Understanding the basic number properties is fundamental in mathematics. These properties form the foundation for algebra, geometry, and higher-level math. They help simplify calculations, solve equations, and comprehend mathematical relationships.
The basic number properties are the commutative, associative, distributive, identity, inverse, and zero properties.
Addition
The order in which you add numbers does not change the sum.
a + b = b + a
Example: 3 + 5 = 5 + 3 = 8
Multiplication
The order in which you multiply numbers does not change the product.
a × b = b × a
Example: 4 × 7 = 7 × 4 = 28
Using the commutative property with subtraction or division. Order matters there: 5 − 3 = 2, but 3 − 5 = −2. And 8 ÷ 2 = 4, but 2 ÷ 8 = 0.25.
Which property does 6 × 9 = 9 × 6 show?
The commutative property of multiplication. The two numbers changed places, and the product stayed 54.
More examples: Commutative Property
Addition
When adding three or more numbers, the way you group them does not change the sum.
(a + b) + c = a + (b + c)
Example: (2 + 3) + 4 = 2 + (3 + 4) = 9
Multiplication
When multiplying three or more numbers, the way you group them does not change the product.
(a × b) × c = a × (b × c)
Example: (2 × 3) × 4 = 2 × (3 × 4) = 24
Commutative changes the order. Associative changes the grouping, where the parentheses go. The numbers stay in the same order.
Regrouping a subtraction or a division. (8 − 4) − 2 = 2, but 8 − (4 − 2) = 6. And (8 ÷ 4) ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 4.
Use the associative property to find 17 + 25 + 75 quickly.
117. Group the last two numbers first: 17 + (25 + 75) = 17 + 100 = 117.
More examples: What is the Associative Property?
This property connects addition and multiplication, showing how multiplication distributes over addition.
a × (b + c) = a × b + a × c
It works with subtraction too: a × (b − c) = a × b − a × c.
Example: 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27
6 × 98 = 588
Multiplying only the first number in the parentheses. 3 × (4 + 5) is not 3 × 4 + 5 = 17. The 3 multiplies both numbers: 3 × 4 + 3 × 5 = 27.
Fill in the blank: 7 × (10 + 3) = 7 × 10 + 7 × ___
3. The 7 multiplies both numbers inside the parentheses: 70 + 21 = 91.
More examples: Distributive Property
Addition (additive identity)
Adding zero to any number does not change the number.
a + 0 = a
Example: 7 + 0 = 7
Multiplication (multiplicative identity)
Multiplying any number by one does not change the number.
a × 1 = a
Example: 9 × 1 = 9
Mixing up 0 and 1. For addition, the number that changes nothing is 0. For multiplication, it is 1, because 9 × 0 = 0, not 9.
What number can you multiply by 254 without changing it?
1. 254 × 1 = 254. That is the identity property of multiplication.
More examples: Identity Property of Multiplication
An inverse undoes a number. Adding the inverse gets you back to 0, and multiplying by the inverse gets you back to 1.
Addition (additive inverse)
A number plus its opposite is 0.
a + (−a) = 0
Example: 8 + (−8) = 0
Multiplication (multiplicative inverse)
A number times its reciprocal is 1. This works for every number except 0.
a × 1a = 1 (a ≠ 0)
Example: 5 × 15 = 1
To find the reciprocal of a fraction, flip it: the reciprocal of 23 is 32, because 23 × 32 = 1.
0 has no reciprocal: no number times 0 gives 1.
Mixing up the opposite and the reciprocal. The opposite of 4 is −4, because 4 + (−4) = 0. The reciprocal of 4 is 14, because 4 × 14 = 1.
What number do you add to −12 to get 0?
12. −12 + 12 = 0. 12 is the additive inverse, or opposite, of −12.
See all the properties in one table: Properties of Real Numbers
Multiplying any number by zero gives zero.
a × 0 = 0
Example: 3,457 × 0 = 0
Think of it as groups: 3,457 groups of nothing is still nothing.
0 divided by a number (other than 0) is 0: 0 ÷ 5 = 0. But you can't divide by 0: 5 ÷ 0 has no answer, because no number times 0 gives 5.
What is 48 × 17 × 0 × 25?
0. As soon as one factor is 0, the whole product is 0. No need to multiply the other numbers.
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Distributive | a × (b + c) = a × b + a × c | |
| Identity | a + 0 = a | a × 1 = a |
| Inverse | a + (−a) = 0 | a × 1a = 1 (a ≠ 0) |
| Zero | (none) | a × 0 = 0 |
Now check what you learned. Every answer is explained, and each time you play you get new questions.