Arithmetic, Logical & Boolean Operators — Cambridge IGCSE CS (0478)

🎯 Syllabus & Goals 3 min

Cambridge 8.1.4 · Arithmetic, logical and Boolean operators Paper 2 · Algorithms, Programming and Logic

By the end of this lesson you can:

Evaluate expressions using + - * / ^, MOD and DIV, with brackets, and write them in Python.
Use the logical (comparison) operators = <> < > <= >= and know their Python forms.
Combine conditions with the Boolean operators AND, OR and NOT, and predict the result.

Textbook: Chapter 8, §8.1.4(f) (pp. 317–318); MOD and DIV also §8.1.7 (pp. 327–328)

Recap / Warm-Up 5 min

Last lesson: totals, counts and strings. Every one of them relied on an operator — + to total, = to compare a character, & to join strings.

Quick starter

29 students are put into teams of 4. How many full teams are there, and how many students are left over?

Reveal the answer

7 full teams (4 × 7 = 28) with 1 student left over. You just did DIV (7) and MOD (1) in your head.

🧠 Key Concept 14 min

A condition such as Total / Count >= 50 AND NOT Absent uses all three families at once.

1 · Arithmetic operators

Pseudocode	Action	Python	Example → result
`+` `-` `*`	Add, subtract, multiply	`+` `-` `*`	`17 * 5` → 85
`/`	Divide (result is REAL)	`/`	`17 / 5` → 3.4
`^`	Raise to the power of	`**`	`2 ^ 3` → 8
`MOD(17, 5)`	Remainder of a division	`17 % 5`	2
`DIV(17, 5)`	Whole-number part (quotient) of a division	`17 // 5`	3

Seventeen counters shared into groups of five. DIV counts the full groups; MOD counts what is left over.

Key point: brackets are worked out first, then ^, then * and /, then + and -. So 3 + 4 * 2 is 11 but (3 + 4) * 2 is 14. Use brackets whenever the order could be misread, e.g. ((X + Y) * 3) / Pi.

Real world: a clock app turns 135 seconds into minutes and seconds with DIV(135, 60) = 2 and MOD(135, 60) = 15. A spreadsheet shades every other row by testing MOD(RowNumber, 2) = 0— that is how any program tests for an even number.

2 · Logical operators

Logical operators compare two values. The result is always TRUE or FALSE, so they are used to decide the path through a program.

Pseudocode	Meaning	Python
`>` / `<`	Greater than / less than	`>` / `<`
`>=` / `<=`	Greater than or equal / less than or equal	`>=` / `<=`
`=`	Equal	`==`
`<>`	Not equal	`!=`

3 · Boolean operators

Boolean operators join comparisons into one more complex condition.

AND needs both sides TRUE; OR needs at least one; NOT flips the value. Python writes them and, or, not.

Examiner tip: write each comparison in full. IF Age >= 13 AND <= 17 is wrong and loses the mark — it must be IF Age >= 13 AND Age <= 17. Also, in Python = assigns and == compares; in pseudocode ← assigns and = compares.

Beyond the syllabus (not examined): with negative numbers, Python's // and % round towards minus infinity, so -7 // 2 is -4. IGCSE questions use positive values.

Worked Example 12 min

Example 1 · Arithmetic — evaluating expressions and splitting teams

First, evaluate each expression by hand, then check with Python.

print(17 + 5, 17 - 5, 17 * 5)
print(17 / 5)       # divide: always gives a REAL (float)
print(2 ** 3)       # power (^ in pseudocode)
print(17 % 5)       # MOD: remainder
print(17 // 5)      # DIV: whole-number part
print(3 + 4 * 2)    # * before +
print((3 + 4) * 2)  # brackets first

Now a real task: a teacher splits a class into teams of a chosen size. Output the number of full teams and how many students are left over.

Cambridge pseudocode

DECLARE Students, TeamSize : INTEGER
DECLARE Teams, LeftOver : INTEGER
OUTPUT "Number of students: "
INPUT Students
OUTPUT "Students per team: "
INPUT TeamSize
Teams ← DIV(Students, TeamSize)
LeftOver ← MOD(Students, TeamSize)
OUTPUT "Full teams: ", Teams
OUTPUT "Students left over: ", LeftOver
IF LeftOver = 0
  THEN
    OUTPUT "Everyone is in a full team"
ENDIF

The same in Python

# Split a class into teams
students = int(input("Number of students: "))
team_size = int(input("Students per team: "))
teams = students // team_size          # DIV
left_over = students % team_size       # MOD
print("Full teams:", teams)
print("Students left over:", left_over)
if left_over == 0:
    print("Everyone is in a full team")

Number of students: 29
Students per team: 4
Full teams: 7
Students left over: 1

Number of students: 24
Students per team: 4
Full teams: 6
Students left over: 0
Everyone is in a full team

DIV(29, 4): 4 goes into 29 seven whole times → 7.DIV throws away the remainder.
MOD(29, 4): 29 − 7 × 4 = 1 → 1.MOD keeps only the remainder.
LeftOver = 0 tests for an exact split.MOD(X, Y) = 0 means "X divides exactly by Y".

Example 2 · Logical and Boolean — a youth-club entry check

A youth club admits people aged 13 to 17 inclusive who are not banned.

Cambridge pseudocode

DECLARE Age : INTEGER
DECLARE Answer : CHAR
DECLARE Banned : BOOLEAN
OUTPUT "Age: "
INPUT Age
OUTPUT "Banned (Y/N)? "
INPUT Answer
Banned ← Answer = 'Y'
IF Age >= 13 AND Age <= 17 AND NOT Banned
  THEN
    OUTPUT "Welcome in"
  ELSE
    OUTPUT "Entry refused"
ENDIF

The same in Python

# Youth club entry: aged 13 to 17 inclusive AND not banned
age = int(input("Age: "))
banned = input("Banned (Y/N)? ") == "Y"
if age >= 13 and age <= 17 and not banned:
    print("Welcome in")
else:
    print("Entry refused")

Banned ← Answer = 'Y' stores the result of a comparison — TRUE or FALSE — in a BOOLEAN variable.

Age	Answer	Age >= 13	Age <= 17	NOT Banned	Whole condition	OUTPUT
15	N	TRUE	TRUE	TRUE	TRUE	Welcome in
15	Y	TRUE	TRUE	FALSE	FALSE	Entry refused
18	N	TRUE	FALSE	TRUE	FALSE	Entry refused

Output — the three test runs

Age: 15
Banned (Y/N)? N
Welcome in

Age: 15
Banned (Y/N)? Y
Entry refused

Age: 18
Banned (Y/N)? N
Entry refused

Try It Yourself 12 min

🟢 Easy

Goal: Without a computer, work out: MOD(23, 6), DIV(23, 6), 2 ^ 5, 10 - 4 / 2, (10 - 4) / 2. Then check them in Python.

🟡 Medium

Goal: Input two numbers. Output a message saying whether they are equal or not equal, and which is larger. Use <> in pseudocode and != in Python.

🔴 Stretch

Goal: A year is a leap year if it divides exactly by 4 and not by 100 — unless it also divides exactly by 400. Write a program that inputs a year and outputs whether it is a leap year. Test with 2024, 1900 and 2000.

Hint

"Divides exactly by 4" is MOD(Year, 4) = 0. The whole rule fits in one condition: (MOD(Year, 4) = 0 AND MOD(Year, 100) <> 0) OR MOD(Year, 400) = 0. Use brackets to make the grouping clear.

📝 Exam Practice 10 min

State[3]

State the value of each variable after these statements run.

Value1 ← MOD(20, 6)
Value2 ← DIV(20, 6)
Value3 ← 4 + 3 ^ 2

Mark scheme

Value1 = 2 (1)
Value2 = 3 (1)
Value3 = 13 (1)

Explain[4]

Explain the difference between a logical operator and a Boolean operator. Give an example of each.

Mark scheme

A logical operator compares two values (1)…
…e.g. >, <>, <= (1).
A Boolean operator combines conditions / TRUE–FALSE values into a more complex expression (1)…
…e.g. AND, OR, NOT (1).

Write[4]

Write pseudocode that inputs a whole number and outputs "Even" or "Odd", and also outputs "Multiple of 5" if the number divides exactly by 5.

Mark scheme

Prompt and input of the number (1).
IF MOD(Number, 2) = 0 (1).
Correct Even / Odd outputs with ELSE and ENDIF (1).
Separate IF MOD(Number, 5) = 0 with its output (1).

Identify[2]

A student writes IF Mark > 40 AND < 60. Identify the error and write the correct condition.

Mark scheme

The variable is missing from the second comparison (1).
IF Mark > 40 AND Mark < 60 (1).

🗝️ Recap & Key Terms 3 min

Arithmetic operators calculate — remember MOD (remainder) and DIV (whole part). Logical operators compare and give TRUE/FALSE. Boolean operators — AND, OR, NOT — join comparisons. Brackets decide the order.

Operator: A special character or word in a programming language that identifies an action to be performed.
Arithmetic operator: An operator that is used to perform calculations.
Logical operator: An operator used to decide the path through a program by comparing values to give TRUE or FALSE.
Boolean operator: An operator (AND, OR, NOT) used with logical operators to form more complex expressions.
MOD: Returns the remainder of a division.
DIV: Returns the quotient — the whole-number part — of a division.

Homework 1 min

Task (≤ 15 min): Write pseudocode that inputs a number of seconds and outputs it as hours, minutes and seconds (e.g. 3725 → 1 h 2 min 5 s), using only DIV and MOD. [4]

Model answer

DECLARE TotalSeconds, Hours, Minutes, Seconds : INTEGER
OUTPUT "Enter a number of seconds: "
INPUT TotalSeconds
Hours ← DIV(TotalSeconds, 3600)
Minutes ← DIV(MOD(TotalSeconds, 3600), 60)
Seconds ← MOD(TotalSeconds, 60)
OUTPUT Hours, " h ", Minutes, " min ", Seconds, " s"

Marks: input (1); hours with DIV 3600 (1); minutes from the remainder (1); seconds with MOD 60 and output (1). Check: 3725 DIV 3600 = 1; 125 DIV 60 = 2; 3725 MOD 60 = 5.

🎯 Syllabus & Goals 3 min

Cambridge 8.1.4 · Arithmetic, logical and Boolean operators Paper 2 · Algorithms, Programming and Logic

By the end of this lesson you can:

Evaluate expressions using + - * / ^, MOD and DIV, with brackets, and write them in Python.
Use the logical (comparison) operators = <> < > <= >= and know their Python forms.
Combine conditions with the Boolean operators AND, OR and NOT, and predict the result.

Textbook: Chapter 8, §8.1.4(f) (pp. 317–318); MOD and DIV also §8.1.7 (pp. 327–328)

Recap / Warm-Up 5 min

Last lesson: totals, counts and strings. Every one of them relied on an operator — + to total, = to compare a character, & to join strings.

Quick starter

29 students are put into teams of 4. How many full teams are there, and how many students are left over?

Reveal the answer

7 full teams (4 × 7 = 28) with 1 student left over. You just did DIV (7) and MOD (1) in your head.

🧠 Key Concept 14 min

A condition such as Total / Count >= 50 AND NOT Absent uses all three families at once.

1 · Arithmetic operators

Pseudocode	Action	Python	Example → result
`+` `-` `*`	Add, subtract, multiply	`+` `-` `*`	`17 * 5` → 85
`/`	Divide (result is REAL)	`/`	`17 / 5` → 3.4
`^`	Raise to the power of	`**`	`2 ^ 3` → 8
`MOD(17, 5)`	Remainder of a division	`17 % 5`	2
`DIV(17, 5)`	Whole-number part (quotient) of a division	`17 // 5`	3

Seventeen counters shared into groups of five. DIV counts the full groups; MOD counts what is left over.

2 · Logical operators

Logical operators compare two values. The result is always TRUE or FALSE, so they are used to decide the path through a program.

Pseudocode	Meaning	Python
`>` / `<`	Greater than / less than	`>` / `<`
`>=` / `<=`	Greater than or equal / less than or equal	`>=` / `<=`
`=`	Equal	`==`
`<>`	Not equal	`!=`

3 · Boolean operators

Boolean operators join comparisons into one more complex condition.

AND needs both sides TRUE; OR needs at least one; NOT flips the value. Python writes them and, or, not.

Beyond the syllabus (not examined): with negative numbers, Python's // and % round towards minus infinity, so -7 // 2 is -4. IGCSE questions use positive values.

Worked Example 12 min

Example 1 · Arithmetic — evaluating expressions and splitting teams

First, evaluate each expression by hand, then check with Python.

print(17 + 5, 17 - 5, 17 * 5)
print(17 / 5)       # divide: always gives a REAL (float)
print(2 ** 3)       # power (^ in pseudocode)
print(17 % 5)       # MOD: remainder
print(17 // 5)      # DIV: whole-number part
print(3 + 4 * 2)    # * before +
print((3 + 4) * 2)  # brackets first

Now a real task: a teacher splits a class into teams of a chosen size. Output the number of full teams and how many students are left over.

Cambridge pseudocode

DECLARE Students, TeamSize : INTEGER
DECLARE Teams, LeftOver : INTEGER
OUTPUT "Number of students: "
INPUT Students
OUTPUT "Students per team: "
INPUT TeamSize
Teams ← DIV(Students, TeamSize)
LeftOver ← MOD(Students, TeamSize)
OUTPUT "Full teams: ", Teams
OUTPUT "Students left over: ", LeftOver
IF LeftOver = 0
  THEN
    OUTPUT "Everyone is in a full team"
ENDIF

The same in Python

# Split a class into teams
students = int(input("Number of students: "))
team_size = int(input("Students per team: "))
teams = students // team_size          # DIV
left_over = students % team_size       # MOD
print("Full teams:", teams)
print("Students left over:", left_over)
if left_over == 0:
    print("Everyone is in a full team")

Number of students: 29
Students per team: 4
Full teams: 7
Students left over: 1

Number of students: 24
Students per team: 4
Full teams: 6
Students left over: 0
Everyone is in a full team

DIV(29, 4): 4 goes into 29 seven whole times → 7.DIV throws away the remainder.
MOD(29, 4): 29 − 7 × 4 = 1 → 1.MOD keeps only the remainder.
LeftOver = 0 tests for an exact split.MOD(X, Y) = 0 means "X divides exactly by Y".

Example 2 · Logical and Boolean — a youth-club entry check

A youth club admits people aged 13 to 17 inclusive who are not banned.

Cambridge pseudocode

DECLARE Age : INTEGER
DECLARE Answer : CHAR
DECLARE Banned : BOOLEAN
OUTPUT "Age: "
INPUT Age
OUTPUT "Banned (Y/N)? "
INPUT Answer
Banned ← Answer = 'Y'
IF Age >= 13 AND Age <= 17 AND NOT Banned
  THEN
    OUTPUT "Welcome in"
  ELSE
    OUTPUT "Entry refused"
ENDIF

The same in Python

# Youth club entry: aged 13 to 17 inclusive AND not banned
age = int(input("Age: "))
banned = input("Banned (Y/N)? ") == "Y"
if age >= 13 and age <= 17 and not banned:
    print("Welcome in")
else:
    print("Entry refused")

Banned ← Answer = 'Y' stores the result of a comparison — TRUE or FALSE — in a BOOLEAN variable.

Age	Answer	Age >= 13	Age <= 17	NOT Banned	Whole condition	OUTPUT
15	N	TRUE	TRUE	TRUE	TRUE	Welcome in
15	Y	TRUE	TRUE	FALSE	FALSE	Entry refused
18	N	TRUE	FALSE	TRUE	FALSE	Entry refused

Output — the three test runs

Age: 15
Banned (Y/N)? N
Welcome in

Age: 15
Banned (Y/N)? Y
Entry refused

Age: 18
Banned (Y/N)? N
Entry refused

Try It Yourself 12 min

🟢 Easy

Goal: Without a computer, work out: MOD(23, 6), DIV(23, 6), 2 ^ 5, 10 - 4 / 2, (10 - 4) / 2. Then check them in Python.

🟡 Medium

Goal: Input two numbers. Output a message saying whether they are equal or not equal, and which is larger. Use <> in pseudocode and != in Python.

🔴 Stretch

Hint

"Divides exactly by 4" is MOD(Year, 4) = 0. The whole rule fits in one condition: (MOD(Year, 4) = 0 AND MOD(Year, 100) <> 0) OR MOD(Year, 400) = 0. Use brackets to make the grouping clear.

📝 Exam Practice 10 min

State[3]

State the value of each variable after these statements run.

Value1 ← MOD(20, 6)
Value2 ← DIV(20, 6)
Value3 ← 4 + 3 ^ 2

Mark scheme

Value1 = 2 (1)
Value2 = 3 (1)
Value3 = 13 (1)

Explain[4]

Explain the difference between a logical operator and a Boolean operator. Give an example of each.

Mark scheme

A logical operator compares two values (1)…
…e.g. >, <>, <= (1).
A Boolean operator combines conditions / TRUE–FALSE values into a more complex expression (1)…
…e.g. AND, OR, NOT (1).

Write[4]

Write pseudocode that inputs a whole number and outputs "Even" or "Odd", and also outputs "Multiple of 5" if the number divides exactly by 5.

Mark scheme

Prompt and input of the number (1).
IF MOD(Number, 2) = 0 (1).
Correct Even / Odd outputs with ELSE and ENDIF (1).
Separate IF MOD(Number, 5) = 0 with its output (1).

Identify[2]

A student writes IF Mark > 40 AND < 60. Identify the error and write the correct condition.

Mark scheme

The variable is missing from the second comparison (1).
IF Mark > 40 AND Mark < 60 (1).

🗝️ Recap & Key Terms 3 min

Operator: A special character or word in a programming language that identifies an action to be performed.
Arithmetic operator: An operator that is used to perform calculations.
Logical operator: An operator used to decide the path through a program by comparing values to give TRUE or FALSE.
Boolean operator: An operator (AND, OR, NOT) used with logical operators to form more complex expressions.
MOD: Returns the remainder of a division.
DIV: Returns the quotient — the whole-number part — of a division.

Homework 1 min

Task (≤ 15 min): Write pseudocode that inputs a number of seconds and outputs it as hours, minutes and seconds (e.g. 3725 → 1 h 2 min 5 s), using only DIV and MOD. [4]

Model answer

DECLARE TotalSeconds, Hours, Minutes, Seconds : INTEGER
OUTPUT "Enter a number of seconds: "
INPUT TotalSeconds
Hours ← DIV(TotalSeconds, 3600)
Minutes ← DIV(MOD(TotalSeconds, 3600), 60)
Seconds ← MOD(TotalSeconds, 60)
OUTPUT Hours, " h ", Minutes, " min ", Seconds, " s"

Marks: input (1); hours with DIV 3600 (1); minutes from the remainder (1); seconds with MOD 60 and output (1). Check: 3725 DIV 3600 = 1; 125 DIV 60 = 2; 3725 MOD 60 = 5.