What Is the Commutative Property?
The commutative property is one of the fundamental properties of multiplication in mathematics. It states that:
“The order of factors in a multiplication problem does not change the product (the answer).”
In simpler terms, you can multiply numbers in any order and still get the same result.
Mathematical Definition:
For any two numbers a and b: a × b = b × a
Visualizing the Commutative Property
Array Model Demonstration
Example: 4 × 5 vs. 5 × 4
- 4 rows of 5 items each:
• • • • • • • • • • • • • • • • • • • •
- Total: 20 items
- 5 rows of 4 items each:
• • • • • • • • • • • • • • • • • • • •
- Total: 20 items
Same total number of items, different arrangement!
Associative Property
In mathematics, multiplication (and addition) is based on the associative property – that is, combining three or more whole numbers to get the same total product, regardless of how you group those numbers when you multiply them together. Therefore:
*When multiplying three or more numbers together, how you group them has no effect on the end product of the three or more numbers.*
As an Example:
You have three whole numbers, i.e., a, b, c. When you multiply a by b and then multiply the result by c, you have: (a × b) × c = a × (b × c). Since this is the basis of how the associative property works, you can also reorganize whole numbers and still arrive at the same answer. Array model example:
Example 1: (2 × 3) × 4 First, 2 rows of 3 = 6 Then, 6 × 4 = 24
Example 2: 2 × (3 × 4)
First, 3 rows of 4 = 12 Then, 2 × 12 = 24
Thus, although grouping 2 and 3 and multiplying them together before multiplying the result with 4 and grouping 3 rows of 4 by 2 and multiplying the two together results in the exact same total 24 unit cube, it’s done using different groupings.
Identity Property
In mathematics, multiplication has several important properties. One of these is called the identity property of multiplication and it states: “When you multiply a number by 1, the result does not change.”
The number 1 is often called the multiplicative identity because it retains the identity of any number that is multiplied by it.
Mathematically speaking, this means:
For any number a:
(a * 1 = a)
and
(1 * a = a).
Examples of the multiplicative identity property: ** (Multiplying by One). 1. Example: **Cookie Portions * 1 * 5 cookies = 5 cookies (1 portion of 5 cookies) * 5 * 1 cookie = 5 cookies (5 single cookies)
2. Example: **Classroom Desks * 1 row of desks = 8 desks * 8 rows of desks = 8 desks
To demonstrate the identity property on a number line:
* 5 * 1 = 5 (After jumping 1 unit 5 times, you end up at 5). * 1 * 5 = 5 (After jumping 5 units 1 time you end up at 5).
Zero Property
One of the most fundamental rules of mathematics is the zero property of multiplication: “the product of any number multiplied by zero will always produce a total of zero.” This property also represents an absolute rule, true for all numbers and in every situation involving mathematics.
Mathematically speaking, if we take a number ‘a’ (the number being multiplied):
a multiplied by 0 = 0
The reverse is also true:
0 multiplied by a = 0.
Some examples of times when this rule applies include:
For cookie jars:
5 x 0 = 0 (five empty jars) 0 x 3 = 0 (zero jars with three cookies in each jar).
For classroom chairs:
0 x 6 = 0 (zero rows of six chairs) 4 x 0 = 0 (four rows of zero chairs).
On the number line, if you made five jumps of zero, you would still end at zero; likewise, if you were to make zero jumps of five, you would still be at zero.
Distributive Property
The distributive property is an important tool in mathematics relating multiplication to addition. The distributive property states:
“Multiplying a number by a group of numbers (sum) is the same as multiplying that number by each number (addend) separately and then adding those products together.”
Definition: For any numbers a, b, and c: a x (b + c) = (a x b) + (a x c)
The Distributive Property will work in both directions – hence both “distributing” or “factoring”.
Area Model Examples Example of 3 x (4 + 2)
1. Combined Area Model: * Total Distance = 4 + 2 = 6 * Area = 3 x 6 = 18 2. Separate Rectangle Models: * Left Rectangle Area = 3 x 4 = 12 * Right Rectangle Area = 3 x 2 = 6 * Area = 12 + 6 = 18
+-----+----+ | 3×4 |3×2| → 3×(4+2) = 3×4 + 3×2 +-----+----+
Real-World Applications 1. Pizza Order Example: 4 x (2 slices + 3 slices) = (4 x 2) + (4 x 3) = 8 + 12 = 20 2. Sales Tax Example: 5 x (2 tax) = (5 x 2) = 10 = $60 3. Gardening Example: 7 x (3 flowers + 5 herbs) = 21 flowers + 35 herbs = 56 plants
Frequently asked questions
What are the four properties of multiplication?
The four properties of multiplication are commutative, associative, distributive, and identity. Commutative says you can swap the order of two factors. Associative says you can re-group three or more factors. Distributive says multiplication "distributes" over a sum inside parentheses. Identity says multiplying by 1 leaves any number unchanged. These four rules are the foundation for the standard multi-digit multiplication algorithm, the way long division undoes multiplication, and most of beginning algebra.
Why does the distributive property matter?
The distributive property is the rule that makes long multiplication work. To compute 7 × 23, you mentally rewrite it as 7 × (20 + 3) = (7 × 20) + (7 × 3) = 140 + 21 = 161 — that is the standard column algorithm, written out as a single line of distribution. The same property is the foundation for expanding algebraic expressions like 3(x + 4) = 3x + 12. Of the four properties, the distributive one is the one to spend the most time practicing.
Is the commutative property true for division too?
No. Division is not commutative — 12 ÷ 3 is 4, but 3 ÷ 12 is 0.25. The order matters. Subtraction is also not commutative: 5 − 3 is 2, but 3 − 5 is −2. Only addition and multiplication are commutative for the numbers students meet in elementary school. Knowing which operations are commutative and which are not is one of the small but important pieces of arithmetic literacy that prevents algebra mistakes later.
What does the identity property do?
The identity property of multiplication says that any number times 1 equals itself: 7 × 1 = 7, 3.4 × 1 = 3.4, x × 1 = x. The "1" is the multiplicative identity — it leaves whatever you multiply it by unchanged. The property looks trivial but is constantly useful in algebra: when you multiply a fraction by a clever form of 1 (like 3/3) to get a common denominator, you are using the identity property. When you cancel common factors, you are using it. The "do-nothing" rule turns out to do a lot of work.
Are these properties the same as the order of operations?
No — they are different rules. The order of operations (PEMDAS / BODMAS) tells you which operations to do FIRST when an expression has several different operations: parentheses, then exponents, then multiplication and division (left to right), then addition and subtraction (left to right). The properties of multiplication tell you how multiplication itself behaves — which moves are valid. The two work together: order of operations tells you the sequence; the properties tell you what each move actually does.