Basic Properties of Numbers Explained — Commutative, Associative, Distributive, Identity & Inverse (JAMB/WAEC)

The basic properties of numbers are a set of fundamental rules that describe how numbers behave under addition and multiplication, forming the foundation for nearly all algebraic manipulation.

This topic — covering the commutative, associative, distributive, identity, and inverse properties — is foundational in JAMB and WAEC Mathematics, since these rules are used, often without being explicitly named, throughout almost every other topic. This lesson explains each property clearly with examples.

Quick takeaways

• The commutative property means the order of numbers doesn’t affect the result of addition or multiplication (a + b = b + a).
• The associative property means the grouping of numbers doesn’t affect the result of addition or multiplication ((a + b) + c = a + (b + c)).
• The distributive property allows multiplication to be “distributed” over addition or subtraction: a(b + c) = ab + ac.
• The identity property states that adding 0 or multiplying by 1 leaves a number unchanged.
• The inverse property states that every number has an additive inverse (which sums to 0) and every non-zero number has a multiplicative inverse (which multiplies to 1).

Timestamps

• 0:00 Introduction
• 0:35 Overview of Number Properties (Commutative Property)
• 4:15 Associative Property
• 6:23 Distributive Property
• 8:09 Identity Property
• 10:15 Inverse Property

What is the commutative property?

The commutative property states that changing the order of numbers being added or multiplied does not change the result:

a + b = b + a and a × b = b × a

For example, 4 + 7 = 7 + 4 = 11, and 3 × 5 = 5 × 3 = 15. Importantly, the commutative property does not apply to subtraction or division — 5 − 3 does not equal 3 − 5, which is a common point of confusion.

What is the associative property?

The associative property states that changing how numbers are grouped (using brackets) during addition or multiplication does not change the result:

(a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)

For example, (2 + 3) + 4 = 2 + (3 + 4) = 9. Like the commutative property, the associative property does not generally hold for subtraction or division.

What is the distributive property?

The distributive property connects multiplication and addition, allowing a number outside brackets to be “distributed” across terms inside the brackets:

a(b + c) = ab + ac

For example, 3(4 + 5) = 3(4) + 3(5) = 12 + 15 = 27, which matches 3 × 9 = 27. This property is especially important in algebra, since it’s the basis for expanding brackets in algebraic expressions.

What is the identity property?

The identity property describes two special numbers that leave any number unchanged under a specific operation:

• Additive identity — adding 0 to any number leaves it unchanged: a + 0 = a
• Multiplicative identity — multiplying any number by 1 leaves it unchanged: a × 1 = a

What is the inverse property?

The inverse property describes numbers that “undo” another number under a specific operation, returning the identity value:

• Additive inverse — every number has an additive inverse, such that their sum is 0 (e.g., the additive inverse of 5 is −5, since 5 + (−5) = 0)
• Multiplicative inverse — every non-zero number has a multiplicative inverse (its reciprocal), such that their product is 1 (e.g., the multiplicative inverse of 4 is 1/4, since 4 × 1/4 = 1)

Note that zero has no multiplicative inverse, since no number multiplied by zero can ever equal 1.

Common mistakes students make with properties of numbers

• Assuming the commutative or associative properties apply to subtraction or division, when they only apply to addition and multiplication
• Confusing the additive inverse (which sums to 0) with the multiplicative inverse (which multiplies to 1)
• Misapplying the distributive property, such as forgetting to distribute a number to every term inside the brackets
• Mixing up the identity property (a number that leaves values unchanged) with the inverse property (a number that “undoes” another back to the identity)

Frequently asked questions

Q: Does the commutative property apply to subtraction? No, the commutative property only applies to addition and multiplication — changing the order of numbers in subtraction or division changes the result.

Q: What is the difference between the identity property and the inverse property? The identity property describes a number (0 for addition, 1 for multiplication) that leaves any value unchanged, while the inverse property describes a number that combines with another to produce that identity value.

Q: What is the distributive property used for? The distributive property allows multiplication to be spread across terms inside brackets, such as a(b + c) = ab + ac, and it’s the foundation for expanding algebraic expressions.

Q: Does every number have a multiplicative inverse? No, every non-zero number has a multiplicative inverse (its reciprocal), but zero does not, since no number multiplied by zero can produce 1.

Want to see these properties applied to formula rearrangement? Read our lesson on changing the subject of a formula →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →