Properties of Numbers

You've been using certain math properties your whole life without even knowing they had names. When you add $3 + 5$ and get 8, and then add $5 + 3$ and still get 8, that's not a coincidence—it's the commutative property at work. These properties are the basic rules that govern how numbers behave, and understanding them will make algebra much easier.

Think of these properties as the "laws of mathematics." Just like traffic laws tell us how cars should behave on the road, mathematical properties tell us how numbers behave when we add, subtract, multiply, and divide them. Once you understand these rules, you'll be able to rearrange and simplify expressions with confidence.

The Commutative Property

The word "commutative" comes from "commute," like when you commute to school or work. The order of your trip doesn't matter—whether you drive from home to school or school to home, you're traveling the same distance. Numbers work the same way with addition and multiplication.

Commutative Property of Addition: The order in which you add numbers doesn't change the sum.

$$a + b = b + a$$

Example: $7 + 4 = 4 + 7$. Both equal 11.

Commutative Property of Multiplication: The order in which you multiply numbers doesn't change the product.

$$a \times b = b \times a$$

Example: $6 \times 3 = 3 \times 6$. Both equal 18.

Here's something important: subtraction and division are NOT commutative. Order matters for these operations.

Example: $10 - 3 = 7$, but $3 - 10 = -7$. Not the same!

Example: $12 \div 4 = 3$, but $4 \div 12 = \frac{1}{3}$. Definitely not the same!

So when you're working with expressions, remember that you can swap numbers around when adding or multiplying, but you need to be careful with subtraction and division.

The Associative Property

"Associate" means to group together, and that's exactly what this property is about—how we group numbers when we're adding or multiplying several of them.

Associative Property of Addition: When adding three or more numbers, the way you group them doesn't change the sum.

$$(a + b) + c = a + (b + c)$$

Example: Let's add $2 + 3 + 4$.

We could group it as $(2 + 3) + 4$: $$5 + 4 = 9$$

Or we could group it as $2 + (3 + 4)$: $$2 + 7 = 9$$

Either way, we get 9.

Associative Property of Multiplication: When multiplying three or more numbers, the way you group them doesn't change the product.

$$(a \times b) \times c = a \times (b \times c)$$

Example: Let's multiply $2 \times 5 \times 3$.

We could do $(2 \times 5) \times 3$: $$10 \times 3 = 30$$

Or we could do $2 \times (5 \times 3)$: $$2 \times 15 = 30$$

Same answer both ways.

Why does this matter? Sometimes grouping numbers differently makes the mental math easier. If you need to calculate $25 \times 17 \times 4$, it's much easier to do $25 \times 4$ first (which equals 100), then multiply by 17, rather than trying to multiply 25 by 17 first.

Just like the commutative property, the associative property does NOT work for subtraction or division.

The Distributive Property

This is probably the most important property you'll use in algebra. The distributive property connects multiplication and addition, and you'll use it constantly when simplifying expressions and solving equations.

Distributive Property: Multiplying a number by a sum is the same as multiplying the number by each term in the sum and then adding the results.

$$a(b + c) = ab + ac$$

Let's see this with actual numbers first.

Example: Calculate $3(4 + 5)$.

Method 1 - Add first: $$3(4 + 5) = 3(9) = 27$$

Method 2 - Distribute the 3: $$3(4 + 5) = 3 \times 4 + 3 \times 5 = 12 + 15 = 27$$

Both methods give us 27.

Now let's see why this is so useful in algebra.

Example: Simplify $4(x + 3)$.

Using the distributive property: $$4(x + 3) = 4x + 12$$

We distributed the 4 to both terms inside the parentheses.

Example: Simplify $-2(3x - 5)$.

Distribute the $-2$ to both terms: $$-2(3x - 5) = -2 \times 3x + (-2) \times (-5)$$ $$= -6x + 10$$

Notice how the negative sign distributed to both terms. This is a common place where students make mistakes, so be careful with those negative signs!

The distributive property also works in reverse. If you see $6x + 9$, you can factor out a 3: $$6x + 9 = 3(2x + 3)$$

This reverse process is called "factoring," and you'll use it a lot when solving more complex equations.

Identity Properties

Identity properties tell us about special numbers that don't change other numbers when we use certain operations.

Additive Identity: Adding zero to any number gives you the same number back.

$$a + 0 = a$$

Example: $7 + 0 = 7$

Zero is called the "additive identity" because adding it doesn't change a number's identity.

Multiplicative Identity: Multiplying any number by one gives you the same number back.

$$a \times 1 = a$$

Example: $15 \times 1 = 15$

One is called the "multiplicative identity" because multiplying by it doesn't change a number's identity.

These might seem obvious, but they're important when simplifying algebraic expressions. For instance, $x + 0 = x$ and $x \times 1 = x$, which helps us simplify expressions and solve equations.

Inverse Properties

Inverse properties deal with opposites—operations that undo each other.

Additive Inverse: Every number has an opposite (called its additive inverse) that, when added to the original number, gives zero.

$$a + (-a) = 0$$

Example: $8 + (-8) = 0$

The additive inverse of 8 is -8. The additive inverse of -3 is 3.

Multiplicative Inverse: Every number (except zero) has a reciprocal (called its multiplicative inverse) that, when multiplied by the original number, gives one.

$$a \times \frac{1}{a} = 1 \text{ where } a \neq 0$$

Example: $5 \times \frac{1}{5} = 1$

The multiplicative inverse of 5 is $\frac{1}{5}$. The multiplicative inverse of $\frac{2}{3}$ is $\frac{3}{2}$.

Why can't zero have a multiplicative inverse? Because $\frac{1}{0}$ is undefined—you can't divide by zero!

Putting It All Together

Let's work through a problem that uses several properties.

Example: Simplify $3(x + 4) + 2x$.

Step 1 - Use the distributive property: $$3(x + 4) + 2x = 3x + 12 + 2x$$

Step 2 - Use the commutative property to rearrange: $$= 3x + 2x + 12$$

Step 3 - Combine like terms: $$= 5x + 12$$

Here's another one.

Example: Simplify $5(2x - 3) - 4(x + 1)$.

Step 1 - Distribute both constants: $$5(2x - 3) - 4(x + 1) = 10x - 15 - 4x - 4$$

Step 2 - Rearrange using commutative property: $$= 10x - 4x - 15 - 4$$

Step 3 - Combine like terms: $$= 6x - 19$$

Practice Problems

Try these on your own, then check your answers below.

Does $9 + 6 = 6 + 9$? Which property does this demonstrate?
Simplify using the distributive property: $7(x + 2)$
What is the multiplicative inverse of $\frac{4}{5}$?
Simplify: $2(3x + 1) + 4x$
Is $(8 - 3) - 2$ the same as $8 - (3 - 2)$?
Simplify: $-3(2x - 4) + 5x$

Solutions

Yes, both equal 15. This demonstrates the commutative property of addition.
$7(x + 2) = 7x + 14$
The multiplicative inverse of $\frac{4}{5}$ is $\frac{5}{4}$
$2(3x + 1) + 4x = 6x + 2 + 4x = 10x + 2$
No! $(8 - 3) - 2 = 5 - 2 = 3$, but $8 - (3 - 2) = 8 - 1 = 7$. Subtraction is not associative.
$-3(2x - 4) + 5x = -6x + 12 + 5x = -x + 12$

Common Mistakes to Avoid

Forgetting to distribute to all terms: When you see $3(x + 5)$, you must distribute the 3 to BOTH $x$ and $5$. The answer is $3x + 15$, not $3x + 5$.

Losing track of negative signs: When distributing a negative number like in $-4(x - 2)$, remember that the negative distributes to both terms: $-4x + 8$.

Thinking subtraction and division are commutative: Remember that $a - b \neq b - a$ and $a \div b \neq b \div a$ in most cases.

Confusing additive and multiplicative inverses: The additive inverse of 5 is -5 (they add to zero). The multiplicative inverse of 5 is $\frac{1}{5}$ (they multiply to one). Don't mix these up!

What's Next?

Now that you understand these fundamental properties, you have the tools you need to manipulate algebraic expressions with confidence. In the next lesson, we'll dive into integers and absolute value, where you'll work more with negative numbers and learn how to measure distance from zero.