Properties of Numbers

You've been using these properties your whole life without knowing they had names. When you add \(3 + 5\) and get 8, then add \(5 + 3\) and still get 8, that's the commutative property at work. The properties on this page are the basic rules that govern how numbers behave, and once they have names you can use them deliberately in algebra.

Think of them as the laws of mathematics. They describe what stays the same when you rearrange, regroup, or distribute numbers across operations. Once you know the rules, you can simplify expressions with confidence — knowing which moves are legal and which aren't.

The Commutative Property

The word "commutative" comes from "commute," like commuting to school or work. The direction of your trip doesn't matter — home to school or school to home, you cover 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 in an expression, you can swap operands around freely when adding or multiplying. Subtraction and division need more care.

The Associative Property

"Associate" means to group together, which is what this property is about: how numbers can be grouped when you're adding or multiplying several of them.

Associative Property of Addition: When adding three or more numbers, grouping doesn't change the sum.

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

Example: Add \(2 + 3 + 4\) two ways.

Group as \((2 + 3) + 4\): $$5 + 4 = 9$$

Or group as \(2 + (3 + 4)\): $$2 + 7 = 9$$

Both give 9.

Associative Property of Multiplication: When multiplying three or more numbers, grouping doesn't change the product.

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

Example: Multiply \(2 \times 5 \times 3\) two ways.

Group as \((2 \times 5) \times 3\): $$10 \times 3 = 30$$

Or as \(2 \times (5 \times 3)\): $$2 \times 15 = 30$$

Same answer.

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.

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

Start with actual numbers.

Example: Calculate \(3(4 + 5)\). Can you predict the result before working it out? Show answer\(3(4 + 5) = 3(9) = 27\). Or distributed: \(3 \times 4 + 3 \times 5 = 12 + 15 = 27\). Both give 27.

The same idea is what makes distribution useful in algebra, where one of the terms in the parentheses is a variable.

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

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

The 4 distributes 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 describe special numbers that leave others unchanged under 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 matter when simplifying algebraic expressions. The facts that \(x + 0 = x\) and \(x \times 1 = x\) come up constantly when you're cleaning up equations.

Inverse Properties

Inverse properties deal with opposites — pairs of numbers that undo each other under a given operation.

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

Here's a problem that uses several properties at once.

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.

1. Does \(9 + 6 = 6 + 9\)? Which property does this demonstrate? Show answerYes, both equal 15. This demonstrates the commutative property of addition.
2. Simplify using the distributive property: \(7(x + 2)\) Show answer\(7(x + 2) = 7x + 14\)
3. What is the multiplicative inverse of \(\frac{4}{5}\)? Show answerThe multiplicative inverse of \(\frac{4}{5}\) is \(\frac{5}{4}\)
4. Simplify: \(2(3x + 1) + 4x\) Show answer\(2(3x + 1) + 4x = 6x + 2 + 4x = 10x + 2\)
5. Is \((8 - 3) - 2\) the same as \(8 - (3 - 2)\)? Show answerNo! \((8 - 3) - 2 = 5 - 2 = 3\), but \(8 - (3 - 2) = 8 - 1 = 7\). Subtraction is not associative.
6. Simplify: \(-3(2x - 4) + 5x\) Show answer\(-3(2x - 4) + 5x = -6x + 12 + 5x = -x + 12\)

What's Next?

A few patterns to keep an eye on as you use these properties. Distribution must reach every term inside the parentheses: \(3(x + 5)\) is \(3x + 15\), not \(3x + 5\). Negatives distribute too: \(-4(x - 2) = -4x + 8\), so the inside subtraction flips to addition. And don't confuse the two kinds of inverse — the additive inverse of 5 is \(-5\) (they add to zero), while the multiplicative inverse of 5 is \(\frac{1}{5}\) (they multiply to one).

These properties show up everywhere from here on out. Up next: types of numbers and absolute value, which look at how integers, fractions, and signed quantities fit together — and how to measure distance from zero.