
  $= +\dfrac{3 \times 8}{4 \times 9} = \dfrac{24}{36}$
3. Step 3
  Simplify by dividing top and bottom by $\gcd(24, 36) = 12$ .
  $= \dfrac{2}{3}$
Answer
$\dfrac{2}{3}$
Examiner tip
Cancel before multiplying where possible — Cambridge mark schemes accept either order but candidates who multiply first more frequently produce arithmetic errors.
8Find a temperature difference across multiple cities
Extended• Adapted from 0580/22 Feb/Mar 2024 Q4• applied, temperature, word problem
Question
At 06{:}00 the temperatures in five cities were $-12{}^{\circ}\text{C}$ , $-3{}^{\circ}\text{C}$ , $+7{}^{\circ}\text{C}$ , $-8{}^{\circ}\text{C}$ and $+4{}^{\circ}\text{C}$ . Find (a) the difference between the highest and lowest temperatures, (b) the mean temperature.
Step-by-step solution
1. Step 1
  (a) Identify the extremes: highest $= +7{}^{\circ}\text{C}$ , lowest $= -12{}^{\circ}\text{C}$ . Difference $= 7 - (-12) = 7 + 12 = 19{}^{\circ}\text{C}$ .
  $7 - (-12) = 19$
2. Step 2
  (b) Sum the five values: $-12 + (-3) + 7 + (-8) + 4 = -12 - 3 + 7 - 8 + 4 = -12$ .
3. Step 3
  Divide by the count: mean $= \dfrac{-12}{5} = -2.4{}^{\circ}\text{C}$ .
  $\text{mean} = \dfrac{-12}{5} = -2.4$
Answer
(a) $19{}^{\circ}\text{C}$ (b) $-2.4{}^{\circ}\text{C}$
Examiner tip
Examiner reports show many candidates write the difference as $-19$ . Distance between two temperatures is always positive — use $|\text{high} - \text{low}|$ or take the larger value first.
9BIDMAS with mixed-sign rationals
Challenge• Adapted from 0580/42 May/Jun 2024 Q5• BIDMAS, order of operations, stretch
Question
Without a calculator, evaluate $\dfrac{(-3)^{2} - 2 \times (-4)}{-7 + (-2)^{3}}$ .
Step-by-step solution
1. Step 1
  Apply BIDMAS. Start with brackets/indices in the numerator: $(-3)^{2} = 9$ .
2. Step 2
  Multiplication: $2 \times (-4) = -8$ . So numerator $= 9 - (-8) = 9 + 8 = 17$ .
  $(-3)^{2} - 2 \times (-4) = 9 + 8 = 17$
3. Step 3
  Denominator: $(-2)^{3} = -8$ . So denominator $= -7 + (-8) = -15$ .
  $-7 + (-2)^{3} = -7 - 8 = -15$
4. Step 4
  Divide.
  $\dfrac{17}{-15} = -\dfrac{17}{15}$
Answer
$-\dfrac{17}{15}$ (or $-1\dfrac{2}{15}$ ).
Examiner tip
The 2024 examiner report notes candidates often compute $-3^{2}$ as $-9$ inside the bracket, losing the indices step. With brackets, $(-3)^{2} = 9$ — square the negative.
10Tracking a fluctuating debt balance
Challenge• Adapted from 0580/42 Oct/Nov 2023 Q3• applied, debt, word problem
Question
On 1 March a small business has a balance of $-\$ 1,200 $(i.e. \$ 1,200 in debt). During March it earns $3,450 in revenue and incurs costs of $2,860. On 1 April the bank applies $2\%$ interest on the new positive balance (or charges $5\%$ on a negative one). Find the balance on 2 April.
Step-by-step solution
1. Step 1
  Compute the balance after March: $-1200 + 3450 - 2860$ .
  $-1200 + 3450 - 2860 = -610$
2. Step 2
  The balance is still negative ( $-\$ 610 $), so the bank charges$ 5%$ interest on the debt.
  $\text{interest} = -610 \times 0.05 = -30.50$
3. Step 3
  Apply the interest. Charging interest increases the debt — add the (negative) interest to the (negative) balance.
  $\text{new balance} = -610 + (-30.50) = -640.50$
Answer
$-\$ 640.50 $(i.e. \$ 640.50 in debt).
Examiner tip
Mark schemes award method marks for the correct sign of each step. Candidates who write the interest as $+\$ 30.50$ (and so reduce the debt) lose two accuracy marks — interest charged on a debt makes the debt worse.

Key Formulae — Directed Numbers

The formulae you need to memorise for directed numbers on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Sign rules for multiplication and division
$(+)\cdot(+)=(+),\ \ (-)\cdot(-)=(+),\ \ (+)\cdot(-)=(-),\ \ (-)\cdot(+)=(-)$
When to use
Whenever you multiply or divide two signed numbers.
Double-sign simplification
$+(-a) = -a,\quad -(-a) = +a$
$a$
any real number
When to use
Use to simplify expressions before adding or subtracting.

Key Definitions and Keywords — Directed Numbers

Definitions to memorise and the exact keywords mark schemes credit for directed numbers answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Directed number
Examiner keyword
A number with a direction (sign) — positive (above zero) or negative (below zero).
Positive number
A number greater than zero. Sometimes written with a leading $+$ .
Negative number
A number less than zero, written with a leading $-$ .
Magnitude (absolute value)
The size of a number ignoring its sign. $|-7| = 7$ .
Additive inverse (opposite)
The number that, when added to a given number, gives zero. The opposite of $5$ is $-5$ .

Common Mistakes and Misconceptions — Directed Numbers

The traps other students keep falling into on directed numbers questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Mishandling double negatives
0580/12 — recurring across recent series
Why it happens
$3 - (-2)$ looks confusing under pressure, leading to $1$ instead of $5$ .
How to avoid it
Always rewrite as $3 + 2 = 5$ before evaluating.
✕Treating $-3^{2}$ and $(-3)^{2}$ as the same
0580/22 examiner report
Why it happens
Without the bracket, the negative is not squared, but students often square it anyway.
How to avoid it
Read the brackets carefully: $(-3)^{2} = 9$ , $-3^{2} = -9$ .
✕Getting the sign wrong when subtracting a larger number from a smaller one
Why it happens
Treating subtraction as commutative.
How to avoid it
Use a number line: $5 - 12$ → start at $5$ , move $12$ left → $-7$ .
✕Multiplying three negatives and getting positive
Why it happens
Students apply "two negatives make a positive" then forget the third.
How to avoid it
Count the negatives: odd count → negative; even count → positive.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Directed Numbers — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Directed Numbers

Short Study Notes in the form of Slides

Short Study Notes — Directed Numbers

Detailed Notes

What you’ll learn

How it’s examined

1Add and subtract directed numbers

2Multiply and divide with sign rules

3Apply directed numbers to temperature change

4Use directed numbers in a banking context

5Evaluate an expression with brackets and mixed signs

6Multiply three or more negative numbers

7Divide a fraction with mixed signs

8Find a temperature difference across multiple cities

9BIDMAS with mixed-sign rationals

10Tracking a fluctuating debt balance

Sign rules for multiplication and division

Double-sign simplification

Directed number

Positive number

Negative number

Magnitude (absolute value)

Additive inverse (opposite)

✕Mishandling double negatives

✕Treating −32-3^{2}−32 and (−3)2(-3)^{2}(−3)2 as the same

✕Getting the sign wrong when subtracting a larger number from a smaller one

✕Multiplying three negatives and getting positive

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Directed Numbers — frequently asked questions

✕Treating $-3^{2}$ and $(-3)^{2}$ as the same