Integers

Edexcel International GCSE Mathematics A· 1.1· 20 min read

1. Core Integer Definitions and Ordering★☆☆☆☆⏱ 5 min

📘 Definition

Integer

Whole numbers including positive values, negative values and zero, with no fractional or decimal components.

Example:

$- 5, 0, 3, 172$ are integers; $0.6, \frac{1}{2}, - 3.2$ are not integers.

Integers follow standard place value rules, where each digit's value depends on its position (units, tens, hundreds, thousands etc.). When ordering integers: negative numbers are smaller the further they are from zero, while positive numbers are larger the further they are from zero.

📐 Worked Example

Order the following integers from smallest to largest: $7, - 2, 0, - 9, 3$

1
1. Separate values into negative integers, zero and positive integers: negatives = $[- 2, - 9]$ , zero = $0$ , positives = $[7, 3]$
2
1. Order negative values from smallest to largest: more negative values come first, so $- 9 < - 2$
3
1. Order positive values from smallest to largest: $3 < 7$
4
1. Combine the lists in order: $- 9, - 2, 0, 3, 7$

Exam tip:

Double-check the order of negative integers: a very common error is writing $- 2$ before $- 9$ when sorting from smallest to largest.

2. Operations with Directed Integers★★☆☆☆⏱ 5 min

📘 Definition

Directed Number

Numbers with a positive or negative sign that indicate direction relative to zero, often used in practical contexts like temperatures, elevation or bank balances.

Adding a negative number = subtracting a positive number: $3 + (- 2) = 3 - 2 = 1$
Subtracting a negative number = adding a positive number: $5 - (- 3) = 5 + 3 = 8$
Multiplying/dividing two numbers with the same sign gives a positive result
Multiplying/dividing two numbers with different signs gives a negative result

📐 Worked Example

The temperature in Oslo at midnight is $- 4^{\circ} C$ . By midday, the temperature rises by $9^{\circ} C$ , then drops by $1 2^{\circ} C$ by 6pm. What is the temperature at 6pm?

1
1. Start with the initial midnight temperature: $- 4^{\circ} C$
2
1. Add the temperature rise at midday: $- 4 + 9 = 5^{\circ} C$
3
1. Subtract the temperature drop by 6pm: $5 - 12 = - 7^{\circ} C$
4
Final answer: $- 7^{\circ} C$

3. Order of Operations (BIDMAS)★★★☆☆⏱ 5 min

📘 Definition

BIDMAS

Standard hierarchy of operations for multi-step calculations: Brackets first, then Indices, then Division and Multiplication (equal priority, left to right), then Addition and Subtraction (equal priority, left to right).

📐 Worked Example

Calculate the value of $12 - 3 \times (4 + 2) \div 2$

1
1. First calculate the value inside the brackets: $4 + 2 = 6$
2
1. Simplify the expression to: $12 - 3 \times 6 \div 2$
3
1. Complete multiplication and division left to right: $3 \times 6 = 18$ , then $18 \div 2 = 9$
4
1. Complete the final subtraction: $12 - 9 = 3$
5
Final answer: $3$

Exam tip:

Never calculate left to right without following BIDMAS: this is one of the most common sources of avoidable mark loss in arithmetic questions.

4. Prime Numbers, Factors and Multiples★★☆☆☆⏱ 5 min

Even number: Integer divisible by 2 (ends in 0, 2, 4, 6, 8)
Odd number: Integer not divisible by 2 (ends in 1, 3, 5, 7, 9)
Prime number: Positive integer with exactly two distinct positive factors (1 and itself; 1 is not prime)
Factor: Integer that divides another integer exactly with no remainder
Multiple: Product of an integer and any positive whole number

📐 Worked Example

List the prime factors, common factors and first 3 common multiples of 12 and 18.

1
1. Prime factors of 12: 2, 3; Prime factors of 18: 2, 3
2
1. List all factors of 12: 1, 2, 3, 4, 6, 12; All factors of 18: 1, 2, 3, 6, 9, 18
3
1. Common factors are values in both lists: 1, 2, 3, 6
4
1. List multiples of 12: 12, 24, 36, 48, 60, 72...; Multiples of 18: 18, 36, 54, 72, 90...
5
1. First 3 common multiples: 36, 72, 108

✓ Quick check

Which of the following is a prime number?
- A) 1
- B) 9
- C) 13
- D) 21
Reveal answer
C —
13 only has two factors: 1 and 13. 1 has only one factor, 9 has factors 1, 3, 9, and 21 has factors 1, 3, 7, 21.

5. Common Pitfalls

Wrong move:

Ordering integers from smallest to largest as $- 2, - 9, 0, 3, 7$

Why:

Assuming larger digit values mean larger numbers for negative integers

Correct move:

More negative numbers are smaller, so the correct order is $- 9, - 2, 0, 3, 7$

Wrong move:

Calculating $5 - (- 3) = 2$

Why:

Treating subtraction of a negative as subtraction of a positive

Correct move:

Subtracting a negative equals adding a positive, so $5 + 3 = 8$

Wrong move:

Classifying 1 as a prime number

Why:

Misremembering the prime number definition as 'number only divisible by 1 and itself' without the 'exactly two distinct factors' requirement

Correct move:

1 only has one factor (itself), so it is not a prime number

Wrong move:

Calculating $2 + 3 \times 4 = 20$ by working left to right

Why:

Ignoring BIDMAS order of operations

Correct move:

Complete multiplication first: $3 \times 4 = 12$ , then add 2 to get 14

Wrong move:

Calculating $(- 4) \times (- 5) = - 20$

Why:

Misapplying sign rules for multiplication of directed numbers

Correct move:

Two negative numbers multiplied give a positive result, so the answer is 20

6. Quick Reference Cheatsheet

Concept	Rule/Definition
Integer	Whole number (positive, negative, zero; no fractions/decimals)
×/÷ Sign Rules	Same sign = positive, Different sign = negative
BIDMAS Order	Brackets > Indices > ×/÷ (left-right) > +/− (left-right)
Prime Number	Exactly 2 factors: 1 and itself; 1 is not prime
Factor	Divides a number exactly with no remainder
Multiple	Product of a number and any positive integer

7. Frequently Asked

Is 1 a prime number?

No. Prime numbers are defined as positive integers with exactly two distinct positive factors: 1 and themselves. 1 only has one factor (itself), so it is not classified as prime.

What is the result of multiplying two negative integers?

Multiplying two negative integers always gives a positive result, e.g. $(- 3) \times (- 4) = 12$ . The same rule applies to division of two negative integers.

Do I need to follow BIDMAS even if there are no brackets in a calculation?

Yes. Always complete multiplication and division (left to right) before addition and subtraction (left to right), even if no brackets are present.

What's Next

Integers are the foundation of all Edexcel IGCSE Maths A content, so mastering these rules is critical for success in every other topic. You can now move on to learning about powers and roots, where you will apply BIDMAS and prime factor knowledge to solve more complex arithmetic problems, as specified in syllabus point 1.4. You can also practice integer calculation questions to build speed and accuracy for your exam, and review other number system topics such as fractions and decimals to strengthen your core arithmetic skills.