Properties

Properties of Integer Operations

What you'll learn

Name and use the closure, commutative, associative, and distributive properties for integers.
Recognise additive identity (0) and multiplicative identity (1).
Use the additive inverse of a number to simplify expressions and verify answers.
Decide which property justifies a given step in a calculation.

Key concepts

Closure property
- Addition: The sum of any two integers is always an integer.
- Multiplication: The product of any two integers is always an integer.
- Subtraction & division: Not always closed (e.g. 3 − 8 = −5 is an integer, but 7 ÷ 2 is not).
Commutative property — Order can be swapped.
- Addition: a + b = b + a (e.g. −3 + 7 = 7 + (−3) = 4).
- Multiplication: a × b = b × a (e.g. (−4) × 5 = 5 × (−4) = −20).
- Subtraction and division are not commutative: 5 − 3 ≠ 3 − 5.
Associative property — Grouping can be changed without changing the result.
- Addition: (a + b) + c = a + (b + c).
- Multiplication: (a × b) × c = a × (b × c).
- Subtraction and division are not associative.
Distributive property — Multiplication distributes over addition and subtraction:
- a × (b + c) = a × b + a × c
- a × (b − c) = a × b − a × c
  Example: −3 × (4 + (−2)) = (−3 × 4) + (−3 × (−2)) = −12 + 6 = −6.
Identities and inverses
- Additive identity: a + 0 = a. Zero added to any integer leaves it unchanged.
- Multiplicative identity: a × 1 = a.
- Additive inverse: For every integer a, there exists −a such that a + (−a) = 0.
  Example: the additive inverse of −8 is 8 because (−8) + 8 = 0.

Worked example

Use the distributive property to find −5 × 37 + (−5) × 63.

Step 1 — notice the common factor −5 in both terms
Step 2 — apply distributive property in reverse (factoring):
         −5 × 37 + (−5) × 63 = (−5) × (37 + 63)
Step 3 — add inside brackets: 37 + 63 = 100
Step 4 — multiply: (−5) × 100 = −500
Answer: −500

Which property is shown by (−2) × 7 = 7 × (−2)?

Both sides equal −14.
The order of the factors was swapped without changing the product.
Property used: Commutative property of multiplication.

Common mistakes

Misconception	What students think	Scientific correction
Confusing additive inverse with **multiplicative in	Confusing additive inverse with multiplicative inverse (reciprocal). In Class 7 integers, we use additive inverse only.	Check the Key concepts and worked example for the NCERT-accurate version.
Applying distributive property incorrectly: writing **a	Applying distributive property incorrectly: writing a × (b + c) = a × b + c (the c must also be multiplied by a).	Check the Key concepts and worked example for the NCERT-accurate version.
Forgetting that a + (−a) = 0 works for negative a a	Forgetting that a + (−a) = 0 works for negative a as well: (−5) + 5 = 0.	Check the Key concepts and worked example for the NCERT-accurate version.

Quick check

Name the property: (−4) + 9 = 9 + (−4). (Commutative property of addition)
What is the additive inverse of −12? (12)
Simplify using distributive property: 6 × 98 + 6 × 2. (6 × 100 = 600)
Is integer division always closed? Give a counter-example if not. (No — e.g. 5 ÷ 2 is not an integer)

Open the Practice tab for graded questions on properties of integers.

Properties

Properties

Properties of Integer Operations

What you'll learn

Key concepts

Worked example

Common mistakes

Quick check

Key Takeaways (TL;DR)

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