If you try to follow a recipe without knowing the basics of cooking, like when to add certain ingredients or how hot the oven should be, you might end up with a dish that doesn’t taste right.
Number properties in math are like the basic rules in cooking — once you understand them, you can avoid common mistakes, get correct results, and approach new math problems with more confidence.
In this guide, we’ll cover the four basic number properties in math: commutative, associative, identity, and distributive.
You’ll find simple definitions, clear examples, answers to common questions, and a fun quiz to test what you’ve learned.
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Number properties in math are rules that tell us how numbers work when we add, multiply, or combine them in different ways.
Number properties apply to all real numbers, including whole numbers, fractions, decimals, and negative numbers.
Just like cooking has its own order for adding ingredients, and football has rules for throwing, running plays, and scoring touchdowns, math has rules for how numbers interact when we add or multiply them.
There are four basic number properties:
Let’s take a closer look at each number property and see how they work!
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We’ve all heard the word commute before, right? Like when people commute to school or work, it means they move from one place to another.
Similarly, the commutative property in math is about moving numbers around.
The commutative property shows we can change the order of numbers when we add or multiply them and still get the same result.
Let’s see how this works for addition and multiplication.
In math, we express the commutative property of addition like so:
a + b = b + a
Here, a and b can be any numbers, like 3 and 8. For example, 3 + 8 can also be written as 8 + 3, and both equal 11.
We express the commutative property of multiplication like so:
a × b = b × a
Again, a and b can be any numbers, such as 4 and 6. In practice, 4 × 6 can also be written as 6 × 4, and they both equal 24.
If we start from the very word associate, we know it means to connect or group things together.
The associative property shows us that when we add or multiply numbers, we can group them with parentheses in any way we want, and we’ll still get the same result.
Now, let’s see how this works for addition and multiplication.
In math, we often show the associative property of addition like so:
(a + b) + c = a + (b + c)
For example, (1 + 7) + 4 can also be written as 1 + (7 + 4). In the first case, we add 1 and 7 to get 8, then add 4 for a total of 12. In the second, we add 7 and 4 first to get 11, then add 1, and we still get 12.
We can express the associative property of multiplication like so:
(a × b) × c = a × (b × c)
For example, (5 × 2) × 8 can also be written as 5 × (2 × 8). If we multiply 5 and 2 first, we get 10, and 10 × 8 equals 80. If we switch the grouping and multiply 2 and 8 first, we get 16, and 5 × 16 also gives us 80.
An identity in math is a number that, when used in an operation with another number, keeps that number’s value the same.
Zero is the identity element for addition (or additive identity) because when it is added to a number, it leaves the number unchanged — that is to say, it preserves the number's identity.
We can express that with the following formula:
a + 0 = a
For example:
5 + 0 = 5
One is the identity element for multiplication (or multiplicative identity) because when it multiplies another number, it leaves the number unchanged — it preserves the number's identity.
For example:
5 × 1 = 5
When we hear distribute, we often think of handing something out to everyone in a group, right? This gives us a good clue about what the distributive property is all about.
The distributive property is a rule in math that lets us multiply a number outside parentheses by each term inside the parentheses. In other words, we’re distributing, or sharing, that number with everything inside the parentheses.
We can express this through a simple formula:
a(b + c) = ab + ac
Let’s say we want to solve 5 × (5 + 2). What if we use the distributive property to break it up like this:
5 × 5 + 5 × 2 = 25 + 10 = 35
Did we still solve 5 × 7? Yes, just in a different way.
So why do it this way? Because splitting the numbers can make the math easier to handle, especially in your head.
That’s the beauty of the distributive property: it gives us more than one way to solve a problem, and sometimes, that new way is simpler and quicker.
The distributive property also works in reverse. We can think of this as undistributing where a is the common factor. This is also called factoring.
For example:
12a + 3b = 3(4a + b)
Now that we know what commutative, associative, identity, and distributive properties are, let's put them side by side and see how they compare.
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Although number properties are straightforward rules, learning about them often raises questions.
We’ve gathered some of the most common ones we hear at Mathnasium, along with answers to clear up any dilemmas.
Students usually start learning about the four main number properties in elementary school.
Yes, they are! The inverse property and the algebraic property of equality are number properties. However, they are usually introduced at a later stage when students start working with more advanced math concepts, like algebra.
Yes, they do! Decimal numbers follow the same commutative, associative, identity, and distributive properties as whole numbers. Let’s see how:
0.4 × 2.5 = 2.5 × 0.4 = 1.0
(0.2 + 0.3) + 0.5 = 0.2 + (0.3 + 0.5) = 1.0
0.75 × 1 = 0.75
0.5(2 + 4) = 0.5 × 2 + 0.5 × 4 = 1 + 2 = 3
No matter if you’re working with whole numbers, decimals, or even fractions, the same rules apply!
Yes, they can! Negative numbers follow the same commutative, associative, identity, and distributive properties as positive numbers. Here’s how:
-3 + (-5) = -5 + (-3) = -8
(-2 × 4) × 3 = -2 × (4 × 3) = -24
-7 + 0 = -7
-2(3 + 5) = -2 × 3 + (-2 × 5) = -6 + (-10) = -16
Not always. Number properties don’t work for subtraction and division in the same way they do for addition and multiplication. Let’s see why:
Subtraction and division are not commutative. Changing the order changes the result.
For example: 8 — 5 ≠ 5 — 8, and 12 ÷ 3 ≠ 3 ÷ 12
Subtraction and division are not associative. Changing the grouping gives a different result.
For example: (10 — 5) — 2 ≠ 10 — (5 — 2), and (16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷2).
For subtraction, 0 acts as the identity because subtracting 0 from any number leaves it unchanged. For example: 9 — 0 = 9
For division, 1 is the identity element, since dividing any number by 1 doesn’t change its value. For example: 9 ÷ 1 = 9.
The distributive property works with subtraction, but not with division.
For example: 3(10 — 4) = 3 × 10 — 3 × 4 = 30 — 12 = 18.
However, 10 ÷ (2 + 3) ≠ (10 ÷ 2) + (10 ÷ 3).
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